diff --git a/doc/html/math.css b/doc/html/math.css new file mode 100644 index 000000000..24c12fa35 --- /dev/null +++ b/doc/html/math.css @@ -0,0 +1,80 @@ +@import url('../../../../doc/src/boostbook.css'); + +/*============================================================================= +Copyright (c) 2004 Joel de Guzman +Copyright (c) 2014 John Maddock +http://spirit.sourceforge.net/ + +Copyright 2013 Niall Douglas additions for colors and alignment. +Copyright 2013 Paul A. Bristow additions for more colors and alignments. + +Distributed under the Boost Software License, Version 1.0. (See accompany- +ing file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +=============================================================================*/ + +/*============================================================================= +Program listings +=============================================================================*/ + + /* Code on paragraphs */ + p tt.computeroutput + { + font-size: 10pt; + } + + pre.synopsis + { + font-size: 10pt; + margin: 1pc 4% 0pc 4%; + padding: 0.5pc 0.5pc 0.5pc 0.5pc; + } + + .programlisting, + .screen + { + font-size: 10pt; + display: block; + margin: 1pc 4% 0pc 4%; + padding: 0.5pc 0.5pc 0.5pc 0.5pc; + + } + @media screen + { + /* Syntax Highlighting */ + .comment { color: green; } + /* .comment { color: #008000; } */ + } + /* Program listings in tables don't get borders */ + td .programlisting, + td .screen + { + margin: 0pc 0pc 0pc 0pc; + padding: 0pc 0pc 0pc 0pc; + } + +/*============================================================================= +Table of contents +=============================================================================*/ + + div.toc + { + margin: 1pc 4% 0pc 4%; + padding: 0.1pc 1pc 0.1pc 1pc; + font-size: 100%; + line-height: 1.15; + } + + .boost-toc + { + float: right; + padding: 0.5pc; + } + + /* Code on toc */ + .toc .computeroutput { font-size: 120% } + + /* No margin on nested menus */ + + .toc dl dl { margin: 0; } + + diff --git a/doc/math.qbk b/doc/math.qbk index 0b19b3205..6232de51f 100644 --- a/doc/math.qbk +++ b/doc/math.qbk @@ -632,6 +632,7 @@ and as a CD ISBN 0-9504833-2-X 978-0-9504833-2-0, Classification 519.2-dc22. [endsect] [section:quadrature Quadrature] [include quadrature/trapezoidal.qbk] +[include quadrature/double_exponential.qbk] [endsect] [endmathpart] [/mathpart roots Root Finding Algorithms] diff --git a/doc/quadrature/double_exponential.qbk b/doc/quadrature/double_exponential.qbk new file mode 100644 index 000000000..14b3d80cb --- /dev/null +++ b/doc/quadrature/double_exponential.qbk @@ -0,0 +1,438 @@ +[/ +Copyright (c) 2017 Nick Thompson +Use, modification and distribution are subject to the +Boost Software License, Version 1.0. (See accompanying file +LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +] + +[section:double_exponential Double-exponential quadrature] + +[section:de_overview Overview] + +[heading Synopsis] + +`` + #include + #include + #include + + namespace boost{ namespace math{ + + template + class tanh_sinh + { + public: + tanh_sinh(size_t max_refinements = 15); + + template + Real integrate(const F f, Real a, Real b, Real tol = sqrt(std::numeric_limits::epsilon()), Real* error = nullptr, Real* L1 = nullptr, size_t* levels = nullptr) const; + template + Real integrate(const F f, Real tol = sqrt(std::numeric_limits::epsilon()), Real* error = nullptr, Real* L1 = nullptr, size_t* levels = nullptr) const; + } + + template + class exp_sinh + { + public: + exp_sinh(size_t max_refinements = 9); + + template + Real integrate(const F f, Real a, Real b, Real tol = sqrt(std::numeric_limits::epsilon()), Real* error = nullptr, Real* L1 = nullptr, size_t* levels = nullptr) const; + template + Real integrate(const F f, Real tol = sqrt(std::numeric_limits::epsilon()), Real* error = nullptr, Real* L1 = nullptr, size_t* levels = nullptr) const; + } + + template + class sinh_sinh + { + public: + sinh_sinh(size_t max_refinements = 9); + + template + Real integrate(const F f, Real tol = sqrt(std::numeric_limits::epsilon()), Real* error = nullptr, Real* L1 = nullptr, size_t* levels = nullptr) const; + } + +}} +`` + +These three integration routines provide robust general purpose quadrature routines, each has a "native" range over which +quadrature is performed, but two of these (tanh_sinh and exp_sinh) also have auxilliary ranges which are handled via a change +of argument on the function being integrated. + +The `integrate` methods which do not specify a range always integrate over the native range of the method, and generally +are the most efficient and produce the smallest code, on the other hand the methods which do specify the bounds of integration +are the most general, and use argument transformations which are generally very robust. The following table summerizes +the ranges supported by each method: + +[table +[[Integrator][Native range][Other supported ranges][Comments]] +[[tanh_sinh] [(-1,1)] [(a,b)[br](a,[infin])[br](-[infin],b)[br](-[infin],[infin])] + [Special care is taken for endpoints at or near zero to ensure that abscissa values are calculated without the loss of precision + that would normally occur. Likewise when transforming to an infinite endpoint, the additional information which tanh_sinh has + internally on abscissa values is used to ensure no loss of precision during the transformation.]] +[[exp_sinh] [(0,[infin])] [(a,[infin])[br](-[infin],0)[br](-[infin],b)] []] +[[sinh_sinh] [(-[infin],[infin])] [][]] +] + +[endsect] + +[section:de_tanh_sinh tanh_sinh] + + template + class tanh_sinh + { + public: + tanh_sinh(size_t max_refinements = 15); + + template + Real integrate(const F f, Real a, Real b, Real tol = sqrt(std::numeric_limits::epsilon()), Real* error = nullptr, Real* L1 = nullptr, size_t* levels = nullptr) const; + template + Real integrate(const F f, Real tol = sqrt(std::numeric_limits::epsilon()), Real* error = nullptr, Real* L1 = nullptr, size_t* levels = nullptr) const; + } + +The tanh-sinh quadrature routine provided by boost is a rapidly convergent numerical integration scheme for holomorphic integrands. +By this we mean that the integrand is the restriction to the real line of a complex-differentiable function which is bounded on the interior of the unit disk /|z| < 1/, +so that it lies within the so-called [@https://en.wikipedia.org/wiki/Hardy_space Hardy space]. +If your integrand obeys these conditions, it can be shown that tanh-sinh integration is optimal, +in the sense that it requires the fewest function evaluations for a given accuracy of any quadrature algorithm for a random element from the Hardy space. +A basic example of how to use the tanh-sinh quadrature is shown below: + + tanh_sinh integrator; + auto f = [](double x) { return 5*x + 7; }; + // Integrate over native bounds of (-1,1): + double Q = integrator.integrate(f); + // Integrate over (0,1) instead: + Q = integrator.integrate(f, 0.0, 1.1); + +The basic idea of tanh-sinh quadrature is that a variable transformation can cause the endpoint derivatives to decay rapidly. +When the derivatives at the endpoints decay much faster than the Bernoulli numbers grow, +the Euler-Maclaurin summation formula tells us that simple trapezoidal quadrature converges exponentially fast. +That means the precision of the result should be doubling with each new level of integration, so the termination condition for +integration is usually set to the square root of machine epsilon. Most typical well behaved integrals should converge +to full machine precision with this termination condition, and in 6 or fewer levels at double precision, or 7 or fewer +levels for quad precision. + + +One very nice property of tanh-sinh quadrature is that it can handle singularities at the endpoints of the integration domain. +For instance, the following integrand, singular at both endpoints, can be efficiently evaluated to 100 binary digits: + + auto f = [](Real x) { return log(x)*log(1-x); }; + Real Q = integrator.integrate(f, (Real) 0, (Real) 1); + +Now onto the caveats: As stated before, the integrands must lie in a Hardy space to ensure rapid convergence. +Attempting to integrate a function which is not bounded on the unit disk by tanh-sinh can lead to very slow convergence. +For example, take the Runge function: + + auto f1 = [](double t) { return 1/(1+25*t*t); }; + Q = integrator.integrate(f1, (double) -1, (double) 1); + +This function has poles at \u00B1 \u2148/5, and as such it is not bounded on the unit disk. +However, the related function + + auto f2 = [](double t) { return 1/(1+0.04*t*t); }; + Q = integrator.integrate(f2, (double) -1, (double) 1); + +has poles outside the unit disk (at \u00B1 5\u2148), and is therefore in the Hardy space. +The integration is performed 22x faster for the second function! +If you do not understand the structure of your integrand in the complex plane, do performance testing before deployment. + +Like the trapezoidal quadrature, the tanh-sinh quadrature produces an estimate of the L[sub 1] norm of the integral along with the requested integral. +This is to establish a scale against which to measure the tolerance, and provides an estimate of the condition number of the summation. +This can be queries as follows: + + + tanh_sinh integrator; + auto f = [](double x) { return 5*x + 7; }; + double termination = std::sqrt(std::numeric_limits::epsilon()); + double error; + double L1; + size_t levels; + double Q = integrator.integrate(f, 0.0, 1.0, termination, &error, &L1, &levels); + double condition_number = L1/std::abs(Q); + +If the condition number is large, the computed integral is worthless: typically one can assume that Q has lost one digit of precision for each order +of magnitude of the condition number. The returned error term is not the actual error in the result, it is the absolute difference between the last +two approximations, and for well behaved integrals, the actual error should be very much smaller than this. +The following table illustrates how the errors and conditioning vary for few sample integrals, in each case the termination condition was set +to the square root of epsilon, and all tests were conducted with type double: + +[table +[[Integral][Range][Error][Actual measured error][Levels][Condition Number][Comments]] +[[`5 * x + 7`][(0,1)][3.5e-15][0][5][1][This trivial case shows just how accurate these methods can be.]] +[[`log(x) * log(x)`][0, 1)][0][0][5][1][This is an example of an integral that Gaussian integrators fail to handle.]] +[[`exp(-x) / sqrt(x)`][(0,+[infin])][8.0e-10][1.1e-15][5][1][Gaussian integrators typically fail to handle the singularities at the endpoints of this one.]] +[[`x * sin(2 * exp(2 * sin(2 * exp(2 * x))))`][(-1,1)][7.2e-16][4.9e-17][9][1.89][This is a truely horrible integral that oscillates wildly and + unpredictably with some very sharp "spikes" in it's graph. The higher number of levels used reflects the difficulty in reaching into some of the more extreme features.]] +[[`x == 0 ? 1 : sin(x) / x`][(-[infin], [infin])][3.0e-1][4.0e-1][15][159][This highly oscillatory integral isn't handled at all well by tanh-sinh quadrature: there is so much + cancellation in the sum that the result is essentially worthless. The argument transformation of the infinite integral behaves somewhat badly as well, in fact + we do ['slightly] better integrating over 2 symmetrical and large finite limits.]] +[[`sqrt(x / (1 - x * x))`][(0,1)][1e-8][1e-8][5][1][This an example of an integral that has all it's area close to a non-zero endpoint, the problem here is that + the function being integrated returns "garbage" values for x very close to 1. We can easily fix this issue by passing a 2 argument functor to the integrator: + the second argument gives the distance to the nearest endpoint, and we can use that information to return accurate values, and thus fix the integral calculation.]] +[[`x < 0.5 ? sqrt(x) / sqrt(1 - x * x) : sqrt(x / ((x + 1) * (xc)))`][(0,1)][0][0][5][1][This is the 2-argument version of the previous integral, the second + argument ['xc] is `1-x` in this case, and we use 1-x[super 2] == (1-x)(1+x) to calculate 1-x[super 2] with greater accuracy.]] +] + +Although the tanh-sinh quadrature can compute integral over infinite domains by variable transformations, these transformations can create a very poorly behaved integrand. +For this reason, double-exponential variable transformations have been provided that allow stable computation over infinite domains; these being the exp-sinh and sinh-sinh quadrature. + +[endsect] + +[section:de_tanh_sinh_2_arg Handling functions with large features near an endpoint with tanh-sinh quadrature] + +Tanh-sinh quadrature has a unique feature which makes it well suited to handling integrals with either singularities or large features of interest +near one or both endpoints, it turns out that when we calculate and store the abscissa values at which we will be evaluating the function, we can equally +well calculate the difference between the abscissa value and the nearest endpoint. This makes it possible to perform quadrature arbitrarily close to +an endpoint, without suffering cancellation error. Note however, that we never actually reach the endpoint, so any endpoint singularity will always +be excluded from the quadrature. + +The tanh_sinh integration routine will use this additional information internally when performing range transformation, so that for example, +if one end of the range is zero (or infinite) then our transformations will get arbitrarily close to the endpoint without precision loss. + +However, there are some integrals which may have all of their area near ['both] endpoints, or else near the non-zero endpoint, and in that situation +there is a very real risk of loss of precision. For example: + + tanh_sinh integrator; + auto f = [](double x) { return sqrt(x / (1 - x * x); }; + double Q = integrator.integrate(f, 0.0, 1.0); + +Results in very low accuracy as all the area of the integral is near 1, and the `1 - x * x` term suffers from cancellation error here. + +However, both of tanh_sinh's integration routines will automatically handle 2 argument functors: in this case the first argument is the abscissa value as +before, while the second is the distance to the nearest endpoint, ie `a - x` or `b - x` if we are integrating over (a,b). You can always differentiate +between these 2 cases because the second argument will be negative if we are nearer to the left endpoint. + +Knowing this, we can rewrite our lambda expression to take advantage of this additional information: + + tanh_sinh integrator; + auto f = [](double x, double xc) { return x <= 0.5 ? sqrt(x) / sqrt(1 - x * x) : sqrt(x / ((x + 1) * (xc))); }; + double Q = integrator.integrate(f, 0.0, 1.0); + +Not only is this form accurate to full machine precision, but it converges on the result faster too. + +[endsect] + +[section:de_sinh_sinh sinh_sinh] + + template + class sinh_sinh + { + public: + sinh_sinh(size_t max_refinements = 9); + + template + Real integrate(const F f, Real tol = sqrt(std::numeric_limits::epsilon()), Real* error = nullptr, Real* L1 = nullptr, size_t* levels = nullptr) const; + } + +The sinh-sinh quadrature allows computation over the entire real line, and is called as follows: + + sinh_sinh integrator; + auto f = [](double x) { return exp(-x*x); }; + double error; + double L1; + double Q = integrator.integrate(f, &error, &L1); + +Note that the limits of integration are understood to be (-\u221E, \u221E). + +Endpoint singularities are supported by sinh-sinh. + +[endsect] + +[section:de_exp_sinh exp_sinh] + + template + class exp_sinh + { + public: + exp_sinh(size_t max_refinements = 9); + + template + Real integrate(const F f, Real a, Real b, Real tol = sqrt(std::numeric_limits::epsilon()), Real* error = nullptr, Real* L1 = nullptr, size_t* levels = nullptr) const; + template + Real integrate(const F f, Real tol = sqrt(std::numeric_limits::epsilon()), Real* error = nullptr, Real* L1 = nullptr, size_t* levels = nullptr) const; + } + +For half-infinite intervals, the exp-sinh quadrature is provided: + + exp_sinh integrator; + auto f = [](double x) { return exp(-3*x); }; + double termination = sqrt(std::numeric_limits::epsilon()); + double error; + double L1; + double Q = integrator.integrate(f, termination, &error, &L1); + + +The native integration range of this integrator is (0, [infin]), but we also support (a, [infin]), (-[infin], 0) and (-[infin], b) via argument transformations. + +Endpoint singularities are supported by exp-sinh. + +[endsect] + +[section:de_tol Setting the Termination Condition for Integration] + +The integrate method for all three double-exponential quadratures supports ['tolerance] argument that acts as the +termination condition for integration. + +The tolerance is met when two subsequent estimates of the integral have absolute error less than `tolerance*L1`. + +It is highly recommended that the tolerance be left at the default value of [radic][epsilon], or something similar. +Since double exponential quadrature converges exponentially fast for functions in Hardy spaces, then once the routine has *proved* that the error is ~[radic][epsilon], +then the error should in fact be ~[epsilon]. + +If you request that the error be ~[epsilon], this tolerance might never be achieved (as the summation is not stabilized ala Kahan), and the routine will simply flounder, +dividing the interval in half in order to increase the precision of the integrand, only to be thwarted by floating point roundoff. + +If for some reason, the default value doesn't quite achieve full precision, then you could try something a little smaller such as +[radic][epsilon]/4 or [epsilon][super 2/3]. However, more likely, you need to check that your function to be integrated is able to +return accurate values, and that there are no other issues with your integration scheme. + +[endsect] + +[section:de_levels Setting the Maximum Interval Halvings and Memory Requirements] + +The max interval halvings is the maximum number of times the interval can be cut in half before giving up. +If the accuracy is not met at that time, the routine returns its best estimate, along with the `error` and `L1`, +which allows the user to decide if another quadrature routine should be employed. + +An example of this is + + double tol = std::sqrt(std::numeric_limits::epsilon()); + size_t max_halvings = 12; + tanh_sinh integrator(max_halvings); + auto f = [](double x) { return 5*x + 7; }; + double error, L1; + double Q = integrator.integrate(f, (double) 0, (double) 1, &error, &L1); + if (error*L1 > 0.01) + { + Q = some_other_quadrature_method(f, (double) 0, (double) 1); + } + +It's important to remember that the number of sample points doubles with each new level, as does the memory footprint +of the integrator object. Further, if the integral is smooth, then the precision will be doubling with each new level, +so that for example, many integrals can achieve 100 decimal digit precision after just 7 levels. That said, abscissa-weight +pairs for new levels are computed only when a new level is actually required (see thread safety), none the less, +you should avoid setting the maximum arbitrarily high "just in case" as the time and space requirements for a large +number of levels can quickly grow out of control. + +[endsect] + +[section:de_thread Thread Safety] + +All three of the double-exponential integrators are thread safe as long as BOOST_MATH_NO_ATOMIC_INT is not set. Since the +integrators store a large amount of fairly hard to compute data, it is recomended that these objects are stored and reused +as much as possible. + +Internally all three of the double-exponential integrators use the same caching strategy: they allocate all the vectors needed +to store the maximum permitted levels, but only populate the first few levels when constructed. This means a minimal amount of memory +is actually allocated when the integrator is first constructed, and already populated levels can be accessed via a lockfree +atomic read, and only populating new levels requires a thread lock. + +In addition, the three built in types (plus `__float128` when available), have the first 7 levels pre-computed: this is generally sufficient for the vast majority +of integrals - even at quad precision - and means that integrators for these types are relatively cheap to construct. + +[endsect] + +[section:de_caveats Caveats] + +A few things to keep in mind while using the tanh-sinh, exp-sinh, and sinh-sinh quadratures: + +These routines are *very* aggressive about approaching the endpoint singularities. +This allows lots of significant digits to be extracted, but also has another problem: Roundoff error can cause the function to be evaluated at the endpoints. +A few ways to avoid this: Narrow up the bounds of integration to say, [a + [epsilon], b - [epsilon]], make sure (a+b)/2 and (b-a)/2 are representable, and finally, +if you think the compromise between accuracy an usability has gone too far in the direction of accuracy, file a ticket. + +Both exp-sinh and sinh-sinh quadratures evaluate the functions they are passed at *very* large argument. +You might understand that x[super 12]exp(-x) is should be zero when x[super 12] overflows, but IEEE floating point arithmetic does not. +Hence `std::pow(x, 12)*std::exp(-x)` is an indeterminate form whenever `std::pow(x, 12)` overflows. +So make sure your functions have the correct limiting behavior; for example + + auto f = [](double x) { + double t = exp(-x); + if(t == 0) + { + return 0; + } + return t*pow(x, 12); + }; + +has the correct behavior for large /x/, but `auto f = [](double x) { return exp(-x)*pow(x, 12); };` does not. + +Oscillatory integrals, such as the sinc integral, are poorly approximated by double-exponential quadrature. +Fortunately the error estimates and L1 norm are massive for these integrals, but nonetheless, oscillatory integrals require different techniques. + +A special mention should be made about integrating through zero: while our range adaptors preserve precision when one endpoint is zero, +things get harder when the origin is neither in the center of the range, nor at an endpoint. Consider integrating: + + 1 / (1 +x^2) + +Over (a, [infin]). As long as `a >= 0` both the tanh_sinh and the exp_sinh integrators will handle this just fine: in fact they provide +a rather efficient method for this kind of integral. However, if we have `a < 0` then we are forced to adapt the range in a way that +produces abscissa values near zero that have an absolute error of [epsilon], and since all of the area of the integral is near zero +both integrators thrash around trying to reach the target accuracy, but never actually get there for `a << 0`. On the other hand, the +simple expedient of breaking the integral into two domains: (a, 0) and (0, b) and integrating each seperately using the tanh-sinh +integrator, works just fine. + +Finally, some endpoint singularities are too strong to be handled by tanh_sinh or equivalent methods, for example consider integrating +the function: + + double p = some_value; + tanh_sinh integrator; + auto f = [&](double x){ return pow(tan(x), p); }; + auto Q = integrator.integrate(f, 0, constants::half_pi()); + +The first problem with this function, is that the singularity is at [pi]/2, so if we're integrating over (0, [pi]/2) then we can never +approach closer to the singularity than [epsilon], and for p less than but close to 1, we need to get ['very] close to the singularity +to find all the area under the function. If we recall the identity [^tan([pi]/2 - x) == 1/tan(x)] then we can rewrite the function like this: + + auto f = [&](double x){ return pow(tan(x), -p); }; + +And now the singularity is at the origin and we can get much closer to it when evaluating the integral: all we have done is swap the +integral endpoints over. + +This actually works just fine for p < 0.95, but after that the tanh_sinh integrator starts thrashing around and is unable to +converge on the integral. The problem is actually a lack of exponent range: if we simply swap type double for something +with a greater exponent range (an 80-bit long double or a quad precision type), then we can get to at least p = 0.99. If we want to go +beyond that, or stick with type double, then we have to get smart. + +The easiest method is to notice that for small x, then [^tan(x) [cong] x], and so we are simply integrating x[super -p]. Therefore we can use +this approximation over (0, small), and integrate numerically from (small, [pi]/2), using [epsilon] as a suitable crossover point +seems sensible: + + double p = some_value; + double crossover = std::numeric_limits::epsilon(); + tanh_sinh integrator; + auto f = [&](double x){ return pow(tan(x), p); }; + auto Q = integrator.integrate(f, crossover, constants::half_pi()) + pow(crossover, 1 - p) / (1 - p); + +There is an alternative, more complex method, which is applicable when we are dealing with expressions which can be simplified +by evaluating by logs. Let's suppose that as in this case, all the area under the graph is infinitely close to zero, now inagine +that we could expand that region out over a much larger range of abscissa values: that's exactly what happens if we perform +argument substitution, replacing `x` by `exp(-x)` (note that we must also multiply by the derivative of `exp(-x)`). +Now the singularity at zero is moved to +[infin], and the [pi]/2 bound to +-log([pi]/2). Initially our argument substituted function looks like: + + auto f = [&](double x){ return exp(-x) * pow(tan(exp(-x)), -p); }; + +Which is hardly any better, as we still run out of exponent range just as before. However, if we replace `tan(exp(-x))` by `exp(-x)` for suitably +small `exp(-x)`, and therefore [^x > -log([epsilon])], we can greatly simplify the expression and evaluate by logs: + + auto f = [&](double x) + { + static const double crossover = -log(std::numeric_limits::epsilon()); + return x > crossover ? exp((p - 1) * x) : exp(-x) * pow(tan(exp(-x)), -p); + }; + +This form integrates just fine over (-log([pi]/2), +[infin]) using either the tanh_sinh or exp_sinh classes. + + +[endsect] + +[section:de_refes References] + +* Hidetosi Takahasi and Masatake Mori, ['Double Exponential Formulas for Numerical Integration] Publ. Res. Inst. Math. Sci., 9 (1974), pp. 721-741. +* Masatake Mori, ['An IMT-Type Double Exponential Formula for Numerical Integration], Publ RIMS, Kyoto Univ. 14 (1978), 713-729. +* David H. Bailey, Karthik Jeyabalan and Xiaoye S. Li ['A Comparison of Three High-Precision Quadrature Schemes] Office of Scientific & Technical Information Technical Reports. +* Tanaka, Ken’ichiro, et al. ['Function classes for double exponential integration formulas.] Numerische Mathematik 111.4 (2009): 631-655. + +[endsect] +[endsect] diff --git a/example/double_exponential.cpp b/example/double_exponential.cpp new file mode 100644 index 000000000..87e21b187 --- /dev/null +++ b/example/double_exponential.cpp @@ -0,0 +1,59 @@ +// Copyright Nick Thompson, 2017 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#include +#include +#include +#include +#include + +using boost::math::quadrature::tanh_sinh; +using boost::math::quadrature::sinh_sinh; +using boost::math::quadrature::exp_sinh; +using boost::math::constants::pi; +using boost::math::constants::half_pi; +using boost::math::constants::half; +using boost::math::constants::third; +using boost::math::constants::root_pi; +using std::log; +using std::cos; +using std::cosh; +using std::exp; +using std::sqrt; + +int main() +{ + std::cout << std::setprecision(std::numeric_limits::digits10); + double tol = sqrt(std::numeric_limits::epsilon()); + // For an integral over a finite domain, use tanh_sinh: + tanh_sinh tanh_integrator(tol, 10); + auto f1 = [](double x) { return log(x)*log(1-x); }; + double Q = tanh_integrator.integrate(f1, (double) 0, (double) 1); + double Q_expected = 2 - pi()*pi()*half()*third(); + + std::cout << "tanh_sinh quadrature of log(x)log(1-x) gives " << Q << std::endl; + std::cout << "The exact integral is " << Q_expected << std::endl; + + // For an integral over the entire real line, use sinh-sinh quadrature: + sinh_sinh sinh_integrator(tol, 10); + auto f2 = [](double t) { return cos(t)/cosh(t);}; + Q = sinh_integrator.integrate(f2); + Q_expected = pi()/cosh(half_pi()); + std::cout << "sinh_sinh quadrature of cos(x)/cosh(x) gives " << Q << std::endl; + std::cout << "The exact integral is " << Q_expected << std::endl; + + // For half-infinite intervals, use exp-sinh. + // Endpoint singularities are handled well: + exp_sinh exp_integrator(tol, 10); + auto f3 = [](double t) { return exp(-t)/sqrt(t); }; + Q = exp_integrator.integrate(f3, 0, std::numeric_limits::infinity()); + Q_expected = root_pi(); + std::cout << "exp_sinh quadrature of exp(-t)/sqrt(t) gives " << Q << std::endl; + std::cout << "The exact integral is " << Q_expected << std::endl; + + + +} diff --git a/include/boost/math/concepts/real_concept.hpp b/include/boost/math/concepts/real_concept.hpp index 3a759955a..d43a3c50e 100644 --- a/include/boost/math/concepts/real_concept.hpp +++ b/include/boost/math/concepts/real_concept.hpp @@ -31,6 +31,8 @@ #include #include #include +#include +#include #if defined(__SGI_STL_PORT) # include #endif @@ -256,6 +258,24 @@ inline real_concept tanh(real_concept a) { return std::tanh(a.value()); } // +// C++11 ism's +// Note that these must not actually call the std:: versions as that precludes using this +// header to test in C++03 mode, call the Boost versions instead: +// +inline boost::math::concepts::real_concept asinh(boost::math::concepts::real_concept a) +{ + return boost::math::asinh(a.value(), boost::math::policies::make_policy(boost::math::policies::overflow_error())); +} +inline boost::math::concepts::real_concept acosh(boost::math::concepts::real_concept a) +{ + return boost::math::acosh(a.value(), boost::math::policies::make_policy(boost::math::policies::overflow_error())); +} +inline boost::math::concepts::real_concept atanh(boost::math::concepts::real_concept a) +{ + return boost::math::atanh(a.value(), boost::math::policies::make_policy(boost::math::policies::overflow_error())); +} + +// // Conversion and truncation routines: // template diff --git a/include/boost/math/concepts/std_real_concept.hpp b/include/boost/math/concepts/std_real_concept.hpp index b8657bc6a..d855d2d55 100644 --- a/include/boost/math/concepts/std_real_concept.hpp +++ b/include/boost/math/concepts/std_real_concept.hpp @@ -226,6 +226,18 @@ inline boost::math::concepts::std_real_concept sqrt(boost::math::concepts::std_r { return std::sqrt(a.value()); } inline boost::math::concepts::std_real_concept tanh(boost::math::concepts::std_real_concept a) { return std::tanh(a.value()); } +// +// C++11 ism's +// Note that these must not actually call the std:: versions as that precludes using this +// header to test in C++03 mode, call the Boost versions instead: +// +inline boost::math::concepts::std_real_concept asinh(boost::math::concepts::std_real_concept a) +{ return boost::math::asinh(a.value(), boost::math::policies::make_policy(boost::math::policies::overflow_error())); } +inline boost::math::concepts::std_real_concept acosh(boost::math::concepts::std_real_concept a) +{ return boost::math::acosh(a.value(), boost::math::policies::make_policy(boost::math::policies::overflow_error())); } +inline boost::math::concepts::std_real_concept atanh(boost::math::concepts::std_real_concept a) +{ return boost::math::atanh(a.value(), boost::math::policies::make_policy(boost::math::policies::overflow_error())); } + } // namespace std @@ -398,4 +410,12 @@ using concepts::llround; } // namespace math } // namespace boost +// +// These must go at the end, as they include stuff that won't compile until +// after std_real_concept has been defined: +// +#include +#include +#include + #endif // BOOST_MATH_STD_REAL_CONCEPT_HPP diff --git a/include/boost/math/quadrature/detail/exp_sinh_detail.hpp b/include/boost/math/quadrature/detail/exp_sinh_detail.hpp new file mode 100644 index 000000000..3ce7a2d21 --- /dev/null +++ b/include/boost/math/quadrature/detail/exp_sinh_detail.hpp @@ -0,0 +1,475 @@ +// Copyright Nick Thompson, 2017 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_QUADRATURE_DETAIL_EXP_SINH_DETAIL_HPP +#define BOOST_MATH_QUADRATURE_DETAIL_EXP_SINH_DETAIL_HPP + +#include +#include +#include +#include +#include +#include +#include + +namespace boost{ namespace math{ namespace quadrature { namespace detail{ + + +// Returns the exp-sinh quadrature of a function f over the open interval (0, infinity) + +template +class exp_sinh_detail +{ + static const int initializer_selector = + !std::numeric_limits::is_specialized || (std::numeric_limits::radix != 2) ? + 0 : + (std::numeric_limits::digits < 30) && (std::numeric_limits::max_exponent <= 128) ? + 1 : + (std::numeric_limits::digits <= std::numeric_limits::digits) && (std::numeric_limits::max_exponent <= std::numeric_limits::max_exponent) ? + 2 : + (std::numeric_limits::digits <= std::numeric_limits::digits) && (std::numeric_limits::max_exponent <= 16384) ? + 3 : +#ifdef BOOST_HAS_FLOAT128 + (std::numeric_limits::digits <= 113) && (std::numeric_limits::max_exponent <= 16384) ? + 4 : +#endif + 0; +public: + exp_sinh_detail(size_t max_refinements); + + template + Real integrate(const F& f, Real* error, Real* L1, const char* function, Real tolerance, std::size_t* levels) const; + +private: + const std::vector& get_abscissa_row(std::size_t n)const + { +#ifndef BOOST_MATH_NO_ATOMIC_INT + if (m_committed_refinements.load() < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements.load() >= n); +#else + if (m_committed_refinements < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements >= n); +#endif + return m_abscissas[n]; + } + const std::vector& get_weight_row(std::size_t n)const + { +#ifndef BOOST_MATH_NO_ATOMIC_INT + if (m_committed_refinements.load() < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements.load() >= n); +#else + if (m_committed_refinements < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements >= n); +#endif + return m_weights[n]; + } + void init(const mpl::int_<0>&); + void init(const mpl::int_<1>&); + void init(const mpl::int_<2>&); + void init(const mpl::int_<3>&); +#ifdef BOOST_HAS_FLOAT128 + void init(const mpl::int_<4>&); +#endif + + void extend_refinements()const + { +#ifndef BOOST_MATH_NO_ATOMIC_INT + boost::detail::lightweight_mutex::scoped_lock guard(m_mutex); +#endif + // + // Check some other thread hasn't got here after we read the atomic variable, but before we got here: + // +#ifndef BOOST_MATH_NO_ATOMIC_INT + if (m_committed_refinements.load() >= m_max_refinements) + return; +#else + if (m_committed_refinements >= m_max_refinements) + return; +#endif + + using std::ldexp; + using std::ceil; + using std::sinh; + using std::cosh; + using std::exp; + std::size_t row = ++m_committed_refinements; + + Real h = ldexp(Real(1), -static_cast(row)); + const Real t_max = m_t_min + m_abscissas[0].size() - 1; + + size_t k = (size_t)boost::math::lltrunc(ceil((t_max - m_t_min) / (2 * h))); + m_abscissas[row].reserve(k); + m_weights[row].reserve(k); + Real arg = m_t_min; + size_t j = 0; + size_t l = 2; + while (arg + l*h < t_max) + { + arg = m_t_min + (2 * j + 1)*h; + Real x = exp(constants::half_pi()*sinh(arg)); + m_abscissas[row].emplace_back(x); + Real w = cosh(arg)*constants::half_pi()*x; + m_weights[row].emplace_back(w); + ++j; + } + } + + Real m_tol, m_t_min; + + mutable std::vector> m_abscissas; + mutable std::vector> m_weights; + std::size_t m_max_refinements; +#ifndef BOOST_MATH_NO_ATOMIC_INT + mutable boost::math::detail::atomic_unsigned_type m_committed_refinements; + mutable boost::detail::lightweight_mutex m_mutex; +#else + mutable unsigned m_committed_refinements; +#endif +}; + +template +exp_sinh_detail::exp_sinh_detail(size_t max_refinements) + : m_abscissas(max_refinements), m_weights(max_refinements), + m_max_refinements(max_refinements) +{ + init(mpl::int_()); +} +template +template +Real exp_sinh_detail::integrate(const F& f, Real* error, Real* L1, const char* function, Real tolerance, std::size_t* levels) const +{ + using std::abs; + using std::floor; + using std::tanh; + using std::sinh; + using std::sqrt; + using boost::math::constants::half; + using boost::math::constants::half_pi; + + Real y_max = f(tools::max_value()); + if(abs(y_max) > tools::epsilon() || !(boost::math::isfinite)(y_max)) + { + return policies::raise_domain_error(function, "The function you are trying to integrate does not go to zero at infinity, and instead evaluates to %1%", y_max, Policy()); + } + + //std::cout << std::setprecision(5*std::numeric_limits::digits10); + + // Get the party started with two estimates of the integral: + Real I0 = 0; + Real L1_I0 = 0; + for(size_t i = 0; i < m_abscissas[0].size(); ++i) + { + Real y = f(m_abscissas[0][i]); + I0 += y*m_weights[0][i]; + L1_I0 += abs(y)*m_weights[0][i]; + } + + //std::cout << "First estimate : " << I0 << std::endl; + Real I1 = I0; + Real L1_I1 = L1_I0; + for (size_t i = 0; i < m_abscissas[1].size(); ++i) + { + Real y = f(m_abscissas[1][i]); + I1 += y*m_weights[1][i]; + L1_I1 += abs(y)*m_weights[1][i]; + } + + I1 *= half(); + L1_I1 *= half(); + Real err = abs(I0 - I1); + //std::cout << "Second estimate: " << I1 << " Error estimate at level " << 1 << " = " << err << std::endl; + + size_t i = 2; + for(; i < m_abscissas.size(); ++i) + { + I0 = I1; + L1_I0 = L1_I1; + + I1 = half()*I0; + L1_I1 = half()*L1_I0; + Real h = (Real) 1/ (Real) (1 << i); + Real sum = 0; + Real absum = 0; + + auto abscissas_row = get_abscissa_row(i); + auto weight_row = get_weight_row(i); + + Real abterm1 = 1; + Real eps = tools::epsilon()*L1_I1; + for(size_t j = 0; j < m_weights[i].size(); ++j) + { + Real x = abscissas_row[j]; + Real y = f(x); + sum += y*weight_row[j]; + Real abterm0 = abs(y)*weight_row[j]; + absum += abterm0; + + // We require two consecutive terms to be < eps in case we hit a zero of f. + // Numerical experiments indicate that we should start this check at ~30 for floats, + // ~60 for doubles, and ~100 for long doubles. + // However, starting the check at x = 10 rather than x = 100 will only save two function evaluations. + if (x > (Real) 100 && abterm0 < eps && abterm1 < eps) + { + break; + } + abterm1 = abterm0; + } + + I1 += sum*h; + L1_I1 += absum*h; + err = abs(I0 - I1); + //std::cout << "Estimate: " << I1 << " Error estimate at level " << i << " = " << err << std::endl; + if (!isfinite(I1)) + { + return policies::raise_evaluation_error(function, "The exp_sinh quadrature evaluated your function at a singular point and resulted in %1%. Please ensure your function evaluates to a finite number of its entire domain.", I1, Policy()); + } + if (err <= tolerance*L1_I1) + { + break; + } + } + + if (error) + { + *error = err; + } + + if(L1) + { + *L1 = L1_I1; + } + + if (levels) + { + *levels = i; + } + + return I1; +} + + +template +void exp_sinh_detail::init(const mpl::int_<0>&) +{ + using std::exp; + using std::log; + using std::sqrt; + using std::cosh; + using std::sinh; + using std::asinh; + using std::ceil; + using boost::math::constants::two_div_pi; + using boost::math::constants::half_pi; + using boost::math::constants::half; + + m_committed_refinements = 4; + // m_t_min is chosen such that x = exp(pi/2 sinh(-t_max)) = very small, but not too small. + // This is a compromise; we wish to approach a singularity at zero without hitting it through roundoff error. + // If we choose the small number as epsilon, we do not get close enough to the singularity to achieve high accuracy. + // If we choose the small number as the min(), then we round off to zero and hit the singularity. + // The logarithmic average of the min() and the epsilon() has been found to be a reasonable compromise, which achieves high accuracy + // but does not evaluate the function at the singularity. + Real tmp = (boost::math::tools::log_min_value() + log(boost::math::tools::epsilon()))*half(); + m_t_min = asinh(two_div_pi()*tmp); + + // t_max is chosen to make g'(t_max) ~ sqrt(max) (g' grows faster than g). + // This will allow some flexibility on the users part; they can at least square a number function without overflow. + // But there is no unique choice; the further out we can evaluate the function, the better we can do on slowly decaying integrands. + const Real t_max = log(2 * two_div_pi()*log(2 * two_div_pi()*sqrt(tools::max_value()))); + + for (size_t i = 0; i <= 4; ++i) + { + Real h = (Real)1 / (Real)(1 << i); + size_t k = (size_t)boost::math::lltrunc(ceil((t_max - m_t_min) / (2 * h))); + m_abscissas[i].reserve(k); + m_weights[i].reserve(k); + Real arg = m_t_min; + size_t j = 0; + size_t l = 2; + if (i == 0) + { + l = 1; + } + while (arg + l*h < t_max) + { + if (i != 0) + { + arg = m_t_min + (2 * j + 1)*h; + } + else + { + arg = m_t_min + j*h; + } + Real x = exp(half_pi()*sinh(arg)); + m_abscissas[i].emplace_back(x); + Real w = cosh(arg)*half_pi()*x; + m_weights[i].emplace_back(w); + ++j; + } + } +/* + std::cout << std::setprecision(18) << m_t_min << std::endl; + for (unsigned i = 0; i < m_abscissas[0].size(); ++i) + std::cout << m_abscissas[0][i] << ", "; + std::cout << std::endl; + */ +} + +template +void exp_sinh_detail::init(const mpl::int_<1>&) +{ + m_abscissas = { + { 3.47876573e-23f, 5.62503650e-09f, 9.95706124e-04f, 9.67438487e-02f, 7.43599217e-01f, 4.14293205e+00f, 1.08086768e+02f, 4.56291316e+05f, 2.70123007e+15f, }, + { 2.41870864e-14f, 1.02534662e-05f, 1.65637566e-02f, 3.11290799e-01f, 1.64691269e+00f, 1.49800773e+01f, 2.57724301e+03f, 2.24833766e+09f, }, + { 3.24983286e-18f, 2.51095186e-11f, 3.82035773e-07f, 1.33717837e-04f, 4.80260650e-03f, 4.41526928e-02f, 1.83045938e-01f, 4.91960276e-01f, 1.10322609e+00f, 2.53681744e+00f, 7.39791792e+00f, 3.59560256e+01f, 4.36061333e+02f, 2.49501460e+04f, 1.89216933e+07f, 1.03348694e+12f, }, + { 1.51941172e-20f, 3.70201714e-16f, 9.67598102e-13f, 4.44773051e-10f, 5.28493928e-08f, 2.19158236e-06f, 4.00799258e-05f, 3.88011529e-04f, 2.29325538e-03f, 9.25182629e-03f, 2.78117501e-02f, 6.67553298e-02f, 1.35173168e-01f, 2.41374946e-01f, 3.94194704e-01f, 6.07196731e-01f, 9.06432514e-01f, 1.34481045e+00f, 2.03268444e+00f, 3.21243032e+00f, 5.46310949e+00f, 1.03365745e+01f, 2.26486752e+01f, 6.03727778e+01f, 2.08220266e+02f, 1.00431239e+03f, 7.47843388e+03f, 9.75279951e+04f, 2.61755592e+06f, 1.77776624e+08f, 3.98255346e+10f, 4.13443763e+13f, 3.07708133e+17f, }, + { 7.99409438e-22f, 2.41624595e-19f, 3.73461321e-17f, 3.19397902e-15f, 1.62042378e-13f, 5.18579386e-12f, 1.10520072e-10f, 1.64548212e-09f, 1.78534009e-08f, 1.46529196e-07f, 9.40168786e-07f, 4.85507733e-06f, 2.07038029e-05f, 7.45799409e-05f, 2.31536599e-04f, 6.30580368e-04f, 1.53035449e-03f, 3.35582040e-03f, 6.73124842e-03f, 1.24856832e-02f, 2.16245309e-02f, 3.52720523e-02f, 5.45995171e-02f, 8.07587788e-02f, 1.14840025e-01f, 1.57867103e-01f, 2.10837078e-01f, 2.74805391e-01f, 3.51015955e-01f, 4.41077540e-01f, 5.47194016e-01f, 6.72466825e-01f, 8.21304567e-01f, 1.00000000e+00f, 1.21757511e+00f, 1.48706221e+00f, 1.82750536e+00f, 2.26717507e+00f, 2.84887335e+00f, 3.63893880e+00f, 4.74299876e+00f, 6.33444194e+00f, 8.70776542e+00f, 1.23825548e+01f, 1.83151803e+01f, 2.83510579e+01f, 4.62437776e+01f, 8.00917327e+01f, 1.48560852e+02f, 2.97989725e+02f, 6.53443372e+02f, 1.58584068e+03f, 4.31897162e+03f, 1.34084311e+04f, 4.83003053e+04f, 2.05969943e+05f, 1.06363880e+06f, 6.82457850e+06f, 5.60117371e+07f, 6.07724622e+08f, 9.04813016e+09f, 1.92834507e+11f, 6.17122515e+12f, 3.13089095e+14f, 2.67765347e+16f, 4.13865153e+18f, }, + { 1.70893932e-22f, 3.56621447e-21f, 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1.26155602e+20f, 2.50480806e+20f, 5.02601059e+20f, 1.01935525e+21f, }, + }; + m_committed_refinements = static_cast(m_weights.size() - 1); + m_t_min = -4.18750000f; + if (m_max_refinements >= m_abscissas.size()) + { + m_abscissas.resize(m_max_refinements + 1); + m_weights.resize(m_max_refinements + 1); + } + else + { + m_max_refinements = m_abscissas.size() - 1; + } +} + +template +void exp_sinh_detail::init(const mpl::int_<2>&) +{ + m_abscissas = { + { 7.241670621354483269e-163, 2.257639733856759198e-60, 1.153241619257215165e-22, 8.747691973876861825e-09, 1.173446923800022477e-03, 1.032756936219208144e-01, 7.719261204224504866e-01, 4.355544675823585545e+00, 1.215101039066652656e+02, 6.228845436711506169e+05, 6.278613977336989392e+15, 9.127414935180233465e+42, 6.091127771174027909e+116, }, + { 4.547459836328942014e-99, 6.678756542928857080e-37, 5.005042973041566360e-14, 1.341318484151208960e-05, 1.833875636365939263e-02, 3.257972971286326131e-01, 1.712014688483495078e+00, 1.613222549264089627e+01, 3.116246745274236447e+03, 3.751603952020919663e+09, 1.132259067258797346e+26, 6.799257464097374238e+70, }, + { 5.314690663257815465e-127, 2.579830034615362946e-77, 3.534801062399966878e-47, 6.733941646704537777e-29, 8.265803726974829043e-18, 4.424914371157762285e-11, 5.390411046738629465e-07, 1.649389713333761449e-04, 5.463728936866216652e-03, 4.787896410534771955e-02, 1.931544616590306846e-01, 5.121421856617965197e-01, 1.144715949265016019e+00, 2.648424684387670480e+00, 7.856804169938798917e+00, 3.944731803343517708e+01, 5.060291993016831194e+02, 3.181117494063683297e+04, 2.820174654949211729e+07, 1.993745099515255184e+12, 1.943469269499068563e+20, 2.858803732300638372e+33, 1.457292199029008637e+55, 8.943565831706355607e+90, 9.016198369791554655e+149, }, + { 8.165631636299519857e-144, 3.658949309353149331e-112, 1.635242513882908826e-87, 2.578381184977746454e-68, 2.305546416275824199e-53, 1.016725540031465162e-41, 1.191823622917539774e-32, 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2.004807079683827669e+118, 1.360407192665237716e+120, 9.862825609807810517e+121, 7.647551788591128099e+123, 6.348802224871730088e+125, 5.649062361980019098e+127, 5.393248003523784781e+129, 5.530897191915703916e+131, 6.099598644640894333e+133, 7.242098433491964504e+135, 9.268083053637375570e+137, 1.279942702416040582e+140, 1.909796626960621302e+142, 3.082540300669885040e+144, 5.388809732384179657e+146, 1.021610251056626535e+149, 2.103005440072790650e+151, 4.706753990348725570e+153, 1.146834128125248991e+156, }, + }; + m_committed_refinements = static_cast(m_weights.size() - 1); + m_t_min = -6.164062500000000000; + if (m_max_refinements >= m_abscissas.size()) + { + m_abscissas.resize(m_max_refinements + 1); + m_weights.resize(m_max_refinements + 1); + } + else + { + m_max_refinements = m_abscissas.size() - 1; + } +} +#if LDBL_MAX_EXP == 16384 +template +void exp_sinh_detail::init(const mpl::int_<3>&) +{ + m_abscissas = { + { 1.02756529896290544244959196973059583e-2497L, 2.56737528671961581475200468317128232e-919L, 1.17417808941462434184174780056564573e-338L, 4.82182886130634548471358754377036370e-125L, 1.85613301660646818149136526457281019e-46L, 1.52174118093087534300657777272732001e-17L, 6.75122240537294392532710530940672267e-07L, 5.94481311616464419075825632567494453e-03L, 2.00100938779037997581424620542543429e-01L, 1.17328605653712546899681147538372171e+00L, 8.18356490677287285512063117929807241e+00L, 5.59865842621965368881982340891928481e+02L, 3.69902944883650290371636082450503730e+07L, 4.05747121124517088709477132072461878e+20L, 1.07723884748977308147226626407207905e+56L, 2.07250561337258237728923403163755392e+152L, 1.09889904624000153879292638133263171e+414L, 3.02463014753652876878705286965250144e+1125L, }, + { 3.16339947894004847541521297710937343e-1515L, 7.02786763812753004271170107158747593e-558L, 1.08465499866859733494552744914276656e-205L, 3.97220624097731753646493300738183748e-76L, 1.84145280968701148636501796762008815e-28L, 6.40886341101610144938011594904313817e-11L, 1.89200275395722467663251234615351291e-04L, 5.04892743776943294143478969573235061e-02L, 5.25946553709738448524487603870477588e-01L, 2.72635049439566736308739858009584752e+00L, 4.20081657572145199006060015525396829e+01L, 3.75221714819472331206969009243353365e+04L, 3.11642465424592079726559332629050577e+12L, 9.61947229245436139236957297153092319e+33L, 2.42007357898805628851519620917402234e+92L, 1.34351470989086331111432001407522043e+251L, 4.34937560839564995558174638985438726e+682L, 4.01820305474077838467315109614612109e+1855L, }, + { 2.20630426387562582340901270736066292e-1945L, 3.20902289525796097195876526710905754e-1180L, 3.99075419312896749376535004306952481e-716L, 1.22680463935505388461578092999320797e-434L, 6.60416836081631149205027836695167940e-264L, 2.36380363265905808053031227647927829e-160L, 1.52216180146092756550894692311532796e-97L, 1.89865679919908959317123677189779258e-59L, 2.43020115474101871630434504433731237e-36L, 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2.85285821381079277498378752581228807e+322L, 4.35767992687500025738868152293904617e+531L, 3.35101732969916400190150631384628168e+876L, 1.39907636805339624393396738787997778e+1445L, 4.39310839133852761600954729262867435e+2382L, }, + { 2.60416027838200926634868701489352534e-2204L, 7.02786036162213986704884019362612079e-1717L, 2.87472449554862557122883045361480216e-1337L, 1.26052689335617094297502885256721406e-1041L, 2.22203737723769125613395083355625930e-811L, 4.59925691253299141912321586723335574e-632L, 2.06110387430222811031505778840893135e-492L, 1.18783333719310840737835079091916047e-383L, 5.99339963191580129696690680497340117e-299L, 5.55156577871295836646608994333019205e-233L, 1.31650343182343962237291034126486362e-181L, 1.35017224106781623208439493207058713e-141L, 1.95458267270042896122641878745090637e-110L, 3.62345406499123808121116200173051239e-86L, 2.87916036185197771976746396107338099e-67L, 1.50975853574758038650941626753584613e-52L, 4.39366381209590686916061999351406334e-41L, 3.72613279781933265810269222029431206e-32L, 3.35074541708050784063416140930256908e-25L, 8.73699691100611067810340427391712629e-20L, 1.44610938754441658588026566732900630e-15L, 2.79712542830915646153531586097835651e-12L, 1.01712318476608889609455916646557919e-09L, 1.00707686902351812525332969334486408e-07L, 3.62377064535647765957086044101238700e-06L, 5.93514765312557852887858990273347709e-05L, 5.27427922038425038994892508697801291e-04L, 2.91716284057748187538183216214780172e-03L, 1.11812753052373081344852811278990196e-02L, 3.23158910999791296389217134204252891e-02L, 7.52823576293971217140453145701166400e-02L, 1.49072381771447883955117212578891471e-01L, 2.61920116954195648982059938116700281e-01L, 4.23044722849733503488984642521144095e-01L, 6.47337815325830827783598163922905049e-01L, 9.63850663681748439752705317539095691e-01L, 1.43180513983798487645924505146418541e+00L, 2.17546288127550398262362304432746644e+00L, 3.47077553215380191942193887548951574e+00L, 5.98762125926090985873603940556086485e+00L, 1.15588346593765285104362265791852722e+01L, 2.60227165873113109321939526707132515e+01L, 7.18972764924010846908346544069757730e+01L, 2.59849827899514040038730358528910272e+02L, 1.33180383652908565595482681440747426e+03L, 1.07264199927746293597346233164919561e+04L, 1.54770252854197591129695964624494039e+05L, 4.73116974059692035499030617203501902e+06L, 3.79850299532978550624970142683299187e+08L, 1.05506900281875210383901752883017567e+11L, 1.44377693764054834189876938092028936e+14L, 1.53216987231286586179401429003390945e+18L, 2.25749845424265693444972830430363021e+23L, 9.75366126404791278425650801377275643e+29L, 3.22710662318561074542283240061745423e+38L, 2.80427989450823486864584373674252213e+49L, 3.11492865697270427645920832671970699e+63L, 3.37385880956075754428852062324609716e+81L, 4.84069913749095116306098688426537434e+104L, 2.62206965911556490036221731139595538e+134L, 3.95724512624896651389226023882699109e+172L, 4.16658233419696139622872515762144259e+221L, 3.67870997815873300310505495683155714e+284L, 2.45353561048417937411865685638919263e+365L, 1.47890836767899626666102902312593029e+469L, 2.66842264577698782467085710268227542e+602L, 3.39406520823137428862537121090858482e+773L, 1.71072929388916771882465310676448508e+993L, 2.17124413154098944294625847065193945e+1275L, 3.67108839134079465802181460380664133e+1637L, 4.72992162468569560939225829187204387e+2102L, }, + { 1.98007625288857013412122004727810624e-2346L, 8.39652992194258659266745309526472882e-2071L, 1.46030671151231956105710426802408027e-1827L, 6.65727248909208458885310717390128801e-1613L, 1.81581117219394388092797723844111009e-1423L, 2.72619674739250219990613941747932927e-1256L, 9.29590026994488020433315206443278096e-1109L, 1.46377917106366800430984793284730939e-978L, 1.15901956513202291149840314964756250e-863L, 2.89578802021929239619753606490350524e-762L, 8.80858736107764685086031513644364894e-673L, 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1.32995815980583547946711112525038194e+929L, 5.12600842672542100652183507817331030e+943L, 3.35203762278831767267193293263751763e+958L, 3.75009417204153826893896307870937057e+973L, 7.23855050824188975713471018223024339e+988L, 2.43146178229439003194650931378387467e+1004L, 1.43376736234835679429446577737976919e+1020L, 1.49738681485939897351147361037410278e+1036L, 2.79474808763788045317059787958854766e+1052L, 9.40752795141195895210630116566470676e+1068L, 5.76455681194134988563523968421696548e+1085L, 6.49098750070655215612470119412325218e+1102L, 1.35603677524062211064037225514112165e+1120L, 5.30731596106421482379267500594247426e+1137L, 3.93020043869734370491941322540535807e+1155L, 5.56226722613029149136813618379760709e+1173L, 1.51990408820680241841102181882025951e+1192L, 8.10228003417607034858018290055212984e+1210L, 8.51521180721994744419289716417963237e+1229L, 1.78329495296502824223674285976432097e+1249L, 7.52325725334531880022021234450122858e+1268L, 6.46447702663797449769360764507187142e+1288L, 1.14411796435953528891736478925455547e+1309L, 4.21851818337788742448241659152628519e+1329L, 3.27808619028067308419286053048979085e+1350L, 5.43188779501763188996015031600338462e+1371L, 1.94236185576822902995555282219996135e+1393L, 1.51711647857248503907031482339088565e+1415L, 2.62035716514505778741231466041668281e+1437L, 1.01340333806343444348385264050569250e+1460L, 8.88786410219005270769408353397020687e+1482L, 1.79062851327433476234422894649930707e+1506L, 8.39642792144232596950658702528918677e+1529L, 9.28628794761707451127390451820966789e+1553L, 2.45536895088636462839367313490581387e+1578L, 1.57354562229875356978426356241845302e+1603L, 2.47847583619098850454983740131691503e+1628L, 9.73157119375940175602769413110414643e+1653L, 9.66320447484532719750186034583884499e+1679L, 2.46232109538721189753853963925622165e+1706L, 1.63417596934387293040006746046570718e+1733L, 2.86767794799456586471233013913198039e+1760L, 1.35110548955493375904544893914917995e+1788L, 1.73592152545700077097814839827674920e+1816L, 6.17893831026015926230407301367680801e+1844L, 6.19169576655007716364858612984691562e+1873L, 1.77539961419331109157892833779912252e+1903L, 1.48102931564535770775910220632570312e+1933L, 3.65523833138677287346435504279580939e+1963L, 2.71500576533505130343548298805675281e+1994L, 6.17538661057993294979124427622665421e+2025L, 4.37773911614056336228335278244614535e+2057L, 9.84696637606677900175890481086698860e+2089L, 7.15679932872021171306956409599896671e+2122L, 1.71206538074954819527676482193571839e+2156L, 1.37358203471630226464631105286036694e+2190L, 3.76701670663930120912543386433302245e+2224L, 3.60043887764831236838085599723119708e+2259L, 1.22311399177140229476034440208616729e+2295L, 1.50662485913882109231872682750679813e+2331L, 6.86720031172136555587682118613084257e+2367L, 1.18233110675109495729851537371527994e+2405L, 7.85186218386652904615339091354671418e+2442L, }, + }; + m_committed_refinements = static_cast(m_weights.size() - 1); + m_t_min = -8.8984375000000L; + if (m_max_refinements >= m_abscissas.size()) + { + m_abscissas.resize(m_max_refinements + 1); + m_weights.resize(m_max_refinements + 1); + } + else + { + m_max_refinements = m_abscissas.size() - 1; + } +} +#endif +#ifdef BOOST_HAS_FLOAT128 +template +void exp_sinh_detail::init(const mpl::int_<4>&) +{ + m_abscissas = { + { 1.02756529896290544244959196973059583e-2497Q, 2.56737528671961581475200468317128232e-919Q, 1.17417808941462434184174780056564573e-338Q, 4.82182886130634548471358754377036370e-125Q, 1.85613301660646818149136526457281019e-46Q, 1.52174118093087534300657777272732001e-17Q, 6.75122240537294392532710530940672267e-07Q, 5.94481311616464419075825632567494453e-03Q, 2.00100938779037997581424620542543429e-01Q, 1.17328605653712546899681147538372171e+00Q, 8.18356490677287285512063117929807241e+00Q, 5.59865842621965368881982340891928481e+02Q, 3.69902944883650290371636082450503730e+07Q, 4.05747121124517088709477132072461878e+20Q, 1.07723884748977308147226626407207905e+56Q, 2.07250561337258237728923403163755392e+152Q, 1.09889904624000153879292638133263171e+414Q, 3.02463014753652876878705286965250144e+1125Q, }, + { 3.16339947894004847541521297710937343e-1515Q, 7.02786763812753004271170107158747593e-558Q, 1.08465499866859733494552744914276656e-205Q, 3.97220624097731753646493300738183748e-76Q, 1.84145280968701148636501796762008815e-28Q, 6.40886341101610144938011594904313817e-11Q, 1.89200275395722467663251234615351291e-04Q, 5.04892743776943294143478969573235061e-02Q, 5.25946553709738448524487603870477588e-01Q, 2.72635049439566736308739858009584752e+00Q, 4.20081657572145199006060015525396829e+01Q, 3.75221714819472331206969009243353365e+04Q, 3.11642465424592079726559332629050577e+12Q, 9.61947229245436139236957297153092319e+33Q, 2.42007357898805628851519620917402234e+92Q, 1.34351470989086331111432001407522043e+251Q, 4.34937560839564995558174638985438726e+682Q, 4.01820305474077838467315109614612109e+1855Q, }, + { 2.20630426387562582340901270736066292e-1945Q, 3.20902289525796097195876526710905754e-1180Q, 3.99075419312896749376535004306952481e-716Q, 1.22680463935505388461578092999320797e-434Q, 6.60416836081631149205027836695167940e-264Q, 2.36380363265905808053031227647927829e-160Q, 1.52216180146092756550894692311532796e-97Q, 1.89865679919908959317123677189779258e-59Q, 2.43020115474101871630434504433731237e-36Q, 2.52466874690605714759066961852641028e-22Q, 8.05169965363444223580324196979926112e-14Q, 1.16754399126294292055540155961164512e-08Q, 1.59887725419859990767684898940272339e-05Q, 1.30653650105064376247657363109776765e-03Q, 1.96025435814529675467866513121889090e-02Q, 1.07798510399525035644857486110150806e-01Q, 3.35726506001932774135392168623644982e-01Q, 7.91344191300036661743190713036639943e-01Q, 1.75712551051579342077493249233109834e+00Q, 4.50532922519182663928094313196688277e+00Q, 1.69634912879730950991827009814964854e+01Q, 1.31564914034011933527149556208430823e+02Q, 3.54551665937177335700906931219692079e+03Q, 7.69629182688413474235641078406238234e+05Q, 5.31334781541946297455601223275400364e+09Q, 1.11397125403497802554344014693570895e+16Q, 2.91323342059626628322620905024841007e+26Q, 4.33470414741675827465254900104887599e+43Q, 8.87077939346041258291094819287939414e+71Q, 4.20552510572288598564618972079522209e+118Q, 3.78998758665619810759092788424597102e+195Q, 2.85285821381079277498378752581228807e+322Q, 4.35767992687500025738868152293904617e+531Q, 3.35101732969916400190150631384628168e+876Q, 1.39907636805339624393396738787997778e+1445Q, 4.39310839133852761600954729262867435e+2382Q, }, + { 2.60416027838200926634868701489352534e-2204Q, 7.02786036162213986704884019362612079e-1717Q, 2.87472449554862557122883045361480216e-1337Q, 1.26052689335617094297502885256721406e-1041Q, 2.22203737723769125613395083355625930e-811Q, 4.59925691253299141912321586723335574e-632Q, 2.06110387430222811031505778840893135e-492Q, 1.18783333719310840737835079091916047e-383Q, 5.99339963191580129696690680497340117e-299Q, 5.55156577871295836646608994333019205e-233Q, 1.31650343182343962237291034126486362e-181Q, 1.35017224106781623208439493207058713e-141Q, 1.95458267270042896122641878745090637e-110Q, 3.62345406499123808121116200173051239e-86Q, 2.87916036185197771976746396107338099e-67Q, 1.50975853574758038650941626753584613e-52Q, 4.39366381209590686916061999351406334e-41Q, 3.72613279781933265810269222029431206e-32Q, 3.35074541708050784063416140930256908e-25Q, 8.73699691100611067810340427391712629e-20Q, 1.44610938754441658588026566732900630e-15Q, 2.79712542830915646153531586097835651e-12Q, 1.01712318476608889609455916646557919e-09Q, 1.00707686902351812525332969334486408e-07Q, 3.62377064535647765957086044101238700e-06Q, 5.93514765312557852887858990273347709e-05Q, 5.27427922038425038994892508697801291e-04Q, 2.91716284057748187538183216214780172e-03Q, 1.11812753052373081344852811278990196e-02Q, 3.23158910999791296389217134204252891e-02Q, 7.52823576293971217140453145701166400e-02Q, 1.49072381771447883955117212578891471e-01Q, 2.61920116954195648982059938116700281e-01Q, 4.23044722849733503488984642521144095e-01Q, 6.47337815325830827783598163922905049e-01Q, 9.63850663681748439752705317539095691e-01Q, 1.43180513983798487645924505146418541e+00Q, 2.17546288127550398262362304432746644e+00Q, 3.47077553215380191942193887548951574e+00Q, 5.98762125926090985873603940556086485e+00Q, 1.15588346593765285104362265791852722e+01Q, 2.60227165873113109321939526707132515e+01Q, 7.18972764924010846908346544069757730e+01Q, 2.59849827899514040038730358528910272e+02Q, 1.33180383652908565595482681440747426e+03Q, 1.07264199927746293597346233164919561e+04Q, 1.54770252854197591129695964624494039e+05Q, 4.73116974059692035499030617203501902e+06Q, 3.79850299532978550624970142683299187e+08Q, 1.05506900281875210383901752883017567e+11Q, 1.44377693764054834189876938092028936e+14Q, 1.53216987231286586179401429003390945e+18Q, 2.25749845424265693444972830430363021e+23Q, 9.75366126404791278425650801377275643e+29Q, 3.22710662318561074542283240061745423e+38Q, 2.80427989450823486864584373674252213e+49Q, 3.11492865697270427645920832671970699e+63Q, 3.37385880956075754428852062324609716e+81Q, 4.84069913749095116306098688426537434e+104Q, 2.62206965911556490036221731139595538e+134Q, 3.95724512624896651389226023882699109e+172Q, 4.16658233419696139622872515762144259e+221Q, 3.67870997815873300310505495683155714e+284Q, 2.45353561048417937411865685638919263e+365Q, 1.47890836767899626666102902312593029e+469Q, 2.66842264577698782467085710268227542e+602Q, 3.39406520823137428862537121090858482e+773Q, 1.71072929388916771882465310676448508e+993Q, 2.17124413154098944294625847065193945e+1275Q, 3.67108839134079465802181460380664133e+1637Q, 4.72992162468569560939225829187204387e+2102Q, }, + { 1.98007625288857013412122004727810624e-2346Q, 8.39652992194258659266745309526472882e-2071Q, 1.46030671151231956105710426802408027e-1827Q, 6.65727248909208458885310717390128801e-1613Q, 1.81581117219394388092797723844111009e-1423Q, 2.72619674739250219990613941747932927e-1256Q, 9.29590026994488020433315206443278096e-1109Q, 1.46377917106366800430984793284730939e-978Q, 1.15901956513202291149840314964756250e-863Q, 2.89578802021929239619753606490350524e-762Q, 8.80858736107764685086031513644364894e-673Q, 8.19416690740938676449696497280686335e-594Q, 4.00916745706227520865721440095991259e-524Q, 1.27020033549323657895766092175564312e-462Q, 2.38869695900235565316537719061693975e-408Q, 1.88391475973801739487501436111488490e-360Q, 3.49894275509125434733587030489899389e-318Q, 7.01615886164307390146523314851863544e-281Q, 5.82310443050787035609701710903434681e-248Q, 6.54836166711216765944679142249433382e-219Q, 2.84149263299034989667159925017962142e-193Q, 1.19812648186130338854458247959762818e-170Q, 1.10911490518050678588403791608480488e-150Q, 4.62754597695358582538782054737574180e-133Q, 1.64173210211161942076907089038989838e-117Q, 8.67192889174269982709959745767681981e-104Q, 1.11812225550444523531457194417899189e-91Q, 5.44369950217028498211921027217139012e-81Q, 1.47074883585505403248443957516006445e-71Q, 3.09730398495823549037183157408809652e-63Q, 6.86204181060118439675162432029019608e-56Q, 2.08386500906124109876681630886775453e-49Q, 1.09560045982532469724721792988430545e-43Q, 1.22547627904244500951228300724497066e-38Q, 3.49796660995417261286766404577301528e-34Q, 2.99152410870631960427539566799345990e-30Q, 8.83190101026086766963628794276374683e-27Q, 1.01998228741819737621697177531967545e-23Q, 5.14543767526486873178240241354085638e-21Q, 1.24976972746216000788747786633668576e-18Q, 1.59270820536117731616569419326771375e-16Q, 1.14889208313232540746001768184091876e-14Q, 5.01568630936388404881919116817433643e-13Q, 1.40588174426346693556351457090253957e-11Q, 2.66552656620728447371725338328298653e-10Q, 3.57952342849715748848195322266581664e-09Q, 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2.47847583619098850454983740131691503e+1628Q, 9.73157119375940175602769413110414643e+1653Q, 9.66320447484532719750186034583884499e+1679Q, 2.46232109538721189753853963925622165e+1706Q, 1.63417596934387293040006746046570718e+1733Q, 2.86767794799456586471233013913198039e+1760Q, 1.35110548955493375904544893914917995e+1788Q, 1.73592152545700077097814839827674920e+1816Q, 6.17893831026015926230407301367680801e+1844Q, 6.19169576655007716364858612984691562e+1873Q, 1.77539961419331109157892833779912252e+1903Q, 1.48102931564535770775910220632570312e+1933Q, 3.65523833138677287346435504279580939e+1963Q, 2.71500576533505130343548298805675281e+1994Q, 6.17538661057993294979124427622665421e+2025Q, 4.37773911614056336228335278244614535e+2057Q, 9.84696637606677900175890481086698860e+2089Q, 7.15679932872021171306956409599896671e+2122Q, 1.71206538074954819527676482193571839e+2156Q, 1.37358203471630226464631105286036694e+2190Q, 3.76701670663930120912543386433302245e+2224Q, 3.60043887764831236838085599723119708e+2259Q, 1.22311399177140229476034440208616729e+2295Q, 1.50662485913882109231872682750679813e+2331Q, 6.86720031172136555587682118613084257e+2367Q, 1.18233110675109495729851537371527994e+2405Q, 7.85186218386652904615339091354671418e+2442Q, }, + }; + m_committed_refinements = static_cast(m_weights.size() - 1); + m_t_min = -8.89265635039934944849389055479809884Q; + if (m_max_refinements >= m_abscissas.size()) + { + m_abscissas.resize(m_max_refinements + 1); + m_weights.resize(m_max_refinements + 1); + } + else + { + m_max_refinements = m_abscissas.size() - 1; + } +} +#endif + +} +} +} +} +#endif diff --git a/include/boost/math/quadrature/detail/sinh_sinh_detail.hpp b/include/boost/math/quadrature/detail/sinh_sinh_detail.hpp new file mode 100644 index 000000000..7dce73273 --- /dev/null +++ b/include/boost/math/quadrature/detail/sinh_sinh_detail.hpp @@ -0,0 +1,463 @@ +// Copyright Nick Thompson, 2017 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_QUADRATURE_DETAIL_SINH_SINH_DETAIL_HPP +#define BOOST_MATH_QUADRATURE_DETAIL_SINH_SINH_DETAIL_HPP + +#include +#include +#include +#include +#include +#include +#include + +namespace boost{ namespace math{ namespace quadrature { namespace detail{ + + +// Returns the sinh-sinh quadrature of a function f over the entire real line + +template +class sinh_sinh_detail +{ + static const int initializer_selector = + !std::numeric_limits::is_specialized || (std::numeric_limits::radix != 2) ? + 0 : + (std::numeric_limits::digits < 30) && (std::numeric_limits::max_exponent <= 128) ? + 1 : + (std::numeric_limits::digits <= std::numeric_limits::digits) && (std::numeric_limits::max_exponent <= std::numeric_limits::max_exponent) ? + 2 : + (std::numeric_limits::digits <= std::numeric_limits::digits) && (std::numeric_limits::max_exponent <= 16384) ? + 3 : +#ifdef BOOST_HAS_FLOAT128 + (std::numeric_limits::digits <= 113) && (std::numeric_limits::max_exponent <= 16384) ? + 4 : +#endif + 0; +public: + sinh_sinh_detail(size_t max_refinements); + + template + Real integrate(const F f, Real tolerance, Real* error, Real* L1, std::size_t* levels) const; + +private: +private: + const std::vector& get_abscissa_row(std::size_t n)const + { +#ifndef BOOST_MATH_NO_ATOMIC_INT + if (m_committed_refinements.load() < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements.load() >= n); +#else + if (m_committed_refinements < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements >= n); +#endif + return m_abscissas[n]; + } + const std::vector& get_weight_row(std::size_t n)const + { +#ifndef BOOST_MATH_NO_ATOMIC_INT + if (m_committed_refinements.load() < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements.load() >= n); +#else + if (m_committed_refinements < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements >= n); +#endif + return m_weights[n]; + } + void init(const mpl::int_<0>&); + void init(const mpl::int_<1>&); + void init(const mpl::int_<2>&); + void init(const mpl::int_<3>&); +#ifdef BOOST_HAS_FLOAT128 + void init(const mpl::int_<4>&); +#endif + + void extend_refinements()const + { +#ifndef BOOST_MATH_NO_ATOMIC_INT + boost::detail::lightweight_mutex::scoped_lock guard(m_mutex); +#endif + // + // Check some other thread hasn't got here after we read the atomic variable, but before we got here: + // +#ifndef BOOST_MATH_NO_ATOMIC_INT + if (m_committed_refinements.load() >= m_max_refinements) + return; +#else + if (m_committed_refinements >= m_max_refinements) + return; +#endif + + using std::ldexp; + using std::ceil; + using std::sinh; + using std::cosh; + using std::exp; + using constants::half_pi; + + std::size_t row = ++m_committed_refinements; + + Real h = ldexp(Real(1), -static_cast(row)); + size_t k = (size_t)boost::math::lltrunc(ceil(m_t_max / (2 * h))); + m_abscissas[row].reserve(k); + m_weights[row].reserve(k); + Real arg = h; + while (arg < m_t_max) + { + Real tmp = half_pi()*sinh(arg); + Real x = sinh(tmp); + m_abscissas[row].emplace_back(x); + Real w = cosh(arg)*half_pi()*cosh(tmp); + m_weights[row].emplace_back(w); + arg += 2 * h; + } + } + + Real m_t_max; + + mutable std::vector> m_abscissas; + mutable std::vector> m_weights; + std::size_t m_max_refinements; +#ifndef BOOST_MATH_NO_ATOMIC_INT + mutable boost::math::detail::atomic_unsigned_type m_committed_refinements; + mutable boost::detail::lightweight_mutex m_mutex; +#else + mutable unsigned m_committed_refinements; +#endif +}; + +template +sinh_sinh_detail::sinh_sinh_detail(size_t max_refinements) + : m_abscissas(max_refinements), m_weights(max_refinements), m_max_refinements(max_refinements) +{ + init(mpl::int_()); +} + +template +template +Real sinh_sinh_detail::integrate(const F f, Real tolerance, Real* error, Real* L1, std::size_t* levels) const +{ + using std::abs; + using std::sqrt; + using boost::math::constants::half; + using boost::math::constants::half_pi; + + static const char* function = "boost::math::quadrature::sinh_sinh<%1%>::integrate"; + + Real y_max = f(boost::math::tools::max_value()); + if(abs(y_max) > boost::math::tools::epsilon() || !(boost::math::isfinite)(y_max)) + { + return policies::raise_domain_error(function, + "The function you are trying to integrate does not go to zero at infinity, and instead evaluates to %1%", y_max, Policy()); + } + + Real y_min = f(-boost::math::tools::max_value()); + if(abs(y_min) > boost::math::tools::epsilon() || !(boost::math::isfinite)(y_min)) + { + return policies::raise_domain_error(function, + "The function you are trying to integrate does not go to zero at -infinity, and instead evaluates to %1%", y_max, Policy()); + } + + // Get the party started with two estimates of the integral: + Real I0 = f(0)*half_pi(); + Real L1_I0 = abs(I0); + for(size_t i = 0; i < m_abscissas[0].size(); ++i) + { + Real x = m_abscissas[0][i]; + Real yp = f(x); + Real ym = f(-x); + I0 += (yp + ym)*m_weights[0][i]; + L1_I0 += (abs(yp)+abs(ym))*m_weights[0][i]; + } + + // Uncomment the estimates to work the convergence on the command line. + // std::cout << std::setprecision(std::numeric_limits::digits10); + // std::cout << "First estimate : " << I0 << std::endl; + Real I1 = I0; + Real L1_I1 = L1_I0; + for (size_t i = 0; i < m_abscissas[1].size(); ++i) + { + Real x= m_abscissas[1][i]; + Real yp = f(x); + Real ym = f(-x); + I1 += (yp + ym)*m_weights[1][i]; + L1_I1 += (abs(yp) + abs(ym))*m_weights[1][i]; + } + + I1 *= half(); + L1_I1 *= half(); + Real err = abs(I0 - I1); + // std::cout << "Second estimate: " << I1 << " Error estimate at level " << 1 << " = " << err << std::endl; + + size_t i = 2; + for(; i <= m_max_refinements; ++i) + { + I0 = I1; + L1_I0 = L1_I1; + + I1 = half()*I0; + L1_I1 = half()*L1_I0; + Real h = (Real) 1/ (Real) (1 << i); + Real sum = 0; + Real absum = 0; + + Real abterm1 = 1; + Real eps = boost::math::tools::epsilon()*L1_I1; + + auto abscissa_row = get_abscissa_row(i); + auto weight_row = get_weight_row(i); + + for(size_t j = 0; j < abscissa_row.size(); ++j) + { + Real x = abscissa_row[j]; + Real yp = f(x); + Real ym = f(-x); + sum += (yp + ym)*weight_row[j]; + Real abterm0 = (abs(yp) + abs(ym))*weight_row[j]; + absum += abterm0; + + // We require two consecutive terms to be < eps in case we hit a zero of f. + if (x > (Real) 100 && abterm0 < eps && abterm1 < eps) + { + break; + } + abterm1 = abterm0; + } + + I1 += sum*h; + L1_I1 += absum*h; + err = abs(I0 - I1); + // std::cout << "Estimate: " << I1 << " Error estimate at level " << i << " = " << err << std::endl; + if (!(boost::math::isfinite)(I1)) + { + const char* err_msg = "The sinh_sinh quadrature evaluated your function at a singular point, leading to the value %1%.\n" + "sinh_sinh quadrature cannot handle singularities in the domain.\n" + "If you are sure your function has no singularities, please submit a bug against boost.math\n"; + return policies::raise_evaluation_error(function, err_msg, I1, Policy()); + } + if (err <= tolerance*L1_I1) + { + break; + } + } + + if (error) + { + *error = err; + } + + if (L1) + { + *L1 = L1_I1; + } + + if (levels) + { + *levels = i; + } + + return I1; +} + +template +void sinh_sinh_detail::init(const mpl::int_<0>&) +{ + using std::log; + using std::sqrt; + using std::cosh; + using std::sinh; + using std::ceil; + using boost::math::constants::two_div_pi; + using boost::math::constants::half_pi; + + m_committed_refinements = 4; + + // t_max is chosen to make g'(t_max) ~ sqrt(max) (g' grows faster than g). + // This will allow some flexibility on the users part; they can at least square a number function without overflow. + // But there is no unique choice; the further out we can evaluate the function, the better we can do on slowly decaying integrands. + m_t_max = log(2 * two_div_pi()*log(2 * two_div_pi()*sqrt(tools::max_value()))); + + for (size_t i = 0; i <= 4; ++i) + { + Real h = (Real)1 / (Real)(1 << i); + size_t k = (size_t)boost::math::lltrunc(ceil(m_t_max / (2 * h))); + m_abscissas[i].reserve(k); + m_weights[i].reserve(k); + // We don't add 0 to the abscissas; that's treated as a special case. + Real arg = h; + while (arg < m_t_max) + { + Real tmp = half_pi()*sinh(arg); + Real x = sinh(tmp); + m_abscissas[i].emplace_back(x); + Real w = cosh(arg)*half_pi()*cosh(tmp); + m_weights[i].emplace_back(w); + + if (i != 0) + { + arg += 2 * h; + } + else + { + arg += h; + } + } + } +} + +template +void sinh_sinh_detail::init(const mpl::int_<1>&) +{ + m_abscissas = { + { 3.08828742e+00f, 1.48993185e+02f, 3.41228925e+06f, 2.06932577e+18f, }, + { 9.13048763e-01f, 1.41578929e+01f, 6.70421552e+03f, 9.64172533e+10f, }, + { 4.07297690e-01f, 1.68206671e+00f, 6.15089799e+00f, 4.00396235e+01f, 7.92920025e+02f, 1.02984971e+05f, 3.03862311e+08f, 1.56544547e+14f, }, + { 1.98135272e-01f, 6.40155674e-01f, 1.24892870e+00f, 2.26608084e+00f, 4.29646270e+00f, 9.13029039e+00f, 2.31110765e+01f, 7.42770603e+01f, 3.26720921e+02f, 2.15948569e+03f, 2.41501526e+04f, 5.31819400e+05f, 2.80058686e+07f, 4.52406508e+09f, 3.08561257e+12f, 1.33882673e+16f, }, + { 9.83967894e-02f, 3.00605618e-01f, 5.19857979e-01f, 7.70362083e-01f, 1.07131137e+00f, 1.45056976e+00f, 1.95077855e+00f, 2.64003177e+00f, 3.63137237e+00f, 5.11991533e+00f, 7.45666098e+00f, 1.13022613e+01f, 1.79641069e+01f, 3.01781070e+01f, 5.40387580e+01f, 1.04107731e+02f, 2.18029520e+02f, 5.02155699e+02f, 1.28862131e+03f, 3.73921687e+03f, 1.24750730e+04f, 4.87639975e+04f, 2.28145658e+05f, 1.30877796e+06f, 9.46084663e+06f, 8.88883120e+07f, 1.12416883e+09f, 1.99127673e+10f, 5.16743469e+11f, 2.06721881e+13f, 1.35061503e+15f, 1.53854066e+17f, }, + { 4.91151004e-02f, 1.48013150e-01f, 2.48938814e-01f, 3.53325424e-01f, 4.62733557e-01f, 5.78912068e-01f, 7.03870253e-01f, 8.39965859e-01f, 9.90015066e-01f, 1.15743257e+00f, 1.34641276e+00f, 1.56216711e+00f, 1.81123885e+00f, 2.10192442e+00f, 2.44484389e+00f, 2.85372075e+00f, 3.34645891e+00f, 3.94664582e+00f, 4.68567310e+00f, 5.60576223e+00f, 6.76433234e+00f, 8.24038318e+00f, 1.01439436e+01f, 1.26302471e+01f, 1.59213040e+01f, 2.03392186e+01f, 2.63584645e+01f, 3.46892633e+01f, 4.64129147e+01f, 6.32055079e+01f, 8.77149726e+01f, 1.24209693e+02f, 1.79718635e+02f, 2.66081728e+02f, 4.03727303e+02f, 6.28811307e+02f, 1.00707984e+03f, 1.66156823e+03f, 2.82965144e+03f, 4.98438627e+03f, 9.10154693e+03f, 1.72689266e+04f, 3.41309958e+04f, 7.04566898e+04f, 1.52340422e+05f, 3.46047978e+05f, 8.28472421e+05f, 2.09759615e+06f, 5.63695080e+06f, 1.61407141e+07f, 4.94473068e+07f, 1.62781052e+08f, 5.78533297e+08f, 2.23083854e+09f, 9.38239131e+09f, 4.32814954e+10f, 2.20307274e+11f, 1.24524507e+12f, 7.86900053e+12f, 5.59953143e+13f, 4.52148695e+14f, 4.17688952e+15f, 4.45286776e+16f, 5.52914285e+17f, 8.07573252e+18f, }, + { 2.45471558e-02f, 7.37246687e-02f, 1.23152531e-01f, 1.73000138e-01f, 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1.489931846492091580e+02, 3.412289247883437102e+06, 2.069325766042617791e+18, 2.087002407609475560e+50, 2.019766160717908151e+137, }, + { 9.130487626376696748e-01, 1.415789294662811592e+01, 6.704215516223276482e+03, 9.641725327150499415e+10, 2.508950760085778485e+30, 1.447263535710337145e+83, }, + { 4.072976900657586902e-01, 1.682066707021148743e+00, 6.150897986386729515e+00, 4.003962351929400222e+01, 7.929200247931026321e+02, 1.029849713330979583e+05, 3.038623109252438574e+08, 1.565445474362494869e+14, 4.042465098430219104e+23, 1.321706827429658179e+39, 4.991231782099557998e+64, 7.352943850359875966e+106, }, + { 1.981352722514781726e-01, 6.401556735005260177e-01, 1.248928698253977663e+00, 2.266080840944321232e+00, 4.296462696702327381e+00, 9.130290387099955696e+00, 2.311107653864279933e+01, 7.427706034324012430e+01, 3.267209207115258917e+02, 2.159485694311818716e+03, 2.415015262896413060e+04, 5.318194002756929158e+05, 2.800586857217043323e+07, 4.524065079794338780e+09, 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1.430959291578558210e+40, 5.873974584984375049e+40, 2.464828549811283787e+41, 1.057649203090855628e+42, 4.642475639281078035e+42, 2.085287118272421779e+43, 9.588439985186632177e+43, 4.514982011246092280e+44, 2.177974048341973204e+45, 1.076720976822900458e+46, 5.457267432929085589e+46, 2.836869270455781134e+47, 1.513103201392011626e+48, 8.283974667225617075e+48, 4.657239491995971344e+49, 2.689796370712836937e+50, 1.596597846911970388e+51, 9.744154538256586629e+51, 6.117238394843313065e+52, 3.952049650585241827e+53, 2.628701592074258213e+54, 1.800990196502679393e+55, 1.271554462563068383e+56, 9.255880104477760711e+56, 6.949737920133919393e+57, 5.385167200769965621e+58, 4.308493668102978774e+59, 3.560951557542178371e+60, 3.041888528384649992e+61, 2.687094441930837189e+62, 2.455920538900000855e+63, 2.323648254168641537e+64, 2.277129741584892331e+65, 2.312633552913224734e+66, 2.435407592981291129e+67, 2.660910388822465246e+68, 3.018105943423533920e+69, 3.555823489510192503e+70, 4.354188877793849013e+71, 5.544975795511813315e+72, 7.348276481909886336e+73, 1.013998025722423261e+75, 1.457911462244607943e+76, 2.185488876819505295e+77, 3.418022153286623008e+78, 5.580843920601835728e+79, 9.519586502799733908e+80, 1.697573578247197786e+82, 3.166906670990180014e+83, 6.185099106418675430e+84, 1.265541134386934377e+86, 2.714828965877756899e+87, 6.110386802964494082e+88, 1.444054086171083239e+90, 3.586083726638388165e+91, 9.365231868063239600e+92, 2.574080116205122449e+94, 7.452134689862302719e+95, 2.274309903836169819e+97, 7.323011134121164749e+98, 2.489816421737932462e+100, 8.946533386359281588e+101, 3.400401372391165979e+103, 1.368288186208928217e+105, 5.834277489829591931e+106, 2.638486937672383424e+108, 1.266728882767139521e+110, 6.462225178314182803e+111, 3.506432320607573604e+113, 2.025608933943268165e+115, 1.247041677084784707e+117, 8.189865188405279038e+118, 5.743610894406099965e+120, 4.305808934084489763e+122, 3.454156966079496755e+124, 2.968316601530352737e+126, 2.735456242372183592e+128, 2.706317176690077847e+130, 2.877679916342060385e+132, 3.292412878268106390e+134, 4.057840961953725969e+136, 5.393783049105737324e+138, 7.741523901672235406e+140, 1.201209962310668456e+143, 2.017456079556807301e+145, 3.672176623483062526e+147, 7.253163798058577630e+149, 1.556591535302570570e+152, 3.634399832790394885e+154, }, + }; + m_committed_refinements = static_cast(m_weights.size() - 1); + m_t_max = 6.114056619798597593; + if (m_max_refinements >= m_abscissas.size()) + { + m_abscissas.resize(m_max_refinements + 1); + m_weights.resize(m_max_refinements + 1); + } + else + { + m_max_refinements = m_abscissas.size() - 1; + } +} + +#if LDBL_MAX_EXP == 16384 +template +void sinh_sinh_detail::init(const mpl::int_<3>&) +{ + m_abscissas = { + { 3.08828741797632286606397498241221385e+00L, 1.48993184649209158013612709175719825e+02L, 3.41228924788343710247727162226946917e+06L, 2.06932576604261779073902718911207249e+18L, 2.08700240760947556038306635808129743e+50L, 2.01976616071790815078008252209994199e+137L, 5.67213444764437168603513205349222232e+373L, 3.06198394306784061113736467298565948e+1016L, }, + { 9.13048762637669674838078869945948356e-01L, 1.41578929466281159169215570833490092e+01L, 6.70421551622327648231120321610008518e+03L, 9.64172532715049941526010563818587232e+10L, 2.50895076008577848514087028569888161e+30L, 1.44726353571033714499079379626715686e+83L, 3.75263401205045334128277886549019357e+226L, 2.57846123932685341261715758547064293e+616L, 1.25169402230987931584130068460134989e+1676L, }, + { 4.07297690065758690150573950473604675e-01L, 1.68206670702114874332932330092838356e+00L, 6.15089798638672951539668935798437533e+00L, 4.00396235192940022205839866641270319e+01L, 7.92920024793102632052902471550336177e+02L, 1.02984971333097958319686053784919407e+05L, 3.03862310925243857437932941891415841e+08L, 1.56544547436249486913719231527597560e+14L, 4.04246509843021910355659662715183671e+23L, 1.32170682742965817915034577861235933e+39L, 4.99123178209955799774620736137366103e+64L, 7.35294385035987596594501676609562008e+106L, 2.45145417229813114714431821340237112e+176L, 1.03030479197459267961759023741376327e+291L, 9.88478945193556730604342445576518136e+479L, 3.74498116076433060427765453233124618e+791L, 1.90292390713026708045766158039087796e+1305L, 1.72712488231809244721915842027236657e+2152L, }, + { 1.98135272251478172615235718398538323e-01L, 6.40155673500526017670785720911319502e-01L, 1.24892869825397766254802900819319987e+00L, 2.26608084094432123181038001270985037e+00L, 4.29646269670232738148137717958679987e+00L, 9.13029038709995569641811827045549232e+00L, 2.31110765386427993316995139635470937e+01L, 7.42770603432401243012214481410240677e+01L, 3.26720920711525891697790966796945063e+02L, 2.15948569431181871552028449867301828e+03L, 2.41501526289641306037808670196966696e+04L, 5.31819400275692915826269282626844010e+05L, 2.80058685721704332307817790755562305e+07L, 4.52406507979433877971977526863358905e+09L, 3.08561257398067712180174566763237112e+12L, 1.33882673301580747789133427708622972e+16L, 6.25461717656234138147247754416652254e+20L, 6.18209853581416475394863999233554548e+26L, 3.07729364978845806686511011060697837e+34L, 2.34895728937010430290698332595743680e+44L, 1.14854319789946975790127332653552673e+57L, 2.25530007001006986846545694404767295e+73L, 1.87791950056919539419801461538703336e+94L, 1.36747388793862427976781689638282239e+121L, 4.24212177246407592514720294750151286e+155L, 8.23473099012134325391170623028607158e+199L, 6.06134211888406631001019923425936351e+256L, 6.32519197436029413652766295658658263e+329L, 3.61942292928205643948383705201563065e+423L, 8.82464433126646489632943769283758364e+543L, 3.35512488018651738642854323317447493e+698L, 1.02411434678684822400261582079656500e+897L, 7.40716670979374017620610271197945784e+1151L, 1.30455147283895162835186634827124464e+1479L, 2.02948723345659016173510134582487641e+1899L, 6.99019443250345564805134094483457477e+2438L, }, + { 9.83967894006732033862249554537934739e-02L, 3.00605617659955035122465185012715805e-01L, 5.19857978994938490000644785900799951e-01L, 7.70362083298887700858583505753306721e-01L, 1.07131136964131183026831222500748063e+00L, 1.45056975808899844502126797109703210e+00L, 1.95077854952036033400296546339240312e+00L, 2.64003177369555146770080103921424631e+00L, 3.63137237366741227331951933364601985e+00L, 5.11991533090335057003526222543242753e+00L, 7.45666098140488328926087222725984231e+00L, 1.13022612688997262443461342203136009e+01L, 1.79641069247277254966592982669611020e+01L, 3.01781070460189822236726995418068214e+01L, 5.40387580031237056709826662540066860e+01L, 1.04107731447746954767459980678311109e+02L, 2.18029520120262807725988909352359830e+02L, 5.02155698625910164594509258651311707e+02L, 1.28862131099822242045586488612672886e+03L, 3.73921687080054832376623401153054636e+03L, 1.24750729702019123239464970064060864e+04L, 4.87639975322669212414357357061002683e+04L, 2.28145658221913012154676335814417144e+05L, 1.30877796006484301664040981436471740e+06L, 9.46084663420966407698078336053299177e+06L, 8.88883120363727962245667776649286152e+07L, 1.12416882897434413447175648471746990e+09L, 1.99127672953214446985509145592687205e+10L, 5.16743469106098464964433058111638245e+11L, 2.06721881420399088776240044545119948e+13L, 1.35061503318410040604770421574064883e+15L, 1.53854066283650818848773859278941683e+17L, 3.29074729054035066052820102585085173e+19L, 1.43729138188449881561963100200796631e+22L, 1.40983244553034728562740932242727657e+25L, 3.45913548027797144127990544649862792e+28L, 2.39872058234095409213724024023509754e+32L, 5.39880660461729295997086364918754801e+36L, 4.61334000258062860955942270137550165e+41L, 1.78768590966790245678294238939560892e+47L, 3.84198437012433853552557148945213178e+53L, 5.75279795570858370009870455748773410e+60L, 7.77181203842728655149182334288749850e+68L, 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9.02071339187528665617085297107535931e+1425L, 2.32776342394968427168699373890370933e+1448L, 1.35351409374730203846799381939288903e+1471L, 1.79625881886715571166142200348072039e+1494L, 5.51189494978629938318487744451579809e+1517L, 3.96270198063441303687686251379779681e+1541L, 6.76492837567767494547614386605795924e+1565L, 2.77991073110506719318274536923711255e+1590L, 2.78805951202715437087943440518839779e+1615L, 6.92116280606414755524296394791409952e+1640L, 4.31380309053366092943951122004431463e+1666L, 6.84923073262139995438889690748178280e+1692L, 2.81137418164430829179092468240459910e+1719L, 3.02821524237655652451836921974619657e+1746L, 8.69048903533522694231765189726868666e+1773L, 6.74828063313306417588175763238310325e+1801L, 1.44026065844141159256872926595644247e+1830L, 8.58426091495439408178371839859019039e+1858L, 1.45211919400936944187225846047638548e+1888L, 7.08718013380970011842483247147601395e+1917L, 1.01475962220405902523641440040104610e+1948L, 4.33542978691578087509838132034723750e+1978L, 5.62285772227295495777726116212032065e+2009L, 2.25285072921565445630031609130087435e+2041L, 2.83839050906274298341210206309759282e+2073L, 1.14501659579022194588844609345877849e+2106L, 1.50629562239998305958408533546634030e+2139L, 6.58342161306998815751663631350733541e+2172L, 9.74198999952210922892973281379349494e+2206L, 4.97552704772088889228035264268694867e+2241L, 8.94330619882842342247706838649878784e+2276L, 5.77072420156026800834371478648673792e+2312L, 1.36388225285320494361282201451878478e+2349L, 1.20508189950290985406298094670730466e+2386L, 4.06413362486490272506766749982212198e+2423L, 5.34308092015215251528601382597439463e+2461L, }, + }; + m_committed_refinements = static_cast(m_weights.size() - 1); + m_t_max = 8.88600744303961370002819023592353264e+00L; + if (m_max_refinements >= m_abscissas.size()) + { + m_abscissas.resize(m_max_refinements + 1); + m_weights.resize(m_max_refinements + 1); + } + else + { + m_max_refinements = m_abscissas.size() - 1; + } +} +#endif +#ifdef BOOST_HAS_FLOAT128 +template +void sinh_sinh_detail::init(const mpl::int_<4>&) +{ + m_abscissas = { + { 3.08828741797632286606397498241221385e+00Q, 1.48993184649209158013612709175719825e+02Q, 3.41228924788343710247727162226946917e+06Q, 2.06932576604261779073902718911207249e+18Q, 2.08700240760947556038306635808129743e+50Q, 2.01976616071790815078008252209994199e+137Q, 5.67213444764437168603513205349222232e+373Q, 3.06198394306784061113736467298565948e+1016Q, }, + { 9.13048762637669674838078869945948356e-01Q, 1.41578929466281159169215570833490092e+01Q, 6.70421551622327648231120321610008518e+03Q, 9.64172532715049941526010563818587232e+10Q, 2.50895076008577848514087028569888161e+30Q, 1.44726353571033714499079379626715686e+83Q, 3.75263401205045334128277886549019357e+226Q, 2.57846123932685341261715758547064293e+616Q, 1.25169402230987931584130068460134989e+1676Q, }, + { 4.07297690065758690150573950473604675e-01Q, 1.68206670702114874332932330092838356e+00Q, 6.15089798638672951539668935798437533e+00Q, 4.00396235192940022205839866641270319e+01Q, 7.92920024793102632052902471550336177e+02Q, 1.02984971333097958319686053784919407e+05Q, 3.03862310925243857437932941891415841e+08Q, 1.56544547436249486913719231527597560e+14Q, 4.04246509843021910355659662715183671e+23Q, 1.32170682742965817915034577861235933e+39Q, 4.99123178209955799774620736137366103e+64Q, 7.35294385035987596594501676609562008e+106Q, 2.45145417229813114714431821340237112e+176Q, 1.03030479197459267961759023741376327e+291Q, 9.88478945193556730604342445576518136e+479Q, 3.74498116076433060427765453233124618e+791Q, 1.90292390713026708045766158039087796e+1305Q, 1.72712488231809244721915842027236657e+2152Q, }, + { 1.98135272251478172615235718398538323e-01Q, 6.40155673500526017670785720911319502e-01Q, 1.24892869825397766254802900819319987e+00Q, 2.26608084094432123181038001270985037e+00Q, 4.29646269670232738148137717958679987e+00Q, 9.13029038709995569641811827045549232e+00Q, 2.31110765386427993316995139635470937e+01Q, 7.42770603432401243012214481410240677e+01Q, 3.26720920711525891697790966796945063e+02Q, 2.15948569431181871552028449867301828e+03Q, 2.41501526289641306037808670196966696e+04Q, 5.31819400275692915826269282626844010e+05Q, 2.80058685721704332307817790755562305e+07Q, 4.52406507979433877971977526863358905e+09Q, 3.08561257398067712180174566763237112e+12Q, 1.33882673301580747789133427708622972e+16Q, 6.25461717656234138147247754416652254e+20Q, 6.18209853581416475394863999233554548e+26Q, 3.07729364978845806686511011060697837e+34Q, 2.34895728937010430290698332595743680e+44Q, 1.14854319789946975790127332653552673e+57Q, 2.25530007001006986846545694404767295e+73Q, 1.87791950056919539419801461538703336e+94Q, 1.36747388793862427976781689638282239e+121Q, 4.24212177246407592514720294750151286e+155Q, 8.23473099012134325391170623028607158e+199Q, 6.06134211888406631001019923425936351e+256Q, 6.32519197436029413652766295658658263e+329Q, 3.61942292928205643948383705201563065e+423Q, 8.82464433126646489632943769283758364e+543Q, 3.35512488018651738642854323317447493e+698Q, 1.02411434678684822400261582079656500e+897Q, 7.40716670979374017620610271197945784e+1151Q, 1.30455147283895162835186634827124464e+1479Q, 2.02948723345659016173510134582487641e+1899Q, 6.99019443250345564805134094483457477e+2438Q, }, + { 9.83967894006732033862249554537934739e-02Q, 3.00605617659955035122465185012715805e-01Q, 5.19857978994938490000644785900799951e-01Q, 7.70362083298887700858583505753306721e-01Q, 1.07131136964131183026831222500748063e+00Q, 1.45056975808899844502126797109703210e+00Q, 1.95077854952036033400296546339240312e+00Q, 2.64003177369555146770080103921424631e+00Q, 3.63137237366741227331951933364601985e+00Q, 5.11991533090335057003526222543242753e+00Q, 7.45666098140488328926087222725984231e+00Q, 1.13022612688997262443461342203136009e+01Q, 1.79641069247277254966592982669611020e+01Q, 3.01781070460189822236726995418068214e+01Q, 5.40387580031237056709826662540066860e+01Q, 1.04107731447746954767459980678311109e+02Q, 2.18029520120262807725988909352359830e+02Q, 5.02155698625910164594509258651311707e+02Q, 1.28862131099822242045586488612672886e+03Q, 3.73921687080054832376623401153054636e+03Q, 1.24750729702019123239464970064060864e+04Q, 4.87639975322669212414357357061002683e+04Q, 2.28145658221913012154676335814417144e+05Q, 1.30877796006484301664040981436471740e+06Q, 9.46084663420966407698078336053299177e+06Q, 8.88883120363727962245667776649286152e+07Q, 1.12416882897434413447175648471746990e+09Q, 1.99127672953214446985509145592687205e+10Q, 5.16743469106098464964433058111638245e+11Q, 2.06721881420399088776240044545119948e+13Q, 1.35061503318410040604770421574064883e+15Q, 1.53854066283650818848773859278941683e+17Q, 3.29074729054035066052820102585085173e+19Q, 1.43729138188449881561963100200796631e+22Q, 1.40983244553034728562740932242727657e+25Q, 3.45913548027797144127990544649862792e+28Q, 2.39872058234095409213724024023509754e+32Q, 5.39880660461729295997086364918754801e+36Q, 4.61334000258062860955942270137550165e+41Q, 1.78768590966790245678294238939560892e+47Q, 3.84198437012433853552557148945213178e+53Q, 5.75279795570858370009870455748773410e+60Q, 7.77181203842728655149182334288749850e+68Q, 1.26967304420408162635234502672130357e+78Q, 3.49567677376573156773252547632961336e+88Q, 2.36251947497169244454694635287097330e+100Q, 6.00214389327365112307366927985587840e+113Q, 9.29071630346415553922675334522602561e+128Q, 1.51444223803384709004933819608345315e+146Q, 4.83290204410990250226310317342789471e+165Q, 6.09623012864338569215560315517200736e+187Q, 6.73889214277659359787962707085479315e+212Q, 1.60917893672694484647176104023542732e+241Q, 2.30724123468764987944582372785293125e+273Q, 6.32636439816284705814531208407868829e+309Q, 1.23274815890184688792156489065830724e+351Q, 7.55519778823055185901318833981345378e+397Q, 7.85730655106759111844772008129590573e+450Q, 9.36328400229780321059086898529742817e+510Q, 1.11332545426270395766246494603539938e+579Q, 1.53432069979625061288117473415446630e+656Q, 3.94622553764411915381186808199466267e+743Q, 4.41554290953392996069257034424439949e+842Q, 7.62037777854651720247173134262034607e+954Q, 1.15650294539918196985498353779647134e+1082Q, 1.50761161209705395018419027031866347e+1226Q, 3.03485088181756935935831645680716816e+1389Q, 3.38561189018537177501905611143498298e+1574Q, 1.64407968993192526511230001818530622e+1784Q, 6.63148743223242467667800355907041029e+2021Q, 1.15911606715877477373516464924821476e+2291Q, }, + { 4.91151003502902493004859475143401791e-02Q, 1.48013149674360733270682895306542340e-01Q, 2.48938813740683685679302156785267041e-01Q, 3.53325423692668437764413545980527251e-01Q, 4.62733556612235325924240102411897269e-01Q, 5.78912068164096306746325958452873475e-01Q, 7.03870253386062779930091622585657886e-01Q, 8.39965859144650568772715366992985538e-01Q, 9.90015066424437614655918533324528544e-01Q, 1.15743257014369913143250540144278056e+00Q, 1.34641275918536176274682426896283388e+00Q, 1.56216711390133555099139972670978661e+00Q, 1.81123885278232337973709010374295886e+00Q, 2.10192441900655030086142289233010995e+00Q, 2.44484388558419793403331221069953203e+00Q, 2.85372074663291502355627105440782037e+00Q, 3.34645891095535078662878051589922609e+00Q, 3.94664582105783838715586741669722603e+00Q, 4.68567310159667852940801410152900804e+00Q, 5.60576223090815117543658276010994803e+00Q, 6.76433233683057420359243855855175747e+00Q, 8.24038317537998522052755780107530197e+00Q, 1.01439435612985772971719760903679427e+01Q, 1.26302471433889247197016543768391586e+01Q, 1.59213039578034525767683420917805218e+01Q, 2.03392186192185718515730871595417962e+01Q, 2.63584644576063375239063530328964372e+01Q, 3.46892633322415240903243772580777748e+01Q, 4.64129146701972896279022543245987086e+01Q, 6.32055079389042420324823016892494512e+01Q, 8.77149726180890637368701810147417239e+01Q, 1.24209692624041149789460094889690346e+02Q, 1.79718634784512755704082576283626343e+02Q, 2.66081728332790018970422286819998479e+02Q, 4.03727302957571284052623415274320934e+02Q, 6.28811306654590870314912279241546792e+02Q, 1.00707983750749059403228978538051896e+03Q, 1.66156822918511428805794334715835287e+03Q, 2.82965144078658259775800007408843245e+03Q, 4.98438626658566913853681230634186899e+03Q, 9.10154692764781089316711776560031531e+03Q, 1.72689265547504972749956316341831556e+04Q, 3.41309957877860119013952370617496464e+04Q, 7.04566897705309280220071327032864763e+04Q, 1.52340421776127912819322591860346971e+05Q, 3.46047978289794741370038030428999755e+05Q, 8.28472420923318300234685980449600520e+05Q, 2.09759614660119394556971994518988102e+06Q, 5.63695079886127323617800640982012165e+06Q, 1.61407141085560724511058572293479599e+07Q, 4.94473067891506035994359725043521250e+07Q, 1.62781051682099135552732962095205423e+08Q, 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4.71463184672247661988055185868884361e+141Q, 6.76265778595971324046101731037889126e+150Q, 3.77863792934416463055998231270977695e+160Q, 8.97819133628477274891556066769979914e+170Q, 9.95914407403494739885505580434362313e+181Q, 5.69630798679205432723450802096176313e+193Q, 1.86743452325707784871951519925288377e+206Q, 3.92719945859412568046939678789391943e+219Q, 5.97262802266570558434581771213406728e+233Q, 7.46293577265068378653739984957785219e+248Q, 8.77620026161886682323882402399676732e+264Q, 1.12240955327637524990782988952974327e+282Q, 1.82091452946462555573948654780251641e+300Q, 4.41446070956615785669448222555093749e+319Q, 1.90397541953366074155779318767572880e+340Q, 1.75902281129585020130874703618203168e+362Q, 4.24169226543235148528312412985493520e+385Q, 3.29481560149200680874049429678412127e+410Q, 1.03135148368973063748422508803572335e+437Q, 1.65119340908368036904973428016231291e+465Q, 1.74266589416784110509614783944413581e+495Q, 1.58844581452329281145953920728196433e+527Q, 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5.34308092015215251528601382597439463e+2461Q, }, + }; + m_committed_refinements = static_cast(m_weights.size() - 1); + m_t_max = 8.88600744303961370002819023592353264e+00Q; + if (m_max_refinements >= m_abscissas.size()) + { + m_abscissas.resize(m_max_refinements + 1); + m_weights.resize(m_max_refinements + 1); + } + else + { + m_max_refinements = m_abscissas.size() - 1; + } +} +#endif + +}}}} +#endif diff --git a/include/boost/math/quadrature/detail/tanh_sinh_constants.hpp b/include/boost/math/quadrature/detail/tanh_sinh_constants.hpp new file mode 100644 index 000000000..6aa5844a8 --- /dev/null +++ b/include/boost/math/quadrature/detail/tanh_sinh_constants.hpp @@ -0,0 +1,228 @@ +// Copyright Nick Thompson, 2017 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_QUADRATURE_DETAIL_TANH_SINH_CONSTANTS_HPP +#define BOOST_MATH_QUADRATURE_DETAIL_TANH_SINH_CONSTANTS_HPP + +#include +#include + +namespace boost { + namespace math { + namespace quadrature { + namespace detail { + + template + inline Real abscissa_at_t(const Real& t) + { + using std::tanh; + using std::sinh; + return tanh(constants::half_pi()*sinh(t)); + } + template + inline Real weight_at_t(const Real& t) + { + using std::cosh; + using std::sinh; + Real cs = cosh(constants::half_pi() * sinh(t)); + return constants::half_pi() * cosh(t) / (cs * cs); + } + template + inline Real abscissa_complement_at_t(const Real& t) + { + using std::cosh; + using std::exp; + using std::sinh; + Real u2 = constants::half_pi() * sinh(t); + return 1 / (exp(u2) *cosh(u2)); + } + template + inline Real t_from_abscissa_complement(const Real& x) + { + using std::log; + using std::sqrt; + Real l = log(sqrt((2 - x) / x)); + return log((sqrt(4 * l * l + constants::pi() * constants::pi()) + 2 * l) / constants::pi()); + }; + + template + class tanh_sinh_detail_constants + { + static const int initializer_selector = + !std::numeric_limits::is_specialized || (std::numeric_limits::radix != 2) ? + 0 : + (std::numeric_limits::digits < 30) && (std::numeric_limits::max_exponent <= 128) ? + 1 : 0; + public: + tanh_sinh_detail_constants(std::size_t max_refinements, std::size_t initial_commit = 4) : m_max_refinements(max_refinements), m_committed_refinements(initial_commit) + { + typedef mpl::int_ tag_type; + init(tag_type()); + } + + void init(const mpl::int_<0>&) + { + using std::tanh; + using std::sinh; + using std::asinh; + using std::atanh; + using boost::math::constants::half_pi; + using boost::math::constants::pi; + using boost::math::constants::two_div_pi; + + m_inital_row_length = 7; + std::size_t first_complement = 0; + m_t_max = m_inital_row_length; + m_t_crossover = t_from_abscissa_complement(Real(0.5f)); + + m_abscissas.assign(m_max_refinements + 1, std::vector()); + m_weights.assign(m_max_refinements + 1, std::vector()); + m_first_complements.assign(m_max_refinements + 1, 0); + // + // First row is special: + // + Real h = m_t_max / m_inital_row_length; + + std::vector temp(m_inital_row_length + 1, Real(0)); + for (std::size_t i = 0; i < m_inital_row_length; ++i) + { + Real t = h * i; + if ((first_complement < 0) && (t < m_t_crossover)) + first_complement = i; + temp[i] = t < m_t_crossover ? abscissa_at_t(t) : -abscissa_complement_at_t(t); + } + temp[m_inital_row_length] = abscissa_complement_at_t(m_t_max); + m_abscissas[0].swap(temp); + m_first_complements[0] = first_complement; + temp.assign(m_inital_row_length + 1, Real(0)); + for (std::size_t i = 0; i < m_inital_row_length; ++i) + temp[i] = weight_at_t(Real(h * i)); + temp[m_inital_row_length] = weight_at_t(m_t_max); + m_weights[0].swap(temp); + + for (std::size_t row = 1; row <= m_committed_refinements; ++row) + { + h /= 2; + first_complement = 0; + + for (Real pos = h; pos < m_t_max; pos += 2 * h) + { + if (pos < m_t_crossover) + ++first_complement; + temp.push_back(pos < m_t_crossover ? abscissa_at_t(pos) : -abscissa_complement_at_t(pos)); + } + m_abscissas[row].swap(temp); + m_first_complements[row] = first_complement; + for (Real pos = h; pos < m_t_max; pos += 2 * h) + temp.push_back(weight_at_t(pos)); + m_weights[row].swap(temp); + } + } + void init(const mpl::int_<1>&) + { + m_inital_row_length = 4; + m_abscissas.reserve(m_max_refinements + 1); + m_weights.reserve(m_max_refinements + 1); + m_first_complements.reserve(m_max_refinements + 1); + m_abscissas = { + { 0.0f, -0.04863203593f, -2.252280754e-05f, -4.294161056e-14f, -1.167648898e-37f, }, + { -0.3257285078f, -0.002485143543f, -1.112433512e-08f, -5.378491591e-23f, }, + { 0.3772097382f, -0.1404309413f, -0.01295943949f, -0.0003117359716f, -7.952628853e-07f, -4.714355182e-11f, -5.415222824e-18f, -2.040300394e-29f, }, + { 0.1943570033f, -0.4608532946f, -0.219392561f, -0.08512073674f, -0.0260331318f, -0.005944493369f, -0.0009348035442f, -9.061530486e-05f, 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2.879701346e-09f, 2.012621022e-09f, 1.398441431e-09f, 9.659485186e-10f, 6.632086347e-10f, 4.52575761e-10f, 3.069270208e-10f, 2.068420354e-10f, 1.385028753e-10f, 9.214056423e-11f, 6.089338706e-11f, 3.997338952e-11f, 2.60619605e-11f, 1.687451934e-11f, 1.084916183e-11f, 6.925528015e-12f, 4.38886519e-12f, 2.760858767e-12f, 1.723764404e-12f, 1.068075044e-12f, 6.56694435e-13f, 4.00598538e-13f, 2.424296605e-13f, 1.455249916e-13f, 8.663812725e-14f, 5.114974901e-14f, 2.99421776e-14f, 1.737681695e-14f, 9.99642401e-15f, 5.699626666e-15f, 3.220432513e-15f, 1.802958964e-15f, 9.999957344e-16f, 5.493978397e-16f, 2.989420886e-16f, 1.610765424e-16f, 8.593209748e-17f, 4.538246827e-17f, 2.372253167e-17f, 1.227167167e-17f, 6.281229049e-18f, 3.180614714e-18f, 1.593049257e-18f, 7.890855159e-19f, 3.864733103e-19f, 1.87127733e-19f, 8.955739455e-20f, 4.235742852e-20f, 1.979436202e-20f, 9.138078558e-21f, 4.166641158e-21f, 1.876075055e-21f, 8.339901949e-22f, 3.659575236e-22f, 1.584785218e-22f, 6.771575694e-23f, 2.854281708e-23f, 1.186583858e-23f, 4.864069936e-24f, 1.965643419e-24f, 7.829165625e-25f, 3.072789229e-25f, 1.188107615e-25f, 4.524619749e-26f, 1.696710187e-26f, 6.263641003e-27f, 2.275790793e-27f, 8.136077716e-28f, 2.861306549e-28f, 9.896184197e-29f, 3.365200893e-29f, 1.124807055e-29f, 3.694460433e-30f, 1.192093301e-30f, 3.777757876e-31f, 1.175436379e-31f, 3.589879078e-32f, 1.075842686e-32f, 3.162835126e-33f, 9.118674189e-34f, 2.577393168e-34f, 7.139829504e-35f, 1.937828921e-35f, }, + }; + m_first_complements = { + 1, 0, 1, 1, 3, 5, 11, 22, + }; + + while (m_abscissas.size() <= m_committed_refinements) + extend_refinements(); + m_committed_refinements = m_abscissas.size() - 1; + + if (m_max_refinements >= m_abscissas.size()) + { + m_abscissas.resize(m_max_refinements + 1); + m_weights.resize(m_max_refinements + 1); + m_first_complements.resize(m_max_refinements + 1); + } + else + { + m_max_refinements = m_abscissas.size() - 1; + } + m_t_max = m_inital_row_length; + m_t_crossover = t_from_abscissa_complement(Real(0.5f)); + } + + void extend_refinements()const + { + ++m_committed_refinements; + std::size_t row = m_committed_refinements; + Real h = ldexp(Real(1), -static_cast(row)); + std::size_t first_complement = 0; + for (Real pos = h; pos < m_t_max; pos += 2 * h) + { + if (pos < m_t_crossover) + ++first_complement; + m_abscissas[row].push_back(pos < m_t_crossover ? abscissa_at_t(pos) : -abscissa_complement_at_t(pos)); + } + m_first_complements[row] = first_complement; + for (Real pos = h; pos < m_t_max; pos += 2 * h) + m_weights[row].push_back(weight_at_t(pos)); + } + + const std::vector& get_abscissa_row(std::size_t n)const + { + if (m_committed_refinements < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements >= n); + return m_abscissas[n]; + } + const std::vector& get_weight_row(std::size_t n)const + { + if (m_committed_refinements < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements >= n); + return m_weights[n]; + } + std::size_t get_first_complement_index(std::size_t n)const + { + if (m_committed_refinements < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements >= n); + return m_first_complements[n]; + } + + protected: + mutable std::vector> m_abscissas; + mutable std::vector> m_weights; + mutable std::vector m_first_complements; + std::size_t m_max_refinements, m_inital_row_length; + mutable std::size_t m_committed_refinements; + Real m_t_max, m_t_crossover; + }; + + + } + } + } +} + +#endif + + diff --git a/include/boost/math/quadrature/detail/tanh_sinh_detail.hpp b/include/boost/math/quadrature/detail/tanh_sinh_detail.hpp new file mode 100644 index 000000000..0b5e60860 --- /dev/null +++ b/include/boost/math/quadrature/detail/tanh_sinh_detail.hpp @@ -0,0 +1,705 @@ +// Copyright Nick Thompson, 2017 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_QUADRATURE_DETAIL_TANH_SINH_DETAIL_HPP +#define BOOST_MATH_QUADRATURE_DETAIL_TANH_SINH_DETAIL_HPP + +#include +#include +#include +#include +#include +#include +#include + +namespace boost{ namespace math{ namespace quadrature { namespace detail{ + + +// Returns the tanh-sinh quadrature of a function f over the open interval (-1, 1) + +template +class tanh_sinh_detail +{ + static const int initializer_selector = + !std::numeric_limits::is_specialized || (std::numeric_limits::radix != 2) ? + 0 : + (std::numeric_limits::digits < 30) && (std::numeric_limits::max_exponent <= 128) ? + 1 : + (std::numeric_limits::digits <= std::numeric_limits::digits) && (std::numeric_limits::max_exponent <= std::numeric_limits::max_exponent) ? + 2 : + (std::numeric_limits::digits <= std::numeric_limits::digits) && (std::numeric_limits::max_exponent <= 16384) ? + 3 : +#ifdef BOOST_HAS_FLOAT128 + (std::numeric_limits::digits <= 113) && (std::numeric_limits::max_exponent <= 16384) ? + 4 : +#endif + 0; +public: + tanh_sinh_detail(size_t max_refinements, const Real& min_complement) : m_max_refinements(max_refinements) + { + typedef mpl::int_ tag_type; + init(min_complement, tag_type()); + } + + template + Real integrate(const F f, Real* error, Real* L1, const char* function, Real left_min_complement, Real right_min_complement, Real tolerance, std::size_t* levels) const; + +private: + const std::vector& get_abscissa_row(std::size_t n)const + { +#ifndef BOOST_MATH_NO_ATOMIC_INT + if (m_committed_refinements.load() < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements.load() >= n); +#else + if (m_committed_refinements < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements >= n); +#endif + return m_abscissas[n]; + } + const std::vector& get_weight_row(std::size_t n)const + { +#ifndef BOOST_MATH_NO_ATOMIC_INT + if (m_committed_refinements.load() < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements.load() >= n); +#else + if (m_committed_refinements < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements >= n); +#endif + return m_weights[n]; + } + std::size_t get_first_complement_index(std::size_t n)const + { +#ifndef BOOST_MATH_NO_ATOMIC_INT + if (m_committed_refinements.load() < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements.load() >= n); +#else + if (m_committed_refinements < n) + extend_refinements(); + BOOST_ASSERT(m_committed_refinements >= n); +#endif + return m_first_complements[n]; + } + + void init(const Real& min_complement, const mpl::int_<0>&); + void init(const Real& min_complement, const mpl::int_<1>&); + void init(const Real& min_complement, const mpl::int_<2>&); + void init(const Real& min_complement, const mpl::int_<3>&); +#ifdef BOOST_HAS_FLOAT128 + void init(const Real& min_complement, const mpl::int_<4>&); +#endif + void prune_to_min_complement(const Real& m); + void extend_refinements()const + { +#ifndef BOOST_MATH_NO_ATOMIC_INT + boost::detail::lightweight_mutex::scoped_lock guard(m_mutex); +#endif + // + // Check some other thread hasn't got here after we read the atomic variable, but before we got here: + // +#ifndef BOOST_MATH_NO_ATOMIC_INT + if (m_committed_refinements.load() >= m_max_refinements) + return; +#else + if (m_committed_refinements >= m_max_refinements) + return; +#endif + + using std::ldexp; + using std::ceil; + ++m_committed_refinements; +#ifndef BOOST_MATH_NO_ATOMIC_INT + std::size_t row = m_committed_refinements.load(); +#else + std::size_t row = m_committed_refinements; +#endif + Real h = ldexp(Real(1), -static_cast(row)); + std::size_t first_complement = 0; + std::size_t n = boost::math::itrunc(ceil((m_t_max - h) / (2 * h))); + m_abscissas[row].reserve(n); + m_weights[row].reserve(n); + for (Real pos = h; pos < m_t_max; pos += 2 * h) + { + if (pos < m_t_crossover) + ++first_complement; + m_abscissas[row].push_back(pos < m_t_crossover ? abscissa_at_t(pos) : -abscissa_complement_at_t(pos)); + } + m_first_complements[row] = first_complement; + for (Real pos = h; pos < m_t_max; pos += 2 * h) + m_weights[row].push_back(weight_at_t(pos)); + } + + static inline Real abscissa_at_t(const Real& t) + { + using std::tanh; + using std::sinh; + return tanh(constants::half_pi()*sinh(t)); + } + static inline Real weight_at_t(const Real& t) + { + using std::cosh; + using std::sinh; + Real cs = cosh(constants::half_pi() * sinh(t)); + return constants::half_pi() * cosh(t) / (cs * cs); + } + static inline Real abscissa_complement_at_t(const Real& t) + { + using std::cosh; + using std::exp; + using std::sinh; + Real u2 = constants::half_pi() * sinh(t); + return 1 / (exp(u2) *cosh(u2)); + } + static inline Real t_from_abscissa_complement(const Real& x) + { + using std::log; + using std::sqrt; + Real l = log(sqrt((2 - x) / x)); + return log((sqrt(4 * l * l + constants::pi() * constants::pi()) + 2 * l) / constants::pi()); + }; + + + mutable std::vector> m_abscissas; + mutable std::vector> m_weights; + mutable std::vector m_first_complements; + std::size_t m_max_refinements, m_inital_row_length; +#ifndef BOOST_MATH_NO_ATOMIC_INT + mutable boost::math::detail::atomic_unsigned_type m_committed_refinements; + mutable boost::detail::lightweight_mutex m_mutex; +#else + mutable unsigned m_committed_refinements; +#endif + Real m_t_max, m_t_crossover; +}; + +template +template +Real tanh_sinh_detail::integrate(const F f, Real* error, Real* L1, const char* function, Real left_min_complement, Real right_min_complement, Real tolerance, std::size_t* levels) const +{ + using std::abs; + using std::fabs; + using std::floor; + using std::tanh; + using std::sinh; + using std::sqrt; + using boost::math::constants::half; + using boost::math::constants::half_pi; + // + // We maintain 4 integer values: + // max_left_position is the logical index of the abscissa value closest to the + // left endpoint of the range that we can call f(x_i) on without rounding error + // inside f(x_i) causing evaluation at the endpoint. + // max_left_index is the actual position in the current row that has a logical index + // no higher than max_left_position. Remember that since we only store odd numbered + // indexes in each row, this may actually be one position to the left of max_left_position + // in the case that is even. Then, if we only evaluate f(-x_i) for abscissa values + // i <= max_left_index we will never evaluate f(-x_i) at the left endpoint. + // max_right_position and max_right_index are defained similarly for the right boundary + // and are used to guard evaluation of f(x_i). + // + // max_left_position and max_right_position start off as the last element in row zero: + // + std::size_t max_left_position(m_abscissas[0].size() - 1); + std::size_t max_left_index, max_right_position(max_left_position), max_right_index; + // + // Decrement max_left_position and max_right_position until the complement + // of the abscissa value is greater than the smallest permitted (as specified + // by the function caller): + // + while (max_left_position && fabs(m_abscissas[0][max_left_position]) < left_min_complement) + --max_left_position; + while (max_right_position && fabs(m_abscissas[0][max_right_position]) < right_min_complement) + --max_right_position; + // + // Assumption: left_min_complement/right_min_complement are sufficiently small that we only + // ever decrement through the stored values that are complements (the negative ones), and + // never ever hit the true abscissa values (positive stored values). + // + BOOST_ASSERT(m_abscissas[0][max_left_position] < 0); + BOOST_ASSERT(m_abscissas[0][max_right_position] < 0); + + + Real h = m_t_max / m_inital_row_length; + Real I0 = half_pi()*f(0, 1); + Real L1_I0 = abs(I0); + for(size_t i = 1; i < m_abscissas[0].size(); ++i) + { + if ((i > max_right_position) && (i > max_left_position)) + break; + Real x = m_abscissas[0][i]; + Real xc = x; + Real w = m_weights[0][i]; + if ((boost::math::signbit)(x)) + { + // We have stored x - 1: + x = 1 + xc; + } + else + xc = x - 1; + Real yp, ym; + yp = i <= max_right_position ? f(x, -xc) : 0; + ym = i <= max_left_position ? f(-x, xc) : 0; + I0 += (yp + ym)*w; + L1_I0 += (abs(yp) + abs(ym))*w; + } + // + // We have: + // k = current row. + // I0 = last integral value. + // I1 = current integral value. + // L1_I0 and L1_I1 are the absolute integral values. + // + size_t k = 1; + Real I1 = I0; + Real L1_I1 = L1_I0; + Real err = 0; + // + // thrash_count is a heuristic - it counts how many time the error has actually increased + // rather than descreased, if this gets too high we abort... + // + unsigned thrash_count = 0; + + while (k < 4 || (k < m_weights.size() && k < m_max_refinements) ) + { + I0 = I1; + L1_I0 = L1_I1; + + I1 = half()*I0; + L1_I1 = half()*L1_I0; + h *= half(); + Real sum = 0; + Real absum = 0; + auto const& abscissa_row = this->get_abscissa_row(k); + auto const& weight_row = this->get_weight_row(k); + std::size_t first_complement_index = this->get_first_complement_index(k); + // + // At the start of each new row we need to update the max left/right indexes + // at which we can evaluate f(x_i). The new logical position is simply twice + // the old value. The new max index is one position to the left of the new + // logical value (remmember each row contains only odd numbered positions). + // Then we have to make a single check, to see if one position to the right + // is also in bounds (this is the new abscissa value in this row which is + // known to be in between a value known to be in bounds, and one known to be + // not in bounds). + // Thus, we filter which abscissa values generate a call to f(x_i), with a single + // floating point comparison per loop. Everything else is integer logic. + // + max_left_index = max_left_position - 1; + max_left_position *= 2; + max_right_index = max_right_position - 1; + max_right_position *= 2; + if ((abscissa_row.size() > max_left_index + 1) && (fabs(abscissa_row[max_left_index + 1]) > left_min_complement)) + { + ++max_left_position; + ++max_left_index; + } + if ((abscissa_row.size() > max_right_index + 1) && (fabs(abscissa_row[max_right_index + 1]) > right_min_complement)) + { + ++max_right_position; + ++max_right_index; + } + + for(size_t j = 0; j < weight_row.size(); ++j) + { + // If both left and right abscissa values are out of bounds at this step + // we can just stop this loop right now: + if ((j > max_left_index) && (j > max_right_index)) + break; + Real x = abscissa_row[j]; + Real xc = x; + Real w = weight_row[j]; + if (j >= first_complement_index) + { + // We have stored x - 1: + BOOST_ASSERT(x < 0); + x = 1 + xc; + } + else + { + BOOST_ASSERT(x >= 0); + xc = x - 1; + } + + Real yp = j > max_right_index ? 0 : f(x, -xc); + Real ym = j > max_left_index ? 0 : f(-x, xc); + Real term = (yp + ym)*w; + sum += term; + + // A question arises as to how accurately we actually need to estimate the L1 integral. + // For simple integrands, computing the L1 norm makes the integration 20% slower, + // but for more complicated integrands, this calculation is not noticeable. + Real abterm = (abs(yp) + abs(ym))*w; + absum += abterm; + } + + I1 += sum*h; + L1_I1 += absum*h; + ++k; + Real last_err = err; + err = abs(I0 - I1); + // std::cout << "Estimate: " << I1 << " Error estimate at level " << k << " = " << err << std::endl; + + if (!(boost::math::isfinite)(I1)) + { + return policies::raise_evaluation_error(function, "The tanh_sinh quadrature evaluated your function at a singular point at got %1%. Please narrow the bounds of integration or check your function for singularities.", I1, Policy()); + } + // + // If the error is increasing, and we're past level 4, something bad is very likely happening: + // + if ((err > last_err) && (k > 4) && (++thrash_count > 1)) + { + // We could raise an evaluation_error, but since we likely have some sort of result, just return the last one + // (ie before the error started going up) + I1 = I0; + L1_I1 = L1_I0; + --k; + err = last_err; + break; + } + // + // Termination condition: + // No more levels are considered once the error is less than the specified tolerance. + // Note however, that we always go down at least 4 levels, otherwise we risk missing + // features of interest in f() - imagine for example a function which flatlines, except + // for a very small "spike". An example would be the incomplete beta integral with large + // parameters. We could keep hunting until we find something, but that would handicap + // integrals which really are zero.... so a compromise then! + // + if (err <= tolerance*L1_I1) + { + break; + } + + } + if (error) + { + *error = err; + } + + if (L1) + { + *L1 = L1_I1; + } + + if (levels) + { + *levels = k; + } + + return I1; +} + +template +void tanh_sinh_detail::init(const Real& min_complement, const mpl::int_<0>&) +{ + using std::tanh; + using std::sinh; + using std::asinh; + using std::atanh; + using std::ceil; + using boost::math::constants::half_pi; + using boost::math::constants::pi; + using boost::math::constants::two_div_pi; + + m_committed_refinements = 4; + // + // Initial row length is one step to the right of the abscissa value + // that would go to the full extent of the requested range, we need this + // to ensure full precision as otherwise we chop off quite a chunk of the + // range in *subsequent* rows. + // + m_inital_row_length = itrunc(ceil(t_from_abscissa_complement(min_complement))); + std::size_t first_complement = 0; + m_t_max = m_inital_row_length; + m_t_crossover = t_from_abscissa_complement(Real(0.5f)); + + m_abscissas.assign(m_max_refinements + 1, std::vector()); + m_weights.assign(m_max_refinements + 1, std::vector()); + m_first_complements.assign(m_max_refinements + 1, 0); + // + // First row is special: + // + Real h = m_t_max / m_inital_row_length; + + std::vector temp(m_inital_row_length + 1, Real(0)); + for (std::size_t i = 0; i < m_inital_row_length; ++i) + { + Real t = h * i; + if (t < m_t_crossover) + ++first_complement; + temp[i] = t < m_t_crossover ? abscissa_at_t(t) : -abscissa_complement_at_t(t); + } + temp[m_inital_row_length] = -abscissa_complement_at_t(m_t_max); + m_abscissas[0].swap(temp); + m_first_complements[0] = first_complement; + temp.assign(m_inital_row_length + 1, Real(0)); + for (std::size_t i = 0; i < m_inital_row_length; ++i) + temp[i] = weight_at_t(Real(h * i)); + temp[m_inital_row_length] = weight_at_t(m_t_max); + m_weights[0].swap(temp); + +#ifndef BOOST_MATH_NO_ATOMIC_INT + for (std::size_t row = 1; row <= m_committed_refinements.load(); ++row) +#else + for (std::size_t row = 1; row <= m_committed_refinements; ++row) +#endif + { + h /= 2; + first_complement = 0; + + for (Real pos = h; pos < m_t_max; pos += 2 * h) + { + if (pos < m_t_crossover) + ++first_complement; + temp.push_back(pos < m_t_crossover ? abscissa_at_t(pos) : -abscissa_complement_at_t(pos)); + } + m_abscissas[row].swap(temp); + m_first_complements[row] = first_complement; + for (Real pos = h; pos < m_t_max; pos += 2 * h) + temp.push_back(weight_at_t(pos)); + m_weights[row].swap(temp); + } +} + +#ifdef __GNUC__ +// Selective warning disabling via: +// #pragma GCC diagnostic ignored "-Wliteral-range" +// #pragma GCC diagnostic ignored "-Woverflow" +// Seems not to work, so we're left with this: +#pragma GCC system_header +#endif + +template +void tanh_sinh_detail::init(const Real& min_complement, const mpl::int_<1>&) +{ + m_inital_row_length = 4; + m_abscissas.reserve(m_max_refinements + 1); + m_weights.reserve(m_max_refinements + 1); + m_first_complements.reserve(m_max_refinements + 1); + m_abscissas = { + { 0.0f, -0.04863203593f, -2.252280754e-05f, -4.294161056e-14f, -1.167648898e-37f, }, + { -0.3257285078f, -0.002485143543f, -1.112433512e-08f, -5.378491591e-23f, }, + { 0.3772097382f, -0.1404309413f, -0.01295943949f, -0.0003117359716f, -7.952628853e-07f, -4.714355182e-11f, -5.415222824e-18f, -2.040300394e-29f, }, + { 0.1943570033f, -0.4608532946f, -0.219392561f, -0.08512073674f, -0.0260331318f, -0.005944493369f, -0.0009348035442f, -9.061530486e-05f, -4.683958779e-06f, -1.072183876e-07f, -8.572949078e-10f, -1.767834693e-12f, -6.374878495e-16f, -2.442937279e-20f, -5.251546473e-26f, -2.789467162e-33f, }, + { 0.09792388529f, 0.2878799327f, 0.4612535439f, -0.3897263425f, -0.2689819652f, -0.1766829945f, -0.1101085972f, -0.06483914248f, -0.03588783578f, 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2.760858767e-12f, 1.723764404e-12f, 1.068075044e-12f, 6.56694435e-13f, 4.00598538e-13f, 2.424296605e-13f, 1.455249916e-13f, 8.663812725e-14f, 5.114974901e-14f, 2.99421776e-14f, 1.737681695e-14f, 9.99642401e-15f, 5.699626666e-15f, 3.220432513e-15f, 1.802958964e-15f, 9.999957344e-16f, 5.493978397e-16f, 2.989420886e-16f, 1.610765424e-16f, 8.593209748e-17f, 4.538246827e-17f, 2.372253167e-17f, 1.227167167e-17f, 6.281229049e-18f, 3.180614714e-18f, 1.593049257e-18f, 7.890855159e-19f, 3.864733103e-19f, 1.87127733e-19f, 8.955739455e-20f, 4.235742852e-20f, 1.979436202e-20f, 9.138078558e-21f, 4.166641158e-21f, 1.876075055e-21f, 8.339901949e-22f, 3.659575236e-22f, 1.584785218e-22f, 6.771575694e-23f, 2.854281708e-23f, 1.186583858e-23f, 4.864069936e-24f, 1.965643419e-24f, 7.829165625e-25f, 3.072789229e-25f, 1.188107615e-25f, 4.524619749e-26f, 1.696710187e-26f, 6.263641003e-27f, 2.275790793e-27f, 8.136077716e-28f, 2.861306549e-28f, 9.896184197e-29f, 3.365200893e-29f, 1.124807055e-29f, 3.694460433e-30f, 1.192093301e-30f, 3.777757876e-31f, 1.175436379e-31f, 3.589879078e-32f, 1.075842686e-32f, 3.162835126e-33f, 9.118674189e-34f, 2.577393168e-34f, 7.139829504e-35f, 1.937828921e-35f, }, + }; + m_first_complements = { + 1, 0, 1, 1, 3, 5, 11, 22, + }; + + m_committed_refinements = static_cast(m_abscissas.size() - 1); + + if (m_max_refinements >= m_abscissas.size()) + { + m_abscissas.resize(m_max_refinements + 1); + m_weights.resize(m_max_refinements + 1); + m_first_complements.resize(m_max_refinements + 1); + } + else + { + m_max_refinements = m_abscissas.size() - 1; + } + m_t_max = static_cast(m_inital_row_length); + m_t_crossover = t_from_abscissa_complement(Real(0.5f)); + + prune_to_min_complement(min_complement); + +} + +template +void tanh_sinh_detail::init(const Real& min_complement, const mpl::int_<2>&) +{ + m_inital_row_length = 5; + m_abscissas.reserve(m_max_refinements + 1); + m_weights.reserve(m_max_refinements + 1); + m_first_complements.reserve(m_max_refinements + 1); + m_abscissas = { + { 0, -0.0486320359272530543, -2.25228075384071351e-05, -4.29416105587824078e-14, -1.16764889750986093e-37, -1.14795299162938991e-101, -1.22565381365848647e-275, }, + { -0.325728507751564174, -0.00248514354277561317, -1.1124335118015332e-08, -5.37849159139368877e-23, -7.9430213192221161e-62, -2.38712281858192662e-167, }, + { 0.377209738164034174, -0.140430941310103365, -0.0129594394926231083, -0.000311735971646790949, -7.95262885287335534e-07, -4.71435518232225751e-11, -5.41522282380723085e-18, -2.04030039435249433e-29, -3.05969013536445003e-48, -2.86219841925875116e-79, -2.00703306915332264e-130, -9.24799327395281672e-215, }, + { 0.194357003324935432, -0.460853294612032231, -0.219392561016799701, -0.0851207367354253891, -0.0260331318043225514, -0.00594449336859785671, -0.000934803544214153575, -9.06153048560001614e-05, -4.68395877947156992e-06, -1.07218387581618094e-07, -8.57294907821677329e-10, -1.76783469283872037e-12, -6.37487849504443965e-16, -2.44293727908521732e-20, -5.25154647301957486e-26, -2.78946716228941774e-33, -1.2781108980938045e-42, -1.3082723368531808e-54, -5.27997812261025438e-70, -9.06191040541810149e-90, -3.79031527880246479e-115, -9.83017699648704421e-148, -1.41780355464723683e-189, -2.67381847626328947e-243, }, + { 0.0979238852878323333, 0.287879932742715915, 0.461253543939585704, -0.389726342499361055, -0.268981965207438489, -0.17668299449359763, -0.110108597215739802, -0.0648391424780153168, -0.0358878357764527081, -0.01854517332266483, -0.00887300755830119777, -0.00389133456249145746, -0.00154579123230226249, -0.000548565564725394158, -0.00017117792712505834, -4.61289943720392527e-05, -1.05179851814963885e-05, -1.98285940456792063e-06, -3.01105847388779655e-07, -3.57609190846577144e-08, -3.2128009016957605e-09, -2.10267137764114881e-10, -9.606066478508465e-12, -2.91902664101751393e-13, -5.58611663883822752e-15, -6.32820713468316824e-17, 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1); + } + else + { + m_max_refinements = m_abscissas.size() - 1; + } + m_t_max = static_cast(m_inital_row_length); + m_t_crossover = t_from_abscissa_complement(Real(0.5)); + + prune_to_min_complement(min_complement); +} + +template +void tanh_sinh_detail::init(const Real& min_complement, const mpl::int_<3>&) +{ + m_inital_row_length = 9; + m_abscissas.reserve(m_max_refinements + 1); + m_weights.reserve(m_max_refinements + 1); + m_first_complements.reserve(m_max_refinements + 1); + m_abscissas = { + { 0.0L, -0.048632035927253054272944637095360333L, -2.2522807538407135100363691311150714e-05L, -4.2941610558782407776948098746194498e-14L, -1.1676488975098609327433648963732896e-37L, -1.1479529916293899121630752460973831e-101L, -1.2256538136584864685624805213855318e-275L, -1.5540928823936461440049038613030612e-748L, -5.3329091650553293055512604419528931e-2034L, -2.9720916290005160834428657415764727e-5528L, }, + { -0.32572850775156417391957990936794786L, 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2.4283486680552345908529800038916585e-4990L, 5.6028531999416699789095738832933157e-5069L, 7.4653403674710397422097751397913229e-5149L, 5.4919841992020489039553213694177019e-5230L, 2.1312688707736268785590779947753119e-5312L, 4.1653791598980828217930742787672032e-5396L, 3.9114639992058990826656399856975019e-5481L, }, + }; + m_first_complements = { + 1, 0, 1, 1, 3, 5, 11, 22, + }; + + m_committed_refinements = static_cast(m_abscissas.size() - 1); + + if (m_max_refinements >= m_abscissas.size()) + { + m_abscissas.resize(m_max_refinements + 1); + m_weights.resize(m_max_refinements + 1); + m_first_complements.resize(m_max_refinements + 1); + } + else + { + m_max_refinements = m_abscissas.size() - 1; + } + m_t_max = static_cast(m_inital_row_length); + m_t_crossover = t_from_abscissa_complement(Real(0.5)); + + prune_to_min_complement(min_complement); +} + +#ifdef BOOST_HAS_FLOAT128 + +template +void tanh_sinh_detail::init(const Real& min_complement, const mpl::int_<4>&) +{ + m_inital_row_length = 9; + m_abscissas.reserve(m_max_refinements + 1); + m_weights.reserve(m_max_refinements + 1); + m_first_complements.reserve(m_max_refinements + 1); + m_abscissas = { + { 0.0Q, -0.048632035927253054272944637095360333Q, -2.2522807538407135100363691311150714e-05Q, -4.2941610558782407776948098746194498e-14Q, -1.1676488975098609327433648963732896e-37Q, -1.1479529916293899121630752460973831e-101Q, -1.2256538136584864685624805213855318e-275Q, -1.5540928823936461440049038613030612e-748Q, -5.3329091650553293055512604419528931e-2034Q, -2.9720916290005160834428657415764727e-5528Q, }, + { -0.32572850775156417391957990936794786Q, -0.0024851435427756131672825807611796319Q, -1.1124335118015331984966677262985097e-08Q, -5.3784915913936887745567608333252923e-23Q, -7.9430213192221161033965793076252082e-62Q, -2.3871228185819266205711383850686192e-167Q, -3.5505659459518255844385870776118831e-454Q, -7.5205359182648710139026226942624688e-1234Q, 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1.7055766393620807246392140051588691e-3825Q, 8.7186526945824113614692955827617228e-3886Q, 5.0057478847913665022916271652366648e-3947Q, 3.118721903551705673676508466953291e-4009Q, 2.0360271187838855169643637345546437e-4072Q, 1.344193072571113944443469925533764e-4136Q, 8.6564521926545774914724471102499854e-4202Q, 5.2420435234912004640647252129478902e-4268Q, 2.8759098307799421718470860096954611e-4335Q, 1.3763877452472741048491095385711433e-4403Q, 5.5298776937738818735087266649244396e-4473Q, 1.7937216962986389293716315095996859e-4543Q, 4.5149004338871753808860898565339903e-4615Q, 8.4705695199542943533046216376760939e-4688Q, 1.1370794062105632021349671352107651e-4761Q, 1.0477206804639390072354388690482908e-4836Q, 6.3526587059143165049479765478742544e-4913Q, 2.4283486680552345908529800038916585e-4990Q, 5.6028531999416699789095738832933157e-5069Q, 7.4653403674710397422097751397913229e-5149Q, 5.4919841992020489039553213694177019e-5230Q, 2.1312688707736268785590779947753119e-5312Q, 4.1653791598980828217930742787672032e-5396Q, 3.9114639992058990826656399856975019e-5481Q, }, + }; + m_first_complements = { + 1, 0, 1, 1, 3, 5, 11, 22, + }; + + m_committed_refinements = static_cast(m_abscissas.size() - 1); + + if (m_max_refinements >= m_abscissas.size()) + { + m_abscissas.resize(m_max_refinements + 1); + m_weights.resize(m_max_refinements + 1); + m_first_complements.resize(m_max_refinements + 1); + } + else + { + m_max_refinements = m_abscissas.size() - 1; + } + m_t_max = static_cast(m_inital_row_length); + m_t_crossover = t_from_abscissa_complement(Real(0.5)); + + prune_to_min_complement(min_complement); +} + +#endif // BOOST_HAS_FLOAT128 + +template +void tanh_sinh_detail::prune_to_min_complement(const Real& m) +{ + // + // If our tables were constructed from pre-computed data, then they will have more values stored than we can ever use, + // and although the table size at this stage won't be too large, if we calculate down to m_max_levels then they will + // grow by a huge amount - doubling in size at each step - so lets prune them down, removing values which will never + // be used: + // + if (m > tools::min_value() * 4) + { + for (unsigned row = 0; (row < m_abscissas.size()) && m_abscissas[row].size(); ++row) + { + typename std::vector::iterator pos = std::lower_bound(m_abscissas[row].begin(), m_abscissas[row].end(), m, [](const Real& a, const Real& b) { using std::fabs; return fabs(a) > fabs(b); }); + if (pos != m_abscissas[row].end()) + { + m_abscissas[row].erase(pos, m_abscissas[row].end()); + m_weights[row].erase(m_weights[row].begin() + m_abscissas[row].size(), m_weights[row].end()); + } + } + } +} + +}}}} // namespaces + +#endif diff --git a/include/boost/math/quadrature/exp_sinh.hpp b/include/boost/math/quadrature/exp_sinh.hpp new file mode 100644 index 000000000..989abbed7 --- /dev/null +++ b/include/boost/math/quadrature/exp_sinh.hpp @@ -0,0 +1,96 @@ +// Copyright Nick Thompson, 2017 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +/* + * This class performs exp-sinh quadrature on half infinite intervals. + * + * References: + * + * 1) Tanaka, Ken’ichiro, et al. "Function classes for double exponential integration formulas." Numerische Mathematik 111.4 (2009): 631-655. + */ + +#ifndef BOOST_MATH_QUADRATURE_EXP_SINH_HPP +#define BOOST_MATH_QUADRATURE_EXP_SINH_HPP + +#include +#include +#include +#include + +namespace boost{ namespace math{ namespace quadrature { + +template > +class exp_sinh +{ +public: + exp_sinh(size_t max_refinements = 9) + : m_imp(std::make_shared>(max_refinements)) {} + + template + Real integrate(const F& f, Real a, Real b, Real tol = boost::math::tools::root_epsilon(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) const; + template + Real integrate(const F& f, Real tol = boost::math::tools::root_epsilon(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) const; + +private: + std::shared_ptr> m_imp; +}; + +template +template +Real exp_sinh::integrate(const F& f, Real a, Real b, Real tolerance, Real* error, Real* L1, std::size_t* levels) const +{ + using std::abs; + using boost::math::constants::half; + using boost::math::quadrature::detail::exp_sinh_detail; + + static const char* function = "boost::math::quadrature::exp_sinh<%1%>::integrate"; + + // Neither limit may be a NaN: + if((boost::math::isnan)(a) || (boost::math::isnan)(b)) + return policies::raise_domain_error(function, "NaN supplied as one limit of integration - sorry I don't know what to do", a, Policy()); + // Right limit is infinite: + if ((boost::math::isfinite)(a) && (b >= boost::math::tools::max_value())) + { + // If a = 0, don't use an additional level of indirection: + if (a == (Real) 0) + { + return m_imp->integrate(f, error, L1, function, tolerance, levels); + } + const auto u = [&](Real t)->Real { return f(t + a); }; + return m_imp->integrate(u, error, L1, function, tolerance, levels); + } + + if ((boost::math::isfinite)(b) && a <= -boost::math::tools::max_value()) + { + const auto u = [&](Real t)->Real { return f(b-t);}; + return m_imp->integrate(u, error, L1, function, tolerance, levels); + } + + // Infinite limits: + if ((a <= -boost::math::tools::max_value()) && (b >= boost::math::tools::max_value())) + { + return policies::raise_domain_error(function, "Use sinh_sinh quadrature for integration over the whole real line; exp_sinh is for half infinite integrals.", a, Policy()); + } + // If we get to here then both ends must necessarily be finite: + return policies::raise_domain_error(function, "Use tanh_sinh quadrature for integration over finite domains; exp_sinh is for half infinite integrals.", a, Policy()); +} + +template +template +Real exp_sinh::integrate(const F& f, Real tolerance, Real* error, Real* L1, std::size_t* levels) const +{ + using std::abs; + using boost::math::constants::half; + using boost::math::quadrature::detail::exp_sinh_detail; + + static const char* function = "boost::math::quadrature::exp_sinh<%1%>::integrate"; + + return m_imp->integrate(f, error, L1, function, tolerance, levels); +} + + +}}} +#endif diff --git a/include/boost/math/quadrature/sinh_sinh.hpp b/include/boost/math/quadrature/sinh_sinh.hpp new file mode 100644 index 000000000..e596f4946 --- /dev/null +++ b/include/boost/math/quadrature/sinh_sinh.hpp @@ -0,0 +1,43 @@ +// Copyright Nick Thompson, 2017 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +/* + * This class performs sinh-sinh quadrature over the entire real line. + * + * References: + * + * 1) Tanaka, Ken’ichiro, et al. "Function classes for double exponential integration formulas." Numerische Mathematik 111.4 (2009): 631-655. + */ + +#ifndef BOOST_MATH_QUADRATURE_SINH_SINH_HPP +#define BOOST_MATH_QUADRATURE_SINH_SINH_HPP + +#include +#include +#include +#include + +namespace boost{ namespace math{ namespace quadrature { + +template > +class sinh_sinh +{ +public: + sinh_sinh(size_t max_refinements = 9) + : m_imp(std::make_shared >(max_refinements)) {} + + template + Real integrate(const F f, Real tol = boost::math::tools::root_epsilon(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) const + { + return m_imp->integrate(f, tol, error, L1, levels); + } + +private: + std::shared_ptr> m_imp; +}; + +}}} +#endif diff --git a/include/boost/math/quadrature/tanh_sinh.hpp b/include/boost/math/quadrature/tanh_sinh.hpp new file mode 100644 index 000000000..df550eeb6 --- /dev/null +++ b/include/boost/math/quadrature/tanh_sinh.hpp @@ -0,0 +1,245 @@ +// Copyright Nick Thompson, 2017 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +/* + * This class performs tanh-sinh quadrature on the real line. + * Tanh-sinh quadrature is exponentially convergent for integrands in Hardy spaces, + * (see https://en.wikipedia.org/wiki/Hardy_space for a formal definition), and is optimal for a random function from that class. + * + * The tanh-sinh quadrature is one of a class of so called "double exponential quadratures"-there is a large family of them, + * but this one seems to be the most commonly used. + * + * As always, there are caveats: For instance, if the function you want to integrate is not holomorphic on the unit disk, + * then the rapid convergence will be spoiled. In this case, a more appropriate quadrature is (say) Romberg, which does not + * require the function to be holomorphic, only differentiable up to some order. + * + * In addition, if you are integrating a periodic function over a period, the trapezoidal rule is better. + * + * References: + * + * 1) Mori, Masatake. "Quadrature formulas obtained by variable transformation and the DE-rule." Journal of Computational and Applied Mathematics 12 (1985): 119-130. + * 2) Bailey, David H., Karthik Jeyabalan, and Xiaoye S. Li. "A comparison of three high-precision quadrature schemes." Experimental Mathematics 14.3 (2005): 317-329. + * 3) Press, William H., et al. "Numerical recipes third edition: the art of scientific computing." Cambridge University Press 32 (2007): 10013-2473. + * + */ + +#ifndef BOOST_MATH_QUADRATURE_TANH_SINH_HPP +#define BOOST_MATH_QUADRATURE_TANH_SINH_HPP + +#include +#include +#include +#include + +namespace boost{ namespace math{ namespace quadrature { + +template > +class tanh_sinh +{ +public: + tanh_sinh(size_t max_refinements = 15, const Real& min_complement = tools::min_value() * 4) + : m_imp(std::make_shared>(max_refinements, min_complement)) {} + + template + auto integrate(const F f, Real a, Real b, Real tolerance = tools::root_epsilon(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) ->decltype(Real(std::declval()(std::declval()))) const; + template + auto integrate(const F f, Real a, Real b, Real tolerance = tools::root_epsilon(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) ->decltype(Real(std::declval()(std::declval(), std::declval()))) const; + + template + auto integrate(const F f, Real tolerance = tools::root_epsilon(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) ->decltype(Real(std::declval()(std::declval()))) const; + template + auto integrate(const F f, Real tolerance = tools::root_epsilon(), Real* error = nullptr, Real* L1 = nullptr, std::size_t* levels = nullptr) ->decltype(Real(std::declval()(std::declval(), std::declval()))) const; + +private: + std::shared_ptr> m_imp; +}; + +template +template +auto tanh_sinh::integrate(const F f, Real a, Real b, Real tolerance, Real* error, Real* L1, std::size_t* levels) ->decltype(Real(std::declval()(std::declval()))) const +{ + BOOST_MATH_STD_USING + using boost::math::constants::half; + using boost::math::quadrature::detail::tanh_sinh_detail; + + static const char* function = "tanh_sinh<%1%>::integrate"; + + if (!(boost::math::isnan)(a) && !(boost::math::isnan)(b)) + { + + // Infinite limits: + if ((a <= -tools::max_value()) && (b >= tools::max_value())) + { + auto u = [&](const Real& t, const Real& tc)->Real + { + Real t_sq = t*t; + Real inv; + if (t > 0.5f) + inv = 1 / ((2 - tc) * tc); + else if(t < -0.5) + inv = 1 / ((2 + tc) * -tc); + else + inv = 1 / (1 - t_sq); + return f(t*inv)*(1 + t_sq)*inv*inv; + }; + Real limit = sqrt(tools::min_value()) * 4; + return m_imp->integrate(u, error, L1, function, limit, limit, tolerance, levels); + } + + // Right limit is infinite: + if ((boost::math::isfinite)(a) && (b >= tools::max_value())) + { + auto u = [&](const Real& t, const Real& tc)->Real + { + Real z, arg; + if (t > -0.5f) + z = 1 / (t + 1); + else + z = -1 / tc; + if (t < 0.5) + arg = 2 * z + a - 1; + else + arg = a + tc / (2 - tc); + return f(arg)*z*z; + }; + Real left_limit = sqrt(tools::min_value()) * 4; + Real Q = 2 * m_imp->integrate(u, error, L1, function, left_limit, tools::min_value(), tolerance, levels); + if (L1) + { + *L1 *= 2; + } + + return Q; + } + + if ((boost::math::isfinite)(b) && (a <= -tools::max_value())) + { + auto v = [&](const Real& t, const Real& tc)->Real + { + Real z; + if (t > -0.5) + z = 1 / (t + 1); + else + z = -1 / tc; + Real arg; + if (t < 0.5) + arg = 2 * z - 1; + else + arg = tc / (2 - tc); + return f(b - arg) * z * z; + }; + + Real left_limit = sqrt(tools::min_value()) * 4; + Real Q = 2 * m_imp->integrate(v, error, L1, function, left_limit, tools::min_value(), tolerance, levels); + if (L1) + { + *L1 *= 2; + } + return Q; + } + + if ((boost::math::isfinite)(a) && (boost::math::isfinite)(b)) + { + if (b <= a) + { + return policies::raise_domain_error(function, "Arguments to integrate are in wrong order; integration over [a,b] must have b > a.", a, Policy()); + } + Real avg = (a + b)*half(); + Real diff = (b - a)*half(); + Real avg_over_diff_m1 = a / diff; + Real avg_over_diff_p1 = b / diff; + bool have_small_left = fabs(a) < 0.5f; + bool have_small_right = fabs(b) < 0.5f; + Real left_min_complement = float_next(avg_over_diff_m1) - avg_over_diff_m1; + if (left_min_complement < tools::min_value()) + left_min_complement = tools::min_value(); + Real right_min_complement = avg_over_diff_p1 - float_prior(avg_over_diff_p1); + if (right_min_complement < tools::min_value()) + right_min_complement = tools::min_value(); + auto u = [&](Real z, Real zc)->Real + { + if (have_small_left && (z < -0.5)) + return f(diff * (avg_over_diff_m1 - zc)); + if (have_small_right && (z > 0.5)) + return f(diff * (avg_over_diff_p1 - zc)); + Real position = avg + diff*z; + BOOST_ASSERT(position != a); + BOOST_ASSERT(position != b); + return f(position); + }; + Real Q = diff*m_imp->integrate(u, error, L1, function, left_min_complement, right_min_complement, tolerance, levels); + + if (L1) + { + *L1 *= diff; + } + return Q; + } + } + return policies::raise_domain_error(function, "The domain of integration is not sensible; please check the bounds.", a, Policy()); +} + +template +template +auto tanh_sinh::integrate(const F f, Real a, Real b, Real tolerance, Real* error, Real* L1, std::size_t* levels) ->decltype(Real(std::declval()(std::declval(), std::declval()))) const +{ + BOOST_MATH_STD_USING + using boost::math::constants::half; + using boost::math::quadrature::detail::tanh_sinh_detail; + + static const char* function = "tanh_sinh<%1%>::integrate"; + + if ((boost::math::isfinite)(a) && (boost::math::isfinite)(b)) + { + if (b <= a) + { + return policies::raise_domain_error(function, "Arguments to integrate are in wrong order; integration over [a,b] must have b > a.", a, Policy()); + } + auto u = [&](Real z, Real zc)->Real + { + if (z < 0) + return f((a - b) * zc / 2 + a, (b - a) * zc / 2); + else + return f((a - b) * zc / 2 + b, (b - a) * zc / 2); + }; + Real diff = (b - a)*half(); + Real left_min_complement = tools::min_value() * 4; + Real right_min_complement = tools::min_value() * 4; + Real Q = diff*m_imp->integrate(u, error, L1, function, left_min_complement, right_min_complement, tolerance, levels); + + if (L1) + { + *L1 *= diff; + } + return Q; + } + return policies::raise_domain_error(function, "The domain of integration is not sensible; please check the bounds.", a, Policy()); +} + +template +template +auto tanh_sinh::integrate(const F f, Real tolerance, Real* error, Real* L1, std::size_t* levels) ->decltype(Real(std::declval()(std::declval()))) const +{ + using boost::math::quadrature::detail::tanh_sinh_detail; + static const char* function = "tanh_sinh<%1%>::integrate"; + Real min_complement = tools::epsilon(); + return m_imp->integrate([&](const Real& arg, const Real&) { return f(arg); }, error, L1, function, min_complement, min_complement, tolerance, levels); +} + +template +template +auto tanh_sinh::integrate(const F f, Real tolerance, Real* error, Real* L1, std::size_t* levels) ->decltype(Real(std::declval()(std::declval(), std::declval()))) const +{ + using boost::math::quadrature::detail::tanh_sinh_detail; + static const char* function = "tanh_sinh<%1%>::integrate"; + Real min_complement = tools::min_value() * 4; + return m_imp->integrate(f, error, L1, function, min_complement, min_complement, tolerance, levels); +} + +} +} +} +#endif diff --git a/include/boost/math/special_functions/detail/bernoulli_details.hpp b/include/boost/math/special_functions/detail/bernoulli_details.hpp index ce9503443..41a59e53c 100644 --- a/include/boost/math/special_functions/detail/bernoulli_details.hpp +++ b/include/boost/math/special_functions/detail/bernoulli_details.hpp @@ -9,67 +9,11 @@ #include #include +#include #include #include #include -#ifdef BOOST_HAS_THREADS - -#ifndef BOOST_NO_CXX11_HDR_ATOMIC -# include -# define BOOST_MATH_ATOMIC_NS std -#if ATOMIC_INT_LOCK_FREE == 2 -typedef std::atomic atomic_counter_type; -typedef int atomic_integer_type; -#elif ATOMIC_SHORT_LOCK_FREE == 2 -typedef std::atomic atomic_counter_type; -typedef short atomic_integer_type; -#elif ATOMIC_LONG_LOCK_FREE == 2 -typedef std::atomic atomic_counter_type; -typedef long atomic_integer_type; -#elif ATOMIC_LLONG_LOCK_FREE == 2 -typedef std::atomic atomic_counter_type; -typedef long long atomic_integer_type; -#else -# define BOOST_MATH_NO_ATOMIC_INT -#endif - -#else // BOOST_NO_CXX11_HDR_ATOMIC -// -// We need Boost.Atomic, but on any platform that supports auto-linking we do -// not need to link against a separate library: -// -#define BOOST_ATOMIC_NO_LIB -#include -# define BOOST_MATH_ATOMIC_NS boost - -namespace boost{ namespace math{ namespace detail{ - -// -// We need a type to use as an atomic counter: -// -#if BOOST_ATOMIC_INT_LOCK_FREE == 2 -typedef boost::atomic atomic_counter_type; -typedef int atomic_integer_type; -#elif BOOST_ATOMIC_SHORT_LOCK_FREE == 2 -typedef boost::atomic atomic_counter_type; -typedef short atomic_integer_type; -#elif BOOST_ATOMIC_LONG_LOCK_FREE == 2 -typedef boost::atomic atomic_counter_type; -typedef long atomic_integer_type; -#elif BOOST_ATOMIC_LLONG_LOCK_FREE == 2 -typedef boost::atomic atomic_counter_type; -typedef long long atomic_integer_type; -#else -# define BOOST_MATH_NO_ATOMIC_INT -#endif - -}}} // namespaces - -#endif // BOOST_NO_CXX11_HDR_ATOMIC - -#endif // BOOST_HAS_THREADS - namespace boost{ namespace math{ namespace detail{ // // Asymptotic expansion for B2n due to diff --git a/include/boost/math/tools/atomic.hpp b/include/boost/math/tools/atomic.hpp new file mode 100644 index 000000000..29a48b824 --- /dev/null +++ b/include/boost/math/tools/atomic.hpp @@ -0,0 +1,93 @@ +/////////////////////////////////////////////////////////////////////////////// +// Copyright 2017 John Maddock +// Distributed under the Boost +// Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) + +#ifndef BOOST_MATH_ATOMIC_DETAIL_HPP +#define BOOST_MATH_ATOMIC_DETAIL_HPP + +#include + +#ifdef BOOST_HAS_THREADS + +#ifndef BOOST_NO_CXX11_HDR_ATOMIC +# include +# define BOOST_MATH_ATOMIC_NS std +namespace boost { + namespace math { + namespace detail { +#if ATOMIC_INT_LOCK_FREE == 2 + typedef std::atomic atomic_counter_type; + typedef std::atomic atomic_unsigned_type; + typedef int atomic_integer_type; + typedef unsigned atomic_unsigned_integer_type; +#elif ATOMIC_SHORT_LOCK_FREE == 2 + typedef std::atomic atomic_counter_type; + typedef std::atomic atomic_unsigned_type; + typedef short atomic_integer_type; + typedef unsigned short atomic_unsigned_type; +#elif ATOMIC_LONG_LOCK_FREE == 2 + typedef std::atomic atomic_unsigned_integer_type; + typedef std::atomic atomic_unsigned_type; + typedef unsigned long atomic_unsigned_type; + typedef long atomic_integer_type; +#elif ATOMIC_LLONG_LOCK_FREE == 2 + typedef std::atomic atomic_unsigned_integer_type; + typedef std::atomic atomic_unsigned_type; + typedef long long atomic_integer_type; + typedef unsigned long long atomic_unsigned_integer_type; +#else +# define BOOST_MATH_NO_ATOMIC_INT +#endif + } + }} +#else // BOOST_NO_CXX11_HDR_ATOMIC +// +// We need Boost.Atomic, but on any platform that supports auto-linking we do +// not need to link against a separate library: +// +#define BOOST_ATOMIC_NO_LIB +#include +# define BOOST_MATH_ATOMIC_NS boost + +namespace boost{ namespace math{ namespace detail{ + +// +// We need a type to use as an atomic counter: +// +#if BOOST_ATOMIC_INT_LOCK_FREE == 2 +typedef boost::atomic atomic_counter_type; +typedef boost::atomic atomic_unsigned_type; +typedef int atomic_integer_type; +typedef unsigned atomic_unsigned_integer_type; +#elif BOOST_ATOMIC_SHORT_LOCK_FREE == 2 +typedef boost::atomic atomic_counter_type; +typedef boost::atomic atomic_unsigned_type; +typedef short atomic_integer_type; +typedef unsigned short atomic_unsigned_integer_type; +#elif BOOST_ATOMIC_LONG_LOCK_FREE == 2 +typedef boost::atomic atomic_counter_type; +typedef boost::atomic atomic_unsigned_type; +typedef long atomic_integer_type; +typedef unsigned long atomic_unsigned_integer_type; +#elif BOOST_ATOMIC_LLONG_LOCK_FREE == 2 +typedef boost::atomic atomic_counter_type; +typedef boost::atomic atomic_unsigned_type; +typedef long long atomic_integer_type; +typedef unsigned long long atomic_unsigned_integer_type; +#else +# define BOOST_MATH_NO_ATOMIC_INT +#endif + +}}} // namespaces + +#endif // BOOST_NO_CXX11_HDR_ATOMIC + +#else // BOOST_HAS_THREADS + +# define BOOST_MATH_NO_ATOMIC_INT + +#endif // BOOST_HAS_THREADS + +#endif // BOOST_MATH_ATOMIC_DETAIL_HPP diff --git a/test/Jamfile.v2 b/test/Jamfile.v2 index ecf1a559d..08b34069b 100644 --- a/test/Jamfile.v2 +++ b/test/Jamfile.v2 @@ -974,6 +974,87 @@ test-suite misc : [ compile mpfr_concept_check.cpp : [ check-target-builds ../config//has_mpfr_class : : no ] off ] [ compile mpreal_concept_check.cpp : [ check-target-builds ../config//has_mpreal : : no ] off ] [ compile e_float_concept_check.cpp : [ check-target-builds ../config//has_e_float : : no ] off ] + + [ run tanh_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : TEST1 [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax sfinae_expr ] : + tanh_sinh_quadrature_test_1 ] + [ run tanh_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : TEST1A [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax sfinae_expr ] : + tanh_sinh_quadrature_test_1a ] + [ run tanh_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : TEST2 [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax sfinae_expr ] : + tanh_sinh_quadrature_test_2 ] + [ run tanh_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : TEST2A [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax sfinae_expr ] : + tanh_sinh_quadrature_test_2a ] + [ run tanh_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : TEST3 [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax sfinae_expr ] : + tanh_sinh_quadrature_test_3 ] + [ run tanh_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : TEST3A [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax sfinae_expr ] : + tanh_sinh_quadrature_test_3a ] + [ run tanh_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : release TEST4 [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax sfinae_expr ] : + tanh_sinh_quadrature_test_4 ] + [ run tanh_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : release TEST5 [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax sfinae_expr ] : + tanh_sinh_quadrature_test_5 ] + [ run tanh_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : TEST6 [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax sfinae_expr ] : + tanh_sinh_quadrature_test_6 ] + [ run tanh_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : TEST6A [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax sfinae_expr ] : + tanh_sinh_quadrature_test_6a ] + [ run tanh_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : release TEST7 [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax sfinae_expr ] : + tanh_sinh_quadrature_test_7 ] + [ run tanh_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : release TEST8 [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax sfinae_expr ] : + tanh_sinh_quadrature_test_8 ] + + [ run sinh_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : release [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax ] ] + [ run exp_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : TEST1 [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax ] : exp_sinh_quadrature_test_1 ] + + [ run exp_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : release TEST2 [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax ] : exp_sinh_quadrature_test_2 ] + [ run exp_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : TEST3 [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax ] : exp_sinh_quadrature_test_3 ] + [ run exp_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : release TEST4 [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax ] : exp_sinh_quadrature_test_4 ] + [ run exp_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : release TEST5 [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax ] : exp_sinh_quadrature_test_5 ] + [ run exp_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : release TEST6 [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax ] : exp_sinh_quadrature_test_6 ] + [ run exp_sinh_quadrature_test.cpp ../../test/build//boost_unit_test_framework + : : : release TEST7 [ check-target-builds ../config//has_float128 "GCC libquadmath and __float128 support" : -lquadmath ] + [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax ] : exp_sinh_quadrature_test_7 ] + + [ run compile_test/exp_sinh_incl_test.cpp compile_test_main : : : [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax ] ] + [ run compile_test/sinh_sinh_incl_test.cpp compile_test_main : : : [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax ] ] + [ run compile_test/tanh_sinh_incl_test.cpp compile_test_main : : : [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax sfinae_expr ] ] + [ compile compile_test/exp_sinh_concept_test.cpp : [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax ] ] + [ compile compile_test/sinh_sinh_concept_test.cpp : [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax ] ] + [ compile compile_test/tanh_sinh_concept_test.cpp : [ requires cxx11_auto_declarations cxx11_lambdas cxx11_smart_ptr cxx11_unified_initialization_syntax sfinae_expr ] ] + ; diff --git a/test/compile_test/exp_sinh_concept_test.cpp b/test/compile_test/exp_sinh_concept_test.cpp new file mode 100644 index 000000000..6ee6c24e6 --- /dev/null +++ b/test/compile_test/exp_sinh_concept_test.cpp @@ -0,0 +1,16 @@ +// Copyright Nick Thompson 2017. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +#include +#include + + +void compile_and_link_test() +{ + boost::math::concepts::std_real_concept a = 0; + auto f = [](boost::math::concepts::std_real_concept x) { return x; }; + boost::math::quadrature::exp_sinh integrator; + integrator.integrate(f, a); +} diff --git a/test/compile_test/exp_sinh_incl_test.cpp b/test/compile_test/exp_sinh_incl_test.cpp new file mode 100644 index 000000000..62f68876c --- /dev/null +++ b/test/compile_test/exp_sinh_incl_test.cpp @@ -0,0 +1,21 @@ +// Copyright Nick Thompson 2017. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void compile_and_link_test() +{ + auto f = [](double x) { return x; }; + boost::math::quadrature::exp_sinh integrator; + check_result(integrator.integrate(f)); +} diff --git a/test/compile_test/sinh_sinh_concept_test.cpp b/test/compile_test/sinh_sinh_concept_test.cpp new file mode 100644 index 000000000..2b204dc3a --- /dev/null +++ b/test/compile_test/sinh_sinh_concept_test.cpp @@ -0,0 +1,15 @@ +// Copyright Nick Thompson 2017. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +#include +#include + + +void compile_and_link_test() +{ + auto f = [](boost::math::concepts::std_real_concept x) { return x; }; + boost::math::quadrature::sinh_sinh integrator; + integrator.integrate(f); +} diff --git a/test/compile_test/sinh_sinh_incl_test.cpp b/test/compile_test/sinh_sinh_incl_test.cpp new file mode 100644 index 000000000..ed6f05e1e --- /dev/null +++ b/test/compile_test/sinh_sinh_incl_test.cpp @@ -0,0 +1,21 @@ +// Copyright Nick Thompson 2017. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void compile_and_link_test() +{ + auto f = [](double x) { return 1/(1+x*x); }; + boost::math::quadrature::sinh_sinh integrator; + check_result(integrator.integrate(f)); +} diff --git a/test/compile_test/tanh_sinh_concept_test.cpp b/test/compile_test/tanh_sinh_concept_test.cpp new file mode 100644 index 000000000..239d1fb16 --- /dev/null +++ b/test/compile_test/tanh_sinh_concept_test.cpp @@ -0,0 +1,17 @@ +// Copyright Nick Thompson 2017. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +#include +#include + + +void compile_and_link_test() +{ + boost::math::concepts::std_real_concept a = 0; + boost::math::concepts::std_real_concept b = 1; + auto f = [](boost::math::concepts::std_real_concept x) { return x; }; + boost::math::quadrature::tanh_sinh integrator; + integrator.integrate(f, a, b); +} diff --git a/test/compile_test/tanh_sinh_incl_test.cpp b/test/compile_test/tanh_sinh_incl_test.cpp new file mode 100644 index 000000000..ef5fff794 --- /dev/null +++ b/test/compile_test/tanh_sinh_incl_test.cpp @@ -0,0 +1,23 @@ +// Copyright Nick Thompson 2017. +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. (See accompanying file +// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) +// +// Basic sanity check that header +// #includes all the files that it needs to. +// +#include +// +// Note this header includes no other headers, this is +// important if this test is to be meaningful: +// +#include "test_compile_result.hpp" + +void compile_and_link_test() +{ + auto f = [](double x) { return x; }; + double a = 0; + double b = 1; + boost::math::quadrature::tanh_sinh integrator; + check_result(integrator.integrate(f, a, b)); +} diff --git a/test/exp_sinh_quadrature_test.cpp b/test/exp_sinh_quadrature_test.cpp new file mode 100644 index 000000000..86acd175f --- /dev/null +++ b/test/exp_sinh_quadrature_test.cpp @@ -0,0 +1,534 @@ +// Copyright Nick Thompson, 2017 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_TEST_MODULE exp_sinh_quadrature_test + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +using std::exp; +using std::cos; +using std::tan; +using std::log; +using std::sqrt; +using std::abs; +using std::sinh; +using std::cosh; +using std::pow; +using std::atan; +using boost::multiprecision::cpp_bin_float_50; +using boost::multiprecision::cpp_bin_float_100; +using boost::multiprecision::cpp_bin_float_quad; +using boost::math::constants::pi; +using boost::math::constants::half_pi; +using boost::math::constants::two_div_pi; +using boost::math::constants::half; +using boost::math::constants::third; +using boost::math::constants::half; +using boost::math::constants::third; +using boost::math::constants::catalan; +using boost::math::constants::ln_two; +using boost::math::constants::root_two; +using boost::math::constants::root_two_pi; +using boost::math::constants::root_pi; +using boost::math::quadrature::exp_sinh; + +#if !defined(TEST1) && !defined(TEST2) && !defined(TEST3) && !defined(TEST4) && !defined(TEST5) && !defined(TEST6) && !defined(TEST7) +# define TEST1 +# define TEST2 +# define TEST3 +# define TEST4 +# define TEST5 +# define TEST6 +# define TEST7 +#endif + +#ifdef BOOST_MSVC +#pragma warning (disable:4127) +#endif + +// +// Coefficient generation code: +// +template +void print_levels(const T& v, const char* suffix) +{ + std::cout << "{\n"; + for (unsigned i = 0; i < v.size(); ++i) + { + std::cout << " { "; + for (unsigned j = 0; j < v[i].size(); ++j) + { + std::cout << v[i][j] << suffix << ", "; + } + std::cout << "},\n"; + } + std::cout << " };\n"; +} + +template +void print_levels(const std::pair& p, const char* suffix = "") +{ + std::cout << " static const std::vector > abscissa = "; + print_levels(p.first, suffix); + std::cout << " static const std::vector > weights = "; + print_levels(p.second, suffix); +} + +template +std::pair>, std::vector> > generate_constants(unsigned max_rows) +{ + using boost::math::constants::half_pi; + using boost::math::constants::two_div_pi; + using boost::math::constants::pi; + auto g = [](Real t) { return exp(half_pi()*sinh(t)); }; + auto w = [](Real t) { return cosh(t)*half_pi()*exp(half_pi()*sinh(t)); }; + + std::vector> abscissa, weights; + + std::vector temp; + + Real tmp = (Real(boost::math::tools::log_min_value()) + log(Real(boost::math::tools::epsilon())))*0.5f; + Real t_min = asinh(two_div_pi()*tmp); + // truncate t_min to an exact binary value: + t_min = floor(t_min * 128) / 128; + + std::cout << "m_t_min = " << t_min << ";\n"; + + // t_max is chosen to make g'(t_max) ~ sqrt(max) (g' grows faster than g). + // This will allow some flexibility on the users part; they can at least square a number function without overflow. + // But there is no unique choice; the further out we can evaluate the function, the better we can do on slowly decaying integrands. + const Real t_max = log(2 * two_div_pi()*log(2 * two_div_pi()*sqrt(Real(boost::math::tools::max_value())))); + + Real h = 1; + for (Real t = t_min; t < t_max; t += h) + { + temp.push_back(g(t)); + } + abscissa.push_back(temp); + temp.clear(); + + for (Real t = t_min; t < t_max; t += h) + { + temp.push_back(w(t * h)); + } + weights.push_back(temp); + temp.clear(); + + for (unsigned row = 1; row < max_rows; ++row) + { + h /= 2; + for (Real t = t_min + h; t < t_max; t += 2 * h) + temp.push_back(g(t)); + abscissa.push_back(temp); + temp.clear(); + } + h = 1; + for (unsigned row = 1; row < max_rows; ++row) + { + h /= 2; + for (Real t = t_min + h; t < t_max; t += 2 * h) + temp.push_back(w(t)); + weights.push_back(temp); + temp.clear(); + } + + return std::make_pair(abscissa, weights); +} + + +template +const exp_sinh& get_integrator() +{ + static const exp_sinh integrator(14); + return integrator; +} + +template +Real get_convergence_tolerance() +{ + return boost::math::tools::root_epsilon(); +} + +template +void test_right_limit_infinite() +{ + std::cout << "Testing right limit infinite for tanh_sinh in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10 * boost::math::tools::epsilon(); + Real Q; + Real Q_expected; + Real error; + Real L1; + auto integrator = get_integrator(); + + // Example 12 + const auto f2 = [](const Real& t) { return exp(-t)/sqrt(t); }; + Q = integrator.integrate(f2, get_convergence_tolerance(), &error, &L1); + Q_expected = root_pi(); + Real tol_mult = 1; + // Multiprecision type have higher error rates, probably evaluation of f() is less accurate: + if (std::numeric_limits::digits10 > std::numeric_limits::digits10) + tol_mult = 12; + else if (std::numeric_limits::digits10 > std::numeric_limits::digits10) + tol_mult = 5; + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol * tol_mult); + // The integrand is strictly positive, so it coincides with the value of the integral: + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol * tol_mult); + + auto f3 = [](Real t)->Real { Real z = exp(-t); if (z == 0) { return z; } return z*cos(t); }; + Q = integrator.integrate(f3, get_convergence_tolerance(), &error, &L1); + Q_expected = half(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + Q = integrator.integrate(f3, 10, std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value(), get_convergence_tolerance(), &error, &L1); + Q_expected = boost::lexical_cast("-6.6976341310426674140007086979326069121526743314567805278252392932e-6"); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 10 * tol); + // Integrating through zero risks precision loss: + Q = integrator.integrate(f3, -10, std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value(), get_convergence_tolerance(), &error, &L1); + Q_expected = boost::lexical_cast("-15232.3213626280525704332288302799653087046646639974940243044623285817777006"); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, std::numeric_limits::digits10 > 30 ? 1000 * tol : tol); + + auto f4 = [](Real t)->Real { return 1/(1+t*t); }; + Q = integrator.integrate(f4, 1, std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value(), get_convergence_tolerance(), &error, &L1); + Q_expected = pi()/4; + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); + Q = integrator.integrate(f4, 20, std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value(), get_convergence_tolerance(), &error, &L1); + Q_expected = boost::lexical_cast("0.0499583957219427614100062870348448814912770804235071744108534548299835954767"); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); + Q = integrator.integrate(f4, 500, std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value(), get_convergence_tolerance(), &error, &L1); + Q_expected = boost::lexical_cast("0.0019999973333397333150476759363217553199063513829126652556286269630"); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); +} + +template +void test_left_limit_infinite() +{ + std::cout << "Testing left limit infinite for 1/(1+t^2) on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10 * boost::math::tools::epsilon(); + Real Q; + Real Q_expected; + Real error; + Real L1; + auto integrator = get_integrator(); + + // Example 11: + auto f1 = [](const Real& t) { return 1/(1+t*t);}; + Q = integrator.integrate(f1, std::numeric_limits::has_infinity ? -std::numeric_limits::infinity() : -boost::math::tools::max_value(), 0, get_convergence_tolerance(), &error, &L1); + Q_expected = half_pi(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); + Q = integrator.integrate(f1, std::numeric_limits::has_infinity ? -std::numeric_limits::infinity() : -boost::math::tools::max_value(), -20, get_convergence_tolerance(), &error, &L1); + Q_expected = boost::lexical_cast("0.0499583957219427614100062870348448814912770804235071744108534548299835954767"); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); + Q = integrator.integrate(f1, std::numeric_limits::has_infinity ? -std::numeric_limits::infinity() : -boost::math::tools::max_value(), -500, get_convergence_tolerance(), &error, &L1); + Q_expected = boost::lexical_cast("0.0019999973333397333150476759363217553199063513829126652556286269630"); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); +} + + +// Some examples of tough integrals from NR, section 4.5.4: +template +void test_nr_examples() +{ + using std::sin; + using std::cos; + using std::pow; + using std::exp; + using std::sqrt; + std::cout << "Testing type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10 * boost::math::tools::epsilon(); + std::cout << std::setprecision(std::numeric_limits::digits10); + Real Q; + Real Q_expected; + Real L1; + Real error; + auto integrator = get_integrator(); + + auto f0 = [] (Real)->Real { return (Real) 0; }; + Q = integrator.integrate(f0, get_convergence_tolerance(), &error, &L1); + Q_expected = 0; + BOOST_CHECK_CLOSE_FRACTION(Q, 0.0f, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, 0.0f, tol); + + auto f = [](const Real& x) { return 1/(1+x*x); }; + Q = integrator.integrate(f, get_convergence_tolerance(), &error, &L1); + Q_expected = half_pi(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); + + auto f1 = [](Real x)->Real { + Real z1 = exp(-x); + if (z1 == 0) + { + return (Real) 0; + } + Real z2 = pow(x, -3*half())*z1; + if (z2 == 0) + { + return (Real) 0; + } + return sin(x*half())*z2; + }; + + Q = integrator.integrate(f1, get_convergence_tolerance(), &error, &L1); + Q_expected = sqrt(pi()*(sqrt((Real) 5) - 2)); + + // The integrand is oscillatory; the accuracy is low. + Real tol_mul = 1; + if (std::numeric_limits::digits10 > 40) + tol_mul = 500000; + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol_mul * tol); + + auto f2 = [](Real x)->Real { return x > boost::math::tools::log_max_value() ? Real(0) : Real(pow(x, -(Real) 2/(Real) 7)*exp(-x*x)); }; + Q = integrator.integrate(f2, get_convergence_tolerance(), &error, &L1); + Q_expected = half()*boost::math::tgamma((Real) 5/ (Real) 14); + tol_mul = 1; + if (std::numeric_limits::is_specialized == false) + tol_mul = 3; + if (std::numeric_limits::digits10 > 40) + tol_mul = 100; + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol_mul * tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol_mul * tol); + + auto f3 = [](Real x)->Real { return (Real) 1/ (sqrt(x)*(1+x)); }; + Q = integrator.integrate(f3, get_convergence_tolerance(), &error, &L1); + Q_expected = pi(); + + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 10*boost::math::tools::epsilon()); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, 10*boost::math::tools::epsilon()); + + auto f4 = [](const Real& t) { return t > boost::math::tools::log_max_value() ? Real(0) : Real(exp(-t*t*half())); }; + Q = integrator.integrate(f4, get_convergence_tolerance(), &error, &L1); + Q_expected = root_two_pi()/2; + tol_mul = 1; + if (std::numeric_limits::digits10 > 40) + tol_mul = 5000; + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol_mul * tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol_mul * tol); + + auto f5 = [](const Real& t) { return 1/cosh(t);}; + Q = integrator.integrate(f5, get_convergence_tolerance(), &error, &L1); + Q_expected = half_pi(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol * 12); // Fails at float precision without higher error rate + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol * 12); +} + +// Definite integrals found in the CRC Handbook of Mathematical Formulas +template +void test_crc() +{ + using std::sin; + using std::pow; + using std::exp; + using std::sqrt; + using std::log; + using std::cos; + std::cout << "Testing integral from CRC handbook on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10 * boost::math::tools::epsilon(); + std::cout << std::setprecision(std::numeric_limits::digits10); + Real Q; + Real Q_expected; + Real L1; + Real error; + auto integrator = get_integrator(); + + auto f0 = [](const Real& x) { return x > boost::math::tools::log_max_value() ? Real(0) : Real(log(x)*exp(-x)); }; + Q = integrator.integrate(f0, get_convergence_tolerance(), &error, &L1); + Q_expected = -boost::math::constants::euler(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + + // Test the integral representation of the gamma function: + auto f1 = [](Real t)->Real { Real x = exp(-t); + if(x == 0) + { + return (Real) 0; + } + return pow(t, (Real) 12 - 1)*x; + }; + + Q = integrator.integrate(f1, get_convergence_tolerance(), &error, &L1); + Q_expected = boost::math::tgamma(12.0f); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + + // Integral representation of the modified bessel function: + // K_5(12) + auto f2 = [](Real t)->Real { + Real x = 12*cosh(t); + if (x > boost::math::tools::log_max_value()) + { + return (Real) 0; + } + return exp(-x)*cosh(5*t); + }; + Q = integrator.integrate(f2, get_convergence_tolerance(), &error, &L1); + Q_expected = boost::math::cyl_bessel_k(5, 12); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + // Laplace transform of cos(at) + Real a = 20; + Real s = 1; + auto f3 = [&](Real t)->Real { + Real x = s * t; + if (x > boost::math::tools::log_max_value()) + { + return (Real) 0; + } + return cos(a * t) * exp(-x); + }; + + // For high oscillation frequency, the quadrature sum is ill-conditioned. + Q = integrator.integrate(f3, get_convergence_tolerance(), &error, &L1); + Q_expected = s/(a*a+s*s); + // Since the integrand is oscillatory, we increase the tolerance: + Real tol_mult = 10; + // Multiprecision type have higher error rates, probably evaluation of f() is less accurate: + if (!boost::is_class::value) + { + if (std::numeric_limits::digits10 > std::numeric_limits::digits10) + tol_mult = 5000; // we should really investigate this more?? + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol_mult*tol); + } + + // + // This one doesn't pass for real_concept.. + // + if (std::numeric_limits::is_specialized) + { + // Laplace transform of J_0(t): + auto f4 = [&](Real t)->Real { + Real x = s * t; + if (x > boost::math::tools::log_max_value()) + { + return (Real)0; + } + return boost::math::cyl_bessel_j(0, t) * exp(-x); + }; + + Q = integrator.integrate(f4, get_convergence_tolerance(), &error, &L1); + Q_expected = 1 / sqrt(1 + s*s); + tol_mult = 3; + // Multiprecision type have higher error rates, probably evaluation of f() is less accurate: + if (std::numeric_limits::digits10 > std::numeric_limits::digits10) + tol_mult = 750; + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol_mult * tol); + } + auto f6 = [](const Real& t) { return t > boost::math::tools::log_max_value() ? Real(0) : Real(exp(-t*t)*log(t));}; + Q = integrator.integrate(f6, get_convergence_tolerance(), &error, &L1); + Q_expected = -boost::math::constants::root_pi()*(boost::math::constants::euler() + 2*ln_two())/4; + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + + // CRC Section 5.5, integral 591 + // The parameter p allows us to control the strength of the singularity. + // Rapid convergence is not guaranteed for this function, as the branch cut makes it non-analytic on a disk. + // This converges only when our test type has an extended exponent range as all the area of the integral + // occurs so close to 0 (or 1) that we need abscissa values exceptionally small to find it. + // "There's a lot of room at the bottom". + // This version is transformed via argument substitution (exp(-x) for x) so that the integral is spread + // over (0, INF). + tol *= boost::math::tools::digits() > 100 ? 100000 : 75; + for (Real pn = 99; pn > 0; pn -= 10) { + Real p = pn / 100; + auto f = [&](Real x)->Real + { + return x > 1000 * boost::math::tools::log_max_value() ? Real(0) : Real(exp(-x * (1 - p) + p * log(-boost::math::expm1(-x)))); + }; + Q = integrator.integrate(f, get_convergence_tolerance(), &error, &L1); + Q_expected = 1 / boost::math::sinc_pi(p*pi()); + /* + std::cout << std::setprecision(std::numeric_limits::max_digits10) << p << std::endl; + std::cout << std::setprecision(std::numeric_limits::max_digits10) << Q << std::endl; + std::cout << std::setprecision(std::numeric_limits::max_digits10) << Q_expected << std::endl; + std::cout << fabs((Q - Q_expected) / Q_expected) << std::endl; + */ + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + } + // and for p < 1: + for (Real p = -0.99; p < 0; p += 0.1) { + auto f = [&](Real x)->Real + { + return x > 1000 * boost::math::tools::log_max_value() ? Real(0) : Real(exp(-p * log(-boost::math::expm1(-x)) - (1 + p) * x)); + }; + Q = integrator.integrate(f, get_convergence_tolerance(), &error, &L1); + Q_expected = 1 / boost::math::sinc_pi(p*pi()); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + } +} + + +BOOST_AUTO_TEST_CASE(exp_sinh_quadrature_test) +{ + // + // Uncomment to generate the coefficients: + // + + /* + std::cout << std::scientific << std::setprecision(8); + print_levels(generate_constants(8), "f"); + std::cout << std::setprecision(18); + print_levels(generate_constants(8), ""); + std::cout << std::setprecision(35); + print_levels(generate_constants(8), "L"); + */ + + +#ifdef TEST1 + test_left_limit_infinite(); + test_right_limit_infinite(); + test_nr_examples(); + test_crc(); +#endif +#ifdef TEST2 + test_left_limit_infinite(); + test_right_limit_infinite(); + test_nr_examples(); + test_crc(); +#endif +#ifdef TEST3 + test_left_limit_infinite(); + test_right_limit_infinite(); + test_nr_examples(); + test_crc(); +#endif +#ifdef TEST4 + test_left_limit_infinite(); + test_right_limit_infinite(); + test_nr_examples(); + test_crc(); +#endif + +#ifdef TEST5 + test_left_limit_infinite(); + test_right_limit_infinite(); + test_nr_examples(); + test_crc(); +#endif + +#ifdef TEST6 + test_left_limit_infinite(); + test_right_limit_infinite(); + test_nr_examples(); + test_crc(); +#endif +#ifdef TEST7 + test_left_limit_infinite(); + test_right_limit_infinite(); + test_nr_examples(); + test_crc(); +#endif +} diff --git a/test/sinh_sinh_quadrature_test.cpp b/test/sinh_sinh_quadrature_test.cpp new file mode 100644 index 000000000..fd6e666c5 --- /dev/null +++ b/test/sinh_sinh_quadrature_test.cpp @@ -0,0 +1,263 @@ +// Copyright Nick Thompson, 2017 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_TEST_MODULE sinh_sinh_quadrature_test + +#include +#include +#include +#include +#include +#include + +#if !BOOST_WORKAROUND(BOOST_MSVC, < 1900) +// MSVC-12 has problems if we include 2 different multiprecision types in the same program, +// basically random things stop compiling, even though they work fine in isolation :( +#include +#endif + +using std::expm1; +using std::exp; +using std::sin; +using std::cos; +using std::atan; +using std::tan; +using std::log; +using std::log1p; +using std::asinh; +using std::atanh; +using std::sqrt; +using std::isnormal; +using std::abs; +using std::sinh; +using std::tanh; +using std::cosh; +using std::pow; +using std::string; +using boost::multiprecision::cpp_bin_float_quad; +using boost::math::quadrature::sinh_sinh; +using boost::math::constants::pi; +using boost::math::constants::pi_sqr; +using boost::math::constants::half_pi; +using boost::math::constants::two_div_pi; +using boost::math::constants::half; +using boost::math::constants::third; +using boost::math::constants::half; +using boost::math::constants::third; +using boost::math::constants::catalan; +using boost::math::constants::ln_two; +using boost::math::constants::root_two; +using boost::math::constants::root_two_pi; +using boost::math::constants::root_pi; +// +// Code for generating the coefficients: +// +template +void print_levels(const T& v, const char* suffix) +{ + std::cout << "{\n"; + for (unsigned i = 0; i < v.size(); ++i) + { + std::cout << " { "; + for (unsigned j = 0; j < v[i].size(); ++j) + { + std::cout << v[i][j] << suffix << ", "; + } + std::cout << "},\n"; + } + std::cout << " };\n"; +} + +template +void print_levels(const std::pair& p, const char* suffix = "") +{ + std::cout << " static const std::vector > abscissa = "; + print_levels(p.first, suffix); + std::cout << " static const std::vector > weights = "; + print_levels(p.second, suffix); +} + +template +std::pair>, std::vector> > generate_constants(unsigned max_rows) +{ + using boost::math::constants::half_pi; + using boost::math::constants::two_div_pi; + using boost::math::constants::pi; + auto g = [](Real t) { return sinh(half_pi()*sinh(t)); }; + auto w = [](Real t) { return cosh(t)*half_pi()*cosh(half_pi()*sinh(t)); }; + + std::vector> abscissa, weights; + + std::vector temp; + + Real t_max = log(2 * two_div_pi()*log(2 * two_div_pi()*sqrt(boost::math::tools::max_value()))); + + std::cout << "m_t_max = " << t_max << ";\n"; + + Real h = 1; + for (Real t = 1; t < t_max; t += h) + { + temp.push_back(g(t)); + } + abscissa.push_back(temp); + temp.clear(); + + for (Real t = 1; t < t_max; t += h) + { + temp.push_back(w(t * h)); + } + weights.push_back(temp); + temp.clear(); + + for (unsigned row = 1; row < max_rows; ++row) + { + h /= 2; + for (Real t = h; t < t_max; t += 2 * h) + temp.push_back(g(t)); + abscissa.push_back(temp); + temp.clear(); + } + h = 1; + for (unsigned row = 1; row < max_rows; ++row) + { + h /= 2; + for (Real t = h; t < t_max; t += 2 * h) + temp.push_back(w(t)); + weights.push_back(temp); + temp.clear(); + } + + return std::make_pair(abscissa, weights); +} + +template +void test_nr_examples() +{ + std::cout << "Testing type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real integration_limit = sqrt(boost::math::tools::epsilon()); + Real tol = 10 * boost::math::tools::epsilon(); + std::cout << std::setprecision(std::numeric_limits::digits10); + Real Q; + Real Q_expected; + Real L1; + Real error; + sinh_sinh integrator(10); + + auto f0 = [](Real)->Real { return (Real) 0; }; + Q = integrator.integrate(f0, integration_limit, &error, &L1); + Q_expected = 0; + BOOST_CHECK_SMALL(Q, tol); + BOOST_CHECK_SMALL(L1, tol); + + // In spite of the poles at \pm i, we still get a doubling of the correct digits at each level of refinement. + auto f1 = [](const Real& t) { return 1/(1+t*t); }; + Q = integrator.integrate(f1, integration_limit, &error, &L1); + Q_expected = pi(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); +#if defined(BOOST_MSVC) && (BOOST_MSVC < 1900) + auto f2 = [](const Real& x) { return fabs(x) > boost::math::tools::log_max_value() ? 0 : exp(-x*x); }; +#else + auto f2 = [](const Real& x) { return exp(-x*x); }; +#endif + Q = integrator.integrate(f2, integration_limit, &error, &L1); + Q_expected = root_pi(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); + + auto f5 = [](const Real& t) { return 1/cosh(t);}; + Q = integrator.integrate(f5, integration_limit, &error, &L1); + Q_expected = pi(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); + + // This oscillatory integral has rapid convergence because the oscillations get swamped by the exponential growth of the denominator, + // none the less the error is slightly higher than for the other cases: + tol *= 10; + auto f8 = [](const Real& t) { return cos(t)/cosh(t);}; + Q = integrator.integrate(f8, integration_limit, &error, &L1); + Q_expected = pi()/cosh(half_pi()); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + // Try again with progressively fewer arguments: + Q = integrator.integrate(f8, integration_limit); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + Q = integrator.integrate(f8); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); +} + +// Test formulas for in the CRC Handbook of Mathematical functions, 32nd edition. +template +void test_crc() +{ + std::cout << "Testing CRC formulas on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real integration_limit = sqrt(boost::math::tools::epsilon()); + Real tol = 10 * boost::math::tools::epsilon(); + std::cout << std::setprecision(std::numeric_limits::digits10); + Real Q; + Real Q_expected; + Real L1; + Real error; + sinh_sinh integrator(10); + + // CRC Definite integral 698: + auto f0 = [](Real x)->Real { + using std::sinh; + if(x == 0) { + return (Real) 1; + } + return x/sinh(x); + }; + Q = integrator.integrate(f0, integration_limit, &error, &L1); + Q_expected = pi_sqr()/2; + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); + + + // CRC Definite integral 695: + auto f1 = [](Real x)->Real { + using std::sin; using std::sinh; + if(x == 0) { + return (Real) 1; + } + return (Real) sin(x)/sinh(x); + }; + Q = integrator.integrate(f1, integration_limit, &error, &L1); + Q_expected = pi()*tanh(half_pi()); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); +} + + +BOOST_AUTO_TEST_CASE(sinh_sinh_quadrature_test) +{ + // + // Uncomment the following to print out the coefficients: + // + /* + std::cout << std::scientific << std::setprecision(8); + print_levels(generate_constants(8), "f"); + std::cout << std::setprecision(18); + print_levels(generate_constants(8), ""); + std::cout << std::setprecision(35); + print_levels(generate_constants(8), "L"); + */ + test_nr_examples(); + test_nr_examples(); + test_nr_examples(); + test_nr_examples(); + test_nr_examples(); +#if !BOOST_WORKAROUND(BOOST_MSVC, < 1900) + test_nr_examples(); +#endif + + test_crc(); + test_crc(); + test_crc(); + test_crc(); + test_crc(); +#if !BOOST_WORKAROUND(BOOST_MSVC, < 1900) + test_crc(); +#endif +} diff --git a/test/tanh_sinh_quadrature_test.cpp b/test/tanh_sinh_quadrature_test.cpp new file mode 100644 index 000000000..a5c0775c6 --- /dev/null +++ b/test/tanh_sinh_quadrature_test.cpp @@ -0,0 +1,933 @@ +// Copyright Nick Thompson, 2017 +// Use, modification and distribution are subject to the +// Boost Software License, Version 1.0. +// (See accompanying file LICENSE_1_0.txt +// or copy at http://www.boost.org/LICENSE_1_0.txt) + +#define BOOST_TEST_MODULE tanh_sinh_quadrature_test + +#include +#include + +#if !defined(BOOST_NO_CXX11_DECLTYPE) && !defined(BOOST_NO_CXX11_TRAILING_RESULT_TYPES) && !defined(BOOST_NO_SFINAE_EXPR) + +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include +#include + +#ifdef BOOST_HAS_FLOAT128 +#include +#endif + +#ifdef _MSC_VER +#pragma warning(disable:4127) // Conditional expression is constant +#endif + +#if !defined(TEST1) && !defined(TEST2) && !defined(TEST3) && !defined(TEST4) && !defined(TEST5) && !defined(TEST6) && !defined(TEST7) && !defined(TEST8)\ + && !defined(TEST1A) && !defined(TEST2A) && !defined(TEST3A) && !defined(TEST6A) +# define TEST1 +# define TEST2 +# define TEST3 +# define TEST4 +# define TEST5 +# define TEST6 +# define TEST7 +# define TEST8 +# define TEST1A +# define TEST2A +# define TEST3A +# define TEST6A +#endif + +using std::expm1; +using std::atan; +using std::tan; +using std::log; +using std::log1p; +using std::asinh; +using std::atanh; +using std::sqrt; +using std::isnormal; +using std::abs; +using std::sinh; +using std::tanh; +using std::cosh; +using std::pow; +using std::exp; +using std::sin; +using std::cos; +using std::string; +using boost::multiprecision::cpp_bin_float_50; +using boost::multiprecision::cpp_bin_float_100; +using boost::multiprecision::cpp_dec_float_50; +using boost::multiprecision::cpp_dec_float_100; +using boost::multiprecision::cpp_bin_float_quad; +using boost::math::sinc_pi; +using boost::math::quadrature::tanh_sinh; +using boost::math::quadrature::detail::tanh_sinh_detail; +using boost::math::constants::pi; +using boost::math::constants::half_pi; +using boost::math::constants::two_div_pi; +using boost::math::constants::two_pi; +using boost::math::constants::half; +using boost::math::constants::third; +using boost::math::constants::half; +using boost::math::constants::third; +using boost::math::constants::catalan; +using boost::math::constants::ln_two; +using boost::math::constants::root_two; +using boost::math::constants::root_two_pi; +using boost::math::constants::root_pi; + +template +void print_levels(const T& v, const char* suffix) +{ + std::cout << "{\n"; + for (unsigned i = 0; i < v.size(); ++i) + { + std::cout << " { "; + for (unsigned j = 0; j < v[i].size(); ++j) + { + std::cout << v[i][j] << suffix << ", "; + } + std::cout << "},\n"; + } + std::cout << " };\n"; +} + +template +void print_complement_indexes(const T& v) +{ + std::cout << "\n {"; + for (unsigned i = 0; i < v.size(); ++i) + { + unsigned index = 0; + while (v[i][index] >= 0) + ++index; + std::cout << index << ", "; + } + std::cout << "};\n"; +} + +template +void print_levels(const std::pair& p, const char* suffix = "") +{ + std::cout << " static const std::vector > abscissa = "; + print_levels(p.first, suffix); + std::cout << " static const std::vector > weights = "; + print_levels(p.second, suffix); + std::cout << " static const std::vector indexes = "; + print_complement_indexes(p.first); +} + +template +std::pair>, std::vector> > generate_constants(unsigned max_index, unsigned max_rows) +{ + using boost::math::constants::half_pi; + using boost::math::constants::two_div_pi; + using boost::math::constants::pi; + auto g = [](Real t) { return tanh(half_pi()*sinh(t)); }; + auto w = [](Real t) { Real cs = cosh(half_pi() * sinh(t)); return half_pi() * cosh(t) / (cs * cs); }; + auto gc = [](Real t) { Real u2 = half_pi() * sinh(t); return 1 / (exp(u2) *cosh(u2)); }; + auto g_inv = [](float x)->float { return asinh(two_div_pi()*atanh(x)); }; + auto gc_inv = [](float x) + { + float l = log(sqrt((2 - x) / x)); + return log((sqrt(4 * l * l + pi() * pi()) + 2 * l) / pi()); + }; + + std::vector> abscissa, weights; + + std::vector temp; + + float t_crossover = gc_inv(0.5f); + + Real h = 1; + for (unsigned i = 0; i < max_index; ++i) + { + temp.push_back((i < t_crossover) ? g(i * h) : -gc(i * h)); + } + abscissa.push_back(temp); + temp.clear(); + + for (unsigned i = 0; i < max_index; ++i) + { + temp.push_back(w(i * h)); + } + weights.push_back(temp); + temp.clear(); + + for (unsigned row = 1; row < max_rows; ++row) + { + h /= 2; + for (Real t = h; t < max_index - 1; t += 2 * h) + temp.push_back((t < t_crossover) ? g(t) : -gc(t)); + abscissa.push_back(temp); + temp.clear(); + } + h = 1; + for (unsigned row = 1; row < max_rows; ++row) + { + h /= 2; + for (Real t = h; t < max_index - 1; t += 2 * h) + temp.push_back(w(t)); + weights.push_back(temp); + temp.clear(); + } + + return std::make_pair(abscissa, weights); +} + +template +const tanh_sinh& get_integrator() +{ + static const tanh_sinh integrator(15); + return integrator; +} + +template +Real get_convergence_tolerance() +{ + return boost::math::tools::root_epsilon(); +} + + +template +void test_linear() +{ + std::cout << "Testing linear functions are integrated properly by tanh_sinh on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10*boost::math::tools::epsilon(); + auto integrator = get_integrator(); + auto f = [](const Real& x) + { + return 5*x + 7; + }; + Real error; + Real L1; + Real Q = integrator.integrate(f, (Real) 0, (Real) 1, get_convergence_tolerance(), &error, &L1); + BOOST_CHECK_CLOSE_FRACTION(Q, 9.5, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, 9.5, tol); +} + + +template +void test_quadratic() +{ + std::cout << "Testing quadratic functions are integrated properly by tanh_sinh on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10*boost::math::tools::epsilon(); + auto integrator = get_integrator(); + auto f = [](const Real& x) { return 5*x*x + 7*x + 12; }; + Real error; + Real L1; + Real Q = integrator.integrate(f, (Real) 0, (Real) 1, get_convergence_tolerance(), &error, &L1); + BOOST_CHECK_CLOSE_FRACTION(Q, (Real) 17 + half()*third(), tol); + BOOST_CHECK_CLOSE_FRACTION(L1, (Real) 17 + half()*third(), tol); +} + + +template +void test_singular() +{ + std::cout << "Testing integration of endpoint singularities on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10*boost::math::tools::epsilon(); + Real error; + Real L1; + auto integrator = get_integrator(); + auto f = [](const Real& x) { return log(x)*log(1-x); }; + Real Q = integrator.integrate(f, (Real) 0, (Real) 1, get_convergence_tolerance(), &error, &L1); + Real Q_expected = 2 - pi()*pi()*half()*third(); + + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); +} + + +// Examples taken from +//http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/quadrature.pdf +template +void test_ca() +{ + std::cout << "Testing integration of C(a) on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10 * boost::math::tools::epsilon(); + Real error; + Real L1; + + auto integrator = get_integrator(); + auto f1 = [](const Real& x) { return atan(x)/(x*(x*x + 1)) ; }; + Real Q = integrator.integrate(f1, (Real) 0, (Real) 1, get_convergence_tolerance(), &error, &L1); + Real Q_expected = pi()*ln_two()/8 + catalan()*half(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); + + auto f2 = [](Real x)->Real { Real t0 = x*x + 1; Real t1 = sqrt(t0); return atan(t1)/(t0*t1); }; + Q = integrator.integrate(f2, (Real) 0 , (Real) 1, get_convergence_tolerance(), &error, &L1); + Q_expected = pi()/4 - pi()/root_two() + 3*atan(root_two())/root_two(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); + + auto f5 = [](Real t)->Real { return t*t*log(t)/((t*t - 1)*(t*t*t*t + 1)); }; + Q = integrator.integrate(f5, (Real) 0 , (Real) 1); + Q_expected = pi()*pi()*(2 - root_two())/32; + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + + + // Oh it suffers on this one. + auto f6 = [](Real t)->Real { Real x = log(t); return x*x; }; + Q = integrator.integrate(f6, (Real) 0 , (Real) 1, get_convergence_tolerance(), &error, &L1); + Q_expected = 2; + + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 50*tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, 50*tol); + + + // Although it doesn't get to the requested tolerance on this integral, the error bounds can be queried and are reasonable: + tol = sqrt(boost::math::tools::epsilon()); + auto f7 = [](const Real& t) { return sqrt(tan(t)); }; + Q = integrator.integrate(f7, (Real) 0 , (Real) half_pi(), get_convergence_tolerance(), &error, &L1); + Q_expected = pi()/root_two(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); + + auto f8 = [](const Real& t) { return log(cos(t)); }; + Q = integrator.integrate(f8, (Real) 0 , half_pi(), get_convergence_tolerance(), &error, &L1); + Q_expected = -pi()*ln_two()*half(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, -Q_expected, tol); +} + + +template +void test_three_quadrature_schemes_examples() +{ + std::cout << "Testing integral in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10 * boost::math::tools::epsilon(); + Real Q; + Real Q_expected; + + auto integrator = get_integrator(); + // Example 1: + auto f1 = [](const Real& t) { return t*boost::math::log1p(t); }; + Q = integrator.integrate(f1, (Real) 0 , (Real) 1); + Q_expected = half()*half(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + + + // Example 2: + auto f2 = [](const Real& t) { return t*t*atan(t); }; + Q = integrator.integrate(f2, (Real) 0 , (Real) 1); + Q_expected = (pi() -2 + 2*ln_two())/12; + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 2 * tol); + + // Example 3: + auto f3 = [](const Real& t) { return exp(t)*cos(t); }; + Q = integrator.integrate(f3, (Real) 0, half_pi()); + Q_expected = boost::math::expm1(half_pi())*half(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + + // Example 4: + auto f4 = [](Real x)->Real { Real t0 = sqrt(x*x + 2); return atan(t0)/(t0*(x*x+1)); }; + Q = integrator.integrate(f4, (Real) 0 , (Real) 1); + Q_expected = 5*pi()*pi()/96; + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + + // Example 5: + auto f5 = [](const Real& t) { return sqrt(t)*log(t); }; + Q = integrator.integrate(f5, (Real) 0 , (Real) 1); + Q_expected = -4/ (Real) 9; + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + + // Example 6: + auto f6 = [](const Real& t) { return sqrt(1 - t*t); }; + Q = integrator.integrate(f6, (Real) 0 , (Real) 1); + Q_expected = pi()/4; + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); +} + + +template +void test_integration_over_real_line() +{ + std::cout << "Testing integrals over entire real line in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10 * boost::math::tools::epsilon(); + Real Q; + Real Q_expected; + Real error; + Real L1; + auto integrator = get_integrator(); + + auto f1 = [](const Real& t) { return 1/(1+t*t);}; + Q = integrator.integrate(f1, std::numeric_limits::has_infinity ? -std::numeric_limits::infinity() : -boost::math::tools::max_value(), std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value(), get_convergence_tolerance(), &error, &L1); + Q_expected = pi(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); + + auto f2 = [](const Real& t) { return exp(-t*t*half()); }; + Q = integrator.integrate(f2, std::numeric_limits::has_infinity ? -std::numeric_limits::infinity() : -boost::math::tools::max_value(), std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value(), get_convergence_tolerance(), &error, &L1); + Q_expected = root_two_pi(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol * 2); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol * 2); + + // This test shows how oscillatory integrals are approximated very poorly by this method: + //std::cout << "Testing sinc integral: \n"; + //Q = integrator.integrate(boost::math::sinc_pi, -std::numeric_limits::infinity(), std::numeric_limits::infinity(), &error, &L1); + //std::cout << "Error estimate of sinc integral: " << error << std::endl; + //std::cout << "L1 norm of sinc integral " << L1 << std::endl; + //Q_expected = pi(); + //BOOST_CHECK_CLOSE(Q, Q_expected, 100*tol); + + auto f4 = [](const Real& t) { return 1/cosh(t);}; + Q = integrator.integrate(f4, std::numeric_limits::has_infinity ? -std::numeric_limits::infinity() : -boost::math::tools::max_value(), std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value(), get_convergence_tolerance(), &error, &L1); + Q_expected = pi(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); + +} + +template +void test_right_limit_infinite() +{ + std::cout << "Testing right limit infinite for tanh_sinh in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10 * boost::math::tools::epsilon(); + Real Q; + Real Q_expected; + Real error; + Real L1; + auto integrator = get_integrator(); + + // Example 11: + auto f1 = [](const Real& t) { return 1/(1+t*t);}; + Q = integrator.integrate(f1, 0, std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value(), get_convergence_tolerance(), &error, &L1); + Q_expected = half_pi(); + BOOST_CHECK_CLOSE(Q, Q_expected, 100*tol); + + // Example 12 + auto f2 = [](const Real& t) { return exp(-t)/sqrt(t); }; + Q = integrator.integrate(f2, 0, std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value(), get_convergence_tolerance(), &error, &L1); + Q_expected = root_pi(); + BOOST_CHECK_CLOSE(Q, Q_expected, 1000*tol); + + auto f3 = [](const Real& t) { return exp(-t)*cos(t); }; + Q = integrator.integrate(f3, 0, std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value(), get_convergence_tolerance(), &error, &L1); + Q_expected = half(); + BOOST_CHECK_CLOSE(Q, Q_expected, 100*tol); + + auto f4 = [](const Real& t) { return 1/(1+t*t); }; + Q = integrator.integrate(f4, 1, std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value(), get_convergence_tolerance(), &error, &L1); + Q_expected = pi()/4; + BOOST_CHECK_CLOSE(Q, Q_expected, 100*tol); +} + +template +void test_left_limit_infinite() +{ + std::cout << "Testing left limit infinite for tanh_sinh in 'A Comparison of Three High Precision Quadrature Schemes' on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10 * boost::math::tools::epsilon(); + Real Q; + Real Q_expected; + auto integrator = get_integrator(); + + // Example 11: + auto f1 = [](const Real& t) { return 1/(1+t*t);}; + Q = integrator.integrate(f1, std::numeric_limits::has_infinity ? -std::numeric_limits::infinity() : -boost::math::tools::max_value(), Real(0)); + Q_expected = half_pi(); + BOOST_CHECK_CLOSE(Q, Q_expected, 100*tol); +} + + +// A horrible function taken from +// http://www.chebfun.org/examples/quad/GaussClenCurt.html +template +void test_horrible() +{ + std::cout << "Testing Trefenthen's horrible integral on type " << boost::typeindex::type_id().pretty_name() << "\n"; + // We only know the integral to double precision, so requesting a higher tolerance doesn't make sense. + Real tol = 10 * std::numeric_limits::epsilon(); + Real Q; + Real Q_expected; + Real error; + Real L1; + auto integrator = get_integrator(); + + auto f = [](Real x)->Real { return x*sin(2*exp(2*sin(2*exp(2*x) ) ) ); }; + Q = integrator.integrate(f, (Real) -1, (Real) 1, get_convergence_tolerance(), &error, &L1); + // NIntegrate[x*Sin[2*Exp[2*Sin[2*Exp[2*x]]]], {x, -1, 1}, WorkingPrecision -> 130, MaxRecursion -> 100] + Q_expected = boost::lexical_cast("0.33673283478172753598559003181355241139806404130031017259552729882281"); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + // Over again without specifying the bounds: + Q = integrator.integrate(f, get_convergence_tolerance(), &error, &L1); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); +} + +// Some examples of tough integrals from NR, section 4.5.4: +template +void test_nr_examples() +{ + std::cout << "Testing singular integrals from NR 4.5.4 on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10 * boost::math::tools::epsilon(); + Real Q; + Real Q_expected; + Real error; + Real L1; + auto integrator = get_integrator(); + + auto f1 = [](Real x)->Real + { + return (sin(x * half()) * exp(-x) / x) / sqrt(x); + }; + Q = integrator.integrate(f1, 0, std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value(), get_convergence_tolerance(), &error, &L1); + Q_expected = sqrt(pi()*(sqrt((Real) 5) - 2)); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 25*tol); + + auto f2 = [](Real x)->Real { return pow(x, -(Real) 2/(Real) 7)*exp(-x*x); }; + Q = integrator.integrate(f2, 0, std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value()); + Q_expected = half()*boost::math::tgamma((Real) 5/ (Real) 14); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol * 3); + +} + +// Test integrand known to fool some termination schemes: +template +void test_early_termination() +{ + std::cout << "Testing Clenshaw & Curtis's example of integrand which fools termination schemes on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10 * boost::math::tools::epsilon(); + Real Q; + Real Q_expected; + Real error; + Real L1; + auto integrator = get_integrator(); + + auto f1 = [](Real x)->Real { return 23*cosh(x)/ (Real) 25 - cos(x) ; }; + Q = integrator.integrate(f1, (Real) -1, (Real) 1, get_convergence_tolerance(), &error, &L1); + Q_expected = 46*sinh((Real) 1)/(Real) 25 - 2*sin((Real) 1); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + // Over again with no bounds: + Q = integrator.integrate(f1); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); +} + +// Test some definite integrals from the CRC handbook, 32nd edition: +template +void test_crc() +{ + std::cout << "Testing CRC formulas on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10 * boost::math::tools::epsilon(); + Real Q; + Real Q_expected; + Real error; + Real L1; + auto integrator = get_integrator(); + + // CRC Definite integral 585 + auto f1 = [](Real x)->Real { Real t = log(1/x); return x*x*t*t*t; }; + Q = integrator.integrate(f1, (Real) 0, (Real) 1, get_convergence_tolerance(), &error, &L1); + Q_expected = (Real) 2/ (Real) 27; + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + + // CRC 636: + auto f2 = [](Real x)->Real { return sqrt(cos(x)); }; + Q = integrator.integrate(f2, (Real) 0, (Real) half_pi(), get_convergence_tolerance(), &error, &L1); + //Q_expected = pow(two_pi(), 3*half())/pow(boost::math::tgamma((Real) 1/ (Real) 4), 2); + Q_expected = boost::lexical_cast("1.198140234735592207439922492280323878227212663215651558263674952946405214143915670835885556489793389375907225"); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + + // CRC Section 5.5, integral 585: + for (int n = 0; n < 3; ++n) { + for (int m = 0; m < 3; ++m) { + auto f = [&](Real x)->Real { return pow(x, Real(m))*pow(log(1/x), Real(n)); }; + Q = integrator.integrate(f, (Real) 0, (Real) 1, get_convergence_tolerance(), &error, &L1); + // Calculation of the tgamma function is not exact, giving spurious failures. + // Casting to cpp_bin_float_100 beforehand fixes most of them. + cpp_bin_float_100 np1 = n + 1; + cpp_bin_float_100 mp1 = m + 1; + Q_expected = boost::lexical_cast((tgamma(np1)/pow(mp1, np1)).str()); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + } + } + + // CRC Section 5.5, integral 591 + // The parameter p allows us to control the strength of the singularity. + // Rapid convergence is not guaranteed for this function, as the branch cut makes it non-analytic on a disk. + // This converges only when our test type has an extended exponent range as all the area of the integral + // occurs so close to 0 (or 1) that we need abscissa values exceptionally small to find it. + // "There's a lot of room at the bottom". + // We also use a 2 argument functor so that 1-x is evaluated accurately: + if (std::numeric_limits::max_exponent > std::numeric_limits::max_exponent) + { + for (Real p = -0.99; p < 1; p += 0.1) { + auto f = [&](Real x, Real cx)->Real + { + //return pow(x, p) / pow(1 - x, p); + return cx < 0 ? exp(p * (log(x) - boost::math::log1p(-x))) : pow(x, p) / pow(cx, p); + }; + Q = integrator.integrate(f, (Real)0, (Real)1, get_convergence_tolerance(), &error, &L1); + Q_expected = 1 / sinc_pi(p*pi()); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 10 * tol); + } + } + // There is an alternative way to evaluate the above integral: by noticing that all the area of the integral + // is near zero for p < 0 and near 1 for p > 0 we can substitute exp(-x) for x and remap the integral to the + // domain (0, INF). Internally we need to expand out the terms and evaluate using logs to avoid spurous overflow, + // this gives us + // for p > 0: + for (Real p = 0.99; p > 0; p -= 0.1) { + auto f = [&](Real x)->Real + { + return exp(-x * (1 - p) + p * log(-boost::math::expm1(-x))); + }; + Q = integrator.integrate(f, 0, boost::math::tools::max_value(), get_convergence_tolerance(), &error, &L1); + Q_expected = 1 / sinc_pi(p*pi()); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 10 * tol); + } + // and for p < 1: + for (Real p = -0.99; p < 0; p += 0.1) { + auto f = [&](Real x)->Real + { + return exp(-p * log(-boost::math::expm1(-x)) - (1 + p) * x); + }; + Q = integrator.integrate(f, 0, boost::math::tools::max_value(), get_convergence_tolerance(), &error, &L1); + Q_expected = 1 / sinc_pi(p*pi()); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 10 * tol); + } + + // CRC Section 5.5, integral 635 + for (int m = 0; m < 10; ++m) { + auto f = [&](Real x)->Real { return 1/(1 + pow(tan(x), m)); }; + Q = integrator.integrate(f, (Real) 0, half_pi(), get_convergence_tolerance(), &error, &L1); + Q_expected = half_pi()/2; + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + } + + // CRC Section 5.5, integral 637: + // + // When h gets very close to 1, the strength of the singularity gradually increases until we + // no longer have enough exponent range to evaluate the integral.... + // We also have to use the 2-argument functor version of the integrator to avoid + // cancellation error, since the singularity is near PI/2. + // + Real limit = std::numeric_limits::max_exponent > std::numeric_limits::max_exponent + ? .95f : .85f; + for (Real h = 0.01; h < limit; h += 0.1) { + auto f = [&](Real x, Real xc)->Real { return xc > 0 ? pow(1/tan(xc), h) : pow(tan(x), h); }; + Q = integrator.integrate(f, (Real) 0, half_pi(), get_convergence_tolerance(), &error, &L1); + Q_expected = half_pi()/cos(h*half_pi()); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + } + // CRC Section 5.5, integral 637: + // + // Over again, but with argument transformation, we want: + // + // Integral of tan(x)^h over (0, PI/2) + // + // Note that the bulk of the area is next to the singularity at PI/2, + // so we'll start by replacing x by PI/2 - x, and that tan(PI/2 - x) == 1/tan(x) + // so we now have: + // + // Integral of 1/tan(x)^h over (0, PI/2) + // + // Which is almost the ideal form, except that when h is very close to 1 + // we run out of exponent range in evaluating the integral arbitrarily close to 0. + // So now we substitute exp(-x) for x: this stretches out the range of the + // integral to (-log(PI/2), +INF) with the singularity at +INF giving: + // + // Integral of exp(-x)/tan(exp(-x))^h over (-log(PI/2), +INF) + // + // We just need a way to evaluate the function without spurious under/overflow + // in the exp terms. Note that for small x: tan(x) ~= x, so making this + // substitution and evaluating by logs we have: + // + // exp(-x)/tan(exp(-x))^h ~= exp((h - 1) * x) for x > -log(epsion); + // + // Here's how that looks in code: + // + for (Real i = 80; i < 100; ++i) { + Real h = i / 100; + auto f = [&](Real x)->Real + { + if (x > -log(boost::math::tools::epsilon())) + return exp((h - 1) * x); + else + { + Real et = exp(-x); + // Need to deal with numeric instability causing et to be greater than PI/2: + return et >= boost::math::constants::half_pi() ? 0 : et * pow(1 / tan(et), h); + } + }; + Q = integrator.integrate(f, -log(half_pi()), boost::math::tools::max_value(), get_convergence_tolerance(), &error, &L1); + Q_expected = half_pi() / cos(h*half_pi()); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, 5 * tol); + } + + // CRC Section 5.5, integral 670: + + auto f3 = [](Real x)->Real { return sqrt(log(1/x)); }; + Q = integrator.integrate(f3, (Real) 0, (Real) 1, get_convergence_tolerance(), &error, &L1); + Q_expected = root_pi()/2; + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + +} + +template +void test_sf() +{ + using std::sqrt; + // Test some special functions that we already know how to evaluate: + std::cout << "Testing special functions on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10 * boost::math::tools::epsilon(); + auto integrator = get_integrator(); + + // incomplete beta: + if (std::numeric_limits::digits10 < 37) // Otherwise too slow + { + Real a(100), b(15); + auto f = [&](Real x)->Real { return boost::math::ibeta_derivative(a, b, x); }; + BOOST_CHECK_CLOSE_FRACTION(integrator.integrate(f, 0, Real(0.25)), boost::math::ibeta(100, 15, Real(0.25)), tol * 10); + // Check some really extreme versions: + a = 1000; + b = 500; + BOOST_CHECK_CLOSE_FRACTION(integrator.integrate(f, 0, 1), Real(1), tol * 10); + // + // This is as extreme as we can get in this domain: otherwise the function has all it's + // area so close to zero we never get in there no matter how many levels we go down: + // + a = Real(1) / 15; + b = 30; + BOOST_CHECK_CLOSE_FRACTION(integrator.integrate(f, 0, 1), Real(1), tol * 25); + } + + Real x = 2, y = 3, z = 0.5, p = 0.25; + // Elliptic integral RC: + Real Q = integrator.integrate([&](const Real& t) { return 1 / (sqrt(t + x) * (t + y)); }, 0, std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value()) / 2; + BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::ellint_rc(x, y), tol); + // Elliptic Integral RJ: + BOOST_CHECK_CLOSE_FRACTION(Real((Real(3) / 2) * integrator.integrate([&](Real t)->Real { return 1 / (sqrt((t + x) * (t + y) * (t + z)) * (t + p)); }, 0, std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value())), boost::math::ellint_rj(x, y, z, p), tol); + + z = 5.5; + if (std::numeric_limits::digits10 > 40) + tol *= 200; + else if (!std::numeric_limits::is_specialized) + tol *= 3; + // tgamma expressed as an integral: + BOOST_CHECK_CLOSE_FRACTION(integrator.integrate([&](Real t)->Real { using std::pow; using std::exp; return t > 10000 ? Real(0) : Real(pow(t, z - 1) * exp(-t)); }, 0, std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value()), + boost::math::tgamma(z), tol); + BOOST_CHECK_CLOSE_FRACTION(integrator.integrate([](const Real& t) { using std::exp; return exp(-t*t); }, std::numeric_limits::has_infinity ? -std::numeric_limits::infinity() : -boost::math::tools::max_value(), std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : boost::math::tools::max_value()), + boost::math::constants::root_pi(), tol); +} + +template +void test_2_arg() +{ + BOOST_MATH_STD_USING + std::cout << "Testing 2 argument functors on type " << boost::typeindex::type_id().pretty_name() << "\n"; + Real tol = 10 * boost::math::tools::epsilon(); + + auto integrator = get_integrator(); + + // + // There are a whole family of integrals of the general form + // x(1-x)^-N ; N < 1 + // which have all the interesting behaviour near the 2 singularities + // and all converge, see: http://www.wolframalpha.com/input/?i=integrate+(x+*+(1-x))%5E-1%2FN+from+0+to+1 + // + Real Q = integrator.integrate([&](const Real& t, const Real & tc)->Real + { + return tc < 0 ? 1 / sqrt(t * (1-t)) : 1 / sqrt(t * tc); + }, 0, 1); + BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::constants::pi(), tol); + Q = integrator.integrate([&](const Real& t, const Real & tc)->Real + { + return tc < 0 ? 1 / boost::math::cbrt(t * (1-t)) : 1 / boost::math::cbrt(t * tc); + }, 0, 1); + BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::pow<2>(boost::math::tgamma(Real(2) / 3)) / boost::math::tgamma(Real(4) / 3), tol * 3); + // + // We can do the same thing with ((1+x)(1-x))^-N ; N < 1 + // + Q = integrator.integrate([&](const Real& t, const Real & tc)->Real + { + return t < 0 ? 1 / sqrt(-tc * (1-t)) : 1 / sqrt((t + 1) * tc); + }, -1, 1); + BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::constants::pi(), tol); + Q = integrator.integrate([&](const Real& t, const Real & tc)->Real + { + return t < 0 ? 1 / sqrt(-tc * (1-t)) : 1 / sqrt((t + 1) * tc); + }); + BOOST_CHECK_CLOSE_FRACTION(Q, boost::math::constants::pi(), tol); + Q = integrator.integrate([&](const Real& t, const Real & tc)->Real + { + return t < 0 ? 1 / boost::math::cbrt(-tc * (1-t)) : 1 / boost::math::cbrt((t + 1) * tc); + }, -1, 1); + BOOST_CHECK_CLOSE_FRACTION(Q, sqrt(boost::math::constants::pi()) * boost::math::tgamma(Real(2) / 3) / boost::math::tgamma(Real(7) / 6), tol * 10); + Q = integrator.integrate([&](const Real& t, const Real & tc)->Real + { + return t < 0 ? 1 / boost::math::cbrt(-tc * (1-t)) : 1 / boost::math::cbrt((t + 1) * tc); + }); + BOOST_CHECK_CLOSE_FRACTION(Q, sqrt(boost::math::constants::pi()) * boost::math::tgamma(Real(2) / 3) / boost::math::tgamma(Real(7) / 6), tol * 10); + // + // These are taken from above, and do not get to full precision via the single arg functor: + // + auto f7 = [](const Real& t, const Real& tc) { return t < 1 ? sqrt(tan(t)) : sqrt(1/tan(tc)); }; + Real error, L1; + Q = integrator.integrate(f7, (Real)0, (Real)half_pi(), get_convergence_tolerance(), &error, &L1); + Real Q_expected = pi() / root_two(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, Q_expected, tol); + + auto f8 = [](const Real& t, const Real& tc) { return t < 1 ? log(cos(t)) : log(sin(tc)); }; + Q = integrator.integrate(f8, (Real)0, half_pi(), get_convergence_tolerance(), &error, &L1); + Q_expected = -pi()*ln_two()*half(); + BOOST_CHECK_CLOSE_FRACTION(Q, Q_expected, tol); + BOOST_CHECK_CLOSE_FRACTION(L1, -Q_expected, tol); +} + + +BOOST_AUTO_TEST_CASE(tanh_sinh_quadrature_test) +{ +#ifdef GENERATE_CONSTANTS + // + // Generate pre-computed coefficients: + std::cout << std::setprecision(35); + print_levels(generate_constants(10, 8), "L"); + +#else + +#ifdef TEST1 + + test_right_limit_infinite(); + test_left_limit_infinite(); + test_linear(); + test_quadratic(); + test_singular(); + test_ca(); + test_three_quadrature_schemes_examples(); + test_horrible(); + test_integration_over_real_line(); + test_nr_examples(); +#endif +#ifdef TEST1A + test_early_termination(); + test_crc(); + test_2_arg(); + +#endif +#ifdef TEST2 + test_right_limit_infinite(); + test_left_limit_infinite(); + test_linear(); + test_quadratic(); + test_singular(); + test_ca(); + test_three_quadrature_schemes_examples(); + test_horrible(); + test_integration_over_real_line(); + test_nr_examples(); + test_early_termination(); + test_sf(); + test_2_arg(); +#endif +#ifdef TEST2A + test_crc(); +#endif + +#ifdef TEST3 + + test_right_limit_infinite(); + test_left_limit_infinite(); + test_linear(); + test_quadratic(); + test_singular(); + test_ca(); + test_three_quadrature_schemes_examples(); + test_horrible(); + test_integration_over_real_line(); + test_nr_examples(); + test_early_termination(); + test_sf(); + test_2_arg(); +#endif +#ifdef TEST3A + test_crc(); + +#endif + +#ifdef TEST4 + + test_right_limit_infinite(); + test_left_limit_infinite(); + test_linear(); + test_quadratic(); + test_singular(); + test_ca(); + test_three_quadrature_schemes_examples(); + test_horrible(); + test_nr_examples(); + test_early_termination(); + test_crc(); + test_sf(); + test_2_arg(); + +#endif +#ifdef TEST5 + + test_sf(); + test_sf(); + test_sf > >(); + +#endif +#ifdef TEST6 + + test_right_limit_infinite(); + test_left_limit_infinite(); + test_linear(); + test_quadratic(); + test_singular(); + test_ca(); + test_three_quadrature_schemes_examples(); + test_horrible(); + test_integration_over_real_line(); + test_nr_examples(); + test_early_termination(); + test_sf(); + test_2_arg(); +#endif +#ifdef TEST6A + test_crc(); + +#endif +#ifdef TEST7 + test_sf(); +#endif +#if defined(TEST8) && defined(BOOST_HAS_FLOAT128) + + test_right_limit_infinite(); + test_left_limit_infinite(); + test_linear(); + test_quadratic(); + test_singular(); + test_ca(); + test_three_quadrature_schemes_examples(); + test_horrible(); + test_integration_over_real_line(); + test_nr_examples(); + test_early_termination(); + test_crc(); + test_sf(); + test_2_arg(); + +#endif +#endif +} + +#else + +int main() { return 0; } + +#endif