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318 lines
9.8 KiB
318 lines
9.8 KiB
// Copyright (c) 2006 Xiaogang Zhang |
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// Copyright (c) 2006 John Maddock |
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// Use, modification and distribution are subject to the |
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// Boost Software License, Version 1.0. (See accompanying file |
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt) |
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// |
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// History: |
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// XZ wrote the original of this file as part of the Google |
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// Summer of Code 2006. JM modified it to fit into the |
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// Boost.Math conceptual framework better, and to correctly |
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// handle the various corner cases. |
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// |
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#ifndef BOOST_MATH_ELLINT_3_HPP |
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#define BOOST_MATH_ELLINT_3_HPP |
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#ifdef _MSC_VER |
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#pragma once |
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#endif |
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#include <boost/math/special_functions/ellint_rf.hpp> |
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#include <boost/math/special_functions/ellint_rj.hpp> |
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#include <boost/math/special_functions/ellint_1.hpp> |
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#include <boost/math/special_functions/ellint_2.hpp> |
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#include <boost/math/special_functions/log1p.hpp> |
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#include <boost/math/constants/constants.hpp> |
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#include <boost/math/policies/error_handling.hpp> |
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#include <boost/math/tools/workaround.hpp> |
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// Elliptic integrals (complete and incomplete) of the third kind |
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// Carlson, Numerische Mathematik, vol 33, 1 (1979) |
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namespace boost { namespace math { |
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namespace detail{ |
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template <typename T, typename Policy> |
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T ellint_pi_imp(T v, T k, T vc, const Policy& pol); |
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// Elliptic integral (Legendre form) of the third kind |
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template <typename T, typename Policy> |
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T ellint_pi_imp(T v, T phi, T k, T vc, const Policy& pol) |
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{ |
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// Note vc = 1-v presumably without cancellation error. |
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T value, x, y, z, p, t; |
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BOOST_MATH_STD_USING |
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using namespace boost::math::tools; |
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using namespace boost::math::constants; |
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static const char* function = "boost::math::ellint_3<%1%>(%1%,%1%,%1%)"; |
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if (abs(k) > 1) |
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{ |
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return policies::raise_domain_error<T>(function, |
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"Got k = %1%, function requires |k| <= 1", k, pol); |
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} |
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T sphi = sin(fabs(phi)); |
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if(v > 1 / (sphi * sphi)) |
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{ |
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// Complex result is a domain error: |
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return policies::raise_domain_error<T>(function, |
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"Got v = %1%, but result is complex for v > 1 / sin^2(phi)", v, pol); |
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} |
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// Special cases first: |
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if(v == 0) |
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{ |
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// A&S 17.7.18 & 19 |
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return (k == 0) ? phi : ellint_f_imp(phi, k, pol); |
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} |
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if(phi == constants::pi<T>() / 2) |
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{ |
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// Have to filter this case out before the next |
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// special case, otherwise we might get an infinity from |
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// tan(phi). |
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// Also note that since we can't represent PI/2 exactly |
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// in a T, this is a bit of a guess as to the users true |
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// intent... |
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// |
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return ellint_pi_imp(v, k, vc, pol); |
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} |
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if(k == 0) |
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{ |
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// A&S 17.7.20: |
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if(v < 1) |
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{ |
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T vcr = sqrt(vc); |
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return atan(vcr * tan(phi)) / vcr; |
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} |
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else if(v == 1) |
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{ |
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return tan(phi); |
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} |
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else |
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{ |
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// v > 1: |
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T vcr = sqrt(-vc); |
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T arg = vcr * tan(phi); |
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return (boost::math::log1p(arg, pol) - boost::math::log1p(-arg, pol)) / (2 * vcr); |
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} |
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} |
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if(v < 0) |
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{ |
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// |
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// If we don't shift to 0 <= v <= 1 we get |
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// cancellation errors later on. Use |
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// A&S 17.7.15/16 to shift to v > 0: |
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// |
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T k2 = k * k; |
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T N = (k2 - v) / (1 - v); |
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T Nm1 = (1 - k2) / (1 - v); |
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T p2 = sqrt(-v * (k2 - v) / (1 - v)); |
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T delta = sqrt(1 - k2 * sphi * sphi); |
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T result = ellint_pi_imp(N, phi, k, Nm1, pol); |
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result *= sqrt(Nm1 * (1 - k2 / N)); |
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result += ellint_f_imp(phi, k, pol) * k2 / p2; |
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result += atan((p2/2) * sin(2 * phi) / delta); |
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result /= sqrt((1 - v) * (1 - k2 / v)); |
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return result; |
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} |
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#if 0 // disabled but retained for future reference: see below. |
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if(v > 1) |
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{ |
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// |
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// If v > 1 we can use the identity in A&S 17.7.7/8 |
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// to shift to 0 <= v <= 1. Unfortunately this |
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// identity appears only to function correctly when |
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// 0 <= phi <= pi/2, but it's when phi is outside that |
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// range that we really need it: That's when |
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// Carlson's formula fails, and what's more the periodicity |
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// reduction used below on phi doesn't work when v > 1. |
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// |
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// So we're stuck... the code is archived here in case |
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// some bright spart can figure out the fix. |
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// |
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T k2 = k * k; |
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T N = k2 / v; |
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T Nm1 = (v - k2) / v; |
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T p1 = sqrt((-vc) * (1 - k2 / v)); |
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T delta = sqrt(1 - k2 * sphi * sphi); |
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// |
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// These next two terms have a large amount of cancellation |
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// so it's not clear if this relation is useable even if |
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// the issues with phi > pi/2 can be fixed: |
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// |
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T result = -ellint_pi_imp(N, phi, k, Nm1); |
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result += ellint_f_imp(phi, k); |
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// |
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// This log term gives the complex result when |
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// n > 1/sin^2(phi) |
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// However that case is dealt with as an error above, |
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// so we should always get a real result here: |
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// |
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result += log((delta + p1 * tan(phi)) / (delta - p1 * tan(phi))) / (2 * p1); |
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return result; |
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} |
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#endif |
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// Carlson's algorithm works only for |phi| <= pi/2, |
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// use the integrand's periodicity to normalize phi |
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// |
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// Xiaogang's original code used a cast to long long here |
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// but that fails if T has more digits than a long long, |
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// so rewritten to use fmod instead: |
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// |
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if(fabs(phi) > 1 / tools::epsilon<T>()) |
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{ |
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if(v > 1) |
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return policies::raise_domain_error<T>( |
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function, |
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"Got v = %1%, but this is only supported for 0 <= phi <= pi/2", v, pol); |
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// |
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// Phi is so large that phi%pi is necessarily zero (or garbage), |
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// just return the second part of the duplication formula: |
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// |
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value = 2 * fabs(phi) * ellint_pi_imp(v, k, vc, pol) / constants::pi<T>(); |
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} |
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else |
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{ |
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T rphi = boost::math::tools::fmod_workaround(T(fabs(phi)), T(constants::pi<T>() / 2)); |
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T m = floor((2 * fabs(phi)) / constants::pi<T>()); |
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int sign = 1; |
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if(boost::math::tools::fmod_workaround(m, T(2)) > 0.5) |
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{ |
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m += 1; |
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sign = -1; |
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rphi = constants::pi<T>() / 2 - rphi; |
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} |
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T sinp = sin(rphi); |
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T cosp = cos(rphi); |
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x = cosp * cosp; |
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t = sinp * sinp; |
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y = 1 - k * k * t; |
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z = 1; |
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if(v * t < 0.5) |
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p = 1 - v * t; |
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else |
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p = x + vc * t; |
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value = sign * sinp * (ellint_rf_imp(x, y, z, pol) + v * t * ellint_rj_imp(x, y, z, p, pol) / 3); |
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if((m > 0) && (vc > 0)) |
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value += m * ellint_pi_imp(v, k, vc, pol); |
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} |
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if (phi < 0) |
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{ |
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value = -value; // odd function |
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} |
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return value; |
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} |
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// Complete elliptic integral (Legendre form) of the third kind |
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template <typename T, typename Policy> |
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T ellint_pi_imp(T v, T k, T vc, const Policy& pol) |
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{ |
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// Note arg vc = 1-v, possibly without cancellation errors |
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BOOST_MATH_STD_USING |
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using namespace boost::math::tools; |
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static const char* function = "boost::math::ellint_pi<%1%>(%1%,%1%)"; |
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if (abs(k) >= 1) |
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{ |
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return policies::raise_domain_error<T>(function, |
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"Got k = %1%, function requires |k| <= 1", k, pol); |
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} |
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if(vc <= 0) |
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{ |
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// Result is complex: |
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return policies::raise_domain_error<T>(function, |
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"Got v = %1%, function requires v < 1", v, pol); |
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} |
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if(v == 0) |
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{ |
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return (k == 0) ? boost::math::constants::pi<T>() / 2 : ellint_k_imp(k, pol); |
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} |
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if(v < 0) |
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{ |
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T k2 = k * k; |
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T N = (k2 - v) / (1 - v); |
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T Nm1 = (1 - k2) / (1 - v); |
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T p2 = sqrt(-v * (k2 - v) / (1 - v)); |
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T result = boost::math::detail::ellint_pi_imp(N, k, Nm1, pol); |
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result *= sqrt(Nm1 * (1 - k2 / N)); |
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result += ellint_k_imp(k, pol) * k2 / p2; |
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result /= sqrt((1 - v) * (1 - k2 / v)); |
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return result; |
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} |
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T x = 0; |
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T y = 1 - k * k; |
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T z = 1; |
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T p = vc; |
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T value = ellint_rf_imp(x, y, z, pol) + v * ellint_rj_imp(x, y, z, p, pol) / 3; |
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return value; |
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} |
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template <class T1, class T2, class T3> |
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inline typename tools::promote_args<T1, T2, T3>::type ellint_3(T1 k, T2 v, T3 phi, const mpl::false_&) |
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{ |
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return boost::math::ellint_3(k, v, phi, policies::policy<>()); |
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} |
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template <class T1, class T2, class Policy> |
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inline typename tools::promote_args<T1, T2>::type ellint_3(T1 k, T2 v, const Policy& pol, const mpl::true_&) |
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{ |
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typedef typename tools::promote_args<T1, T2>::type result_type; |
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typedef typename policies::evaluation<result_type, Policy>::type value_type; |
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return policies::checked_narrowing_cast<result_type, Policy>( |
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detail::ellint_pi_imp( |
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static_cast<value_type>(v), |
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static_cast<value_type>(k), |
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static_cast<value_type>(1-v), |
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pol), "boost::math::ellint_3<%1%>(%1%,%1%)"); |
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} |
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} // namespace detail |
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template <class T1, class T2, class T3, class Policy> |
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inline typename tools::promote_args<T1, T2, T3>::type ellint_3(T1 k, T2 v, T3 phi, const Policy& pol) |
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{ |
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typedef typename tools::promote_args<T1, T2, T3>::type result_type; |
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typedef typename policies::evaluation<result_type, Policy>::type value_type; |
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return policies::checked_narrowing_cast<result_type, Policy>( |
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detail::ellint_pi_imp( |
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static_cast<value_type>(v), |
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static_cast<value_type>(phi), |
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static_cast<value_type>(k), |
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static_cast<value_type>(1-v), |
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pol), "boost::math::ellint_3<%1%>(%1%,%1%,%1%)"); |
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} |
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template <class T1, class T2, class T3> |
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typename detail::ellint_3_result<T1, T2, T3>::type ellint_3(T1 k, T2 v, T3 phi) |
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{ |
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typedef typename policies::is_policy<T3>::type tag_type; |
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return detail::ellint_3(k, v, phi, tag_type()); |
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} |
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template <class T1, class T2> |
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inline typename tools::promote_args<T1, T2>::type ellint_3(T1 k, T2 v) |
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{ |
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return ellint_3(k, v, policies::policy<>()); |
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} |
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}} // namespaces |
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#endif // BOOST_MATH_ELLINT_3_HPP |
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