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view tango/tango/math/Elliptic.d @ 183:3cdf4b0c75a1 trunk
[svn r199] Fixed: still some small issues with string literals implicitly converting to different pointer types. Should be fixed now!
author | lindquist |
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date | Wed, 07 May 2008 20:22:42 +0200 |
parents | 1700239cab2e |
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/** * Elliptic integrals. * The functions are named similarly to the names used in Mathematica. * * License: BSD style: $(LICENSE) * Copyright: Based on the CEPHES math library, which is * Copyright (C) 1994 Stephen L. Moshier (moshier@world.std.com). * Authors: Stephen L. Moshier (original C code). Conversion to D by Don Clugston * * References: * $(LINK http://en.wikipedia.org/wiki/Elliptic_integral) * * Eric W. Weisstein. "Elliptic Integral of the First Kind." From MathWorld--A Wolfram Web Resource. $(LINK http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html) * * $(LINK http://www.netlib.org/cephes/ldoubdoc.html) * * Macros: * TABLE_SV = <table border=1 cellpadding=4 cellspacing=0> * <caption>Special Values</caption> * $0</table> * SVH = $(TR $(TH $1) $(TH $2)) * SV = $(TR $(TD $1) $(TD $2)) * GAMMA = Γ * INTEGRATE = $(BIG ∫<sub>$(SMALL $1)</sub><sup>$2</sup>) * POWER = $1<sup>$2</sup> * NAN = $(RED NAN) */ /** * Macros: * TABLE_SV = <table border=1 cellpadding=4 cellspacing=0> * <caption>Special Values</caption> * $0</table> * SVH = $(TR $(TH $1) $(TH $2)) * SV = $(TR $(TD $1) $(TD $2)) * * NAN = $(RED NAN) * SUP = <span style="vertical-align:super;font-size:smaller">$0</span> * GAMMA = Γ * INTEGRAL = ∫ * INTEGRATE = $(BIG ∫<sub>$(SMALL $1)</sub><sup>$2</sup>) * POWER = $1<sup>$2</sup> * BIGSUM = $(BIG Σ <sup>$2</sup><sub>$(SMALL $1)</sub>) * CHOOSE = $(BIG () <sup>$(SMALL $1)</sup><sub>$(SMALL $2)</sub> $(BIG )) */ module tango.math.Elliptic; import tango.math.Math; import tango.math.IEEE; /* These functions are based on code from: Cephes Math Library, Release 2.3: October, 1995 Copyright 1984, 1987, 1995 by Stephen L. Moshier */ /** * Incomplete elliptic integral of the first kind * * Approximates the integral * F(phi | m) = $(INTEGRATE 0, phi) dt/ (sqrt( 1- m $(POWER sin, 2) t)) * * of amplitude phi and modulus m, using the arithmetic - * geometric mean algorithm. */ real ellipticF(real phi, real m ) { real a, b, c, e, temp, t, K; int d, mod, sign, npio2; if( m == 0.0L ) return phi; a = 1.0L - m; if( a == 0.0L ) { if ( fabs(phi) >= PI_2 ) return real.infinity; return log( tan( 0.5L*(PI_2 + phi) ) ); } npio2 = cast(int)floor( phi/PI_2 ); if ( npio2 & 1 ) npio2 += 1; if ( npio2 ) { K = ellipticKComplete( a ); phi = phi - npio2 * PI_2; } else K = 0.0L; if( phi < 0.0L ){ phi = -phi; sign = -1; } else sign = 0; b = sqrt(a); t = tan( phi ); if( fabs(t) > 10.0L ) { /* Transform the amplitude */ e = 1.0L/(b*t); /* ... but avoid multiple recursions. */ if( fabs(e) < 10.0L ){ e = atan(e); if( npio2 == 0 ) K = ellipticKComplete( a ); temp = K - ellipticF( e, m ); goto done; } } a = 1.0L; c = sqrt(m); d = 1; mod = 0; while( fabs(c/a) > real.epsilon ) { temp = b/a; phi = phi + atan(t*temp) + mod * PI; mod = cast(int)((phi + PI_2)/PI); t = t * ( 1.0L + temp )/( 1.0L - temp * t * t ); c = 0.5L * ( a - b ); temp = sqrt( a * b ); a = 0.5L * ( a + b ); b = temp; d += d; } temp = (atan(t) + mod * PI)/(d * a); done: if ( sign < 0 ) temp = -temp; temp += npio2 * K; return temp; } /** * Incomplete elliptic integral of the second kind * * Approximates the integral * * E(phi | m) = $(INTEGRATE 0, phi) sqrt( 1- m $(POWER sin, 2) t) dt * * of amplitude phi and modulus m, using the arithmetic - * geometric mean algorithm. */ real ellipticE(real phi, real m ) { real a, b, c, e, temp, t, E; int d, mod, npio2, sign; if ( m == 0.0L ) return phi; real lphi = phi; npio2 = cast(int)floor( lphi/PI_2 ); if( npio2 & 1 ) npio2 += 1; lphi = lphi - npio2 * PI_2; if( lphi < 0.0L ){ lphi = -lphi; sign = -1; } else { sign = 1; } a = 1.0L - m; E = ellipticEComplete( a ); if( a == 0.0L ) { temp = sin( lphi ); goto done; } t = tan( lphi ); b = sqrt(a); if ( fabs(t) > 10.0L ) { /* Transform the amplitude */ e = 1.0L/(b*t); /* ... but avoid multiple recursions. */ if( fabs(e) < 10.0L ){ e = atan(e); temp = E + m * sin( lphi ) * sin( e ) - ellipticE( e, m ); goto done; } } c = sqrt(m); a = 1.0L; d = 1; e = 0.0L; mod = 0; while( fabs(c/a) > real.epsilon ) { temp = b/a; lphi = lphi + atan(t*temp) + mod * PI; mod = cast(int)((lphi + PI_2)/PI); t = t * ( 1.0L + temp )/( 1.0L - temp * t * t ); c = 0.5L*( a - b ); temp = sqrt( a * b ); a = 0.5L*( a + b ); b = temp; d += d; e += c * sin(lphi); } temp = E / ellipticKComplete( 1.0L - m ); temp *= (atan(t) + mod * PI)/(d * a); temp += e; done: if( sign < 0 ) temp = -temp; temp += npio2 * E; return temp; } /** * Complete elliptic integral of the first kind * * Approximates the integral * * K(m) = $(INTEGRATE 0, &pi/2) dt/ (sqrt( 1- m $(POWER sin, 2) t)) * * where m = 1 - x, using the approximation * * P(x) - log x Q(x). * * The argument x is used rather than m so that the logarithmic * singularity at x = 1 will be shifted to the origin; this * preserves maximum accuracy. * * x must be in the range * 0 <= x <= 1 * * This is equivalent to ellipticF(PI_2, 1-x). * * K(0) = &pi/2. */ real ellipticKComplete(real x) in { // assert(x>=0.0L && x<=1.0L); } body{ const real [] P = [ 0x1.62e42fefa39ef35ap+0, // 1.3862943611198906189 0x1.8b90bfbe8ed811fcp-4, // 0.096573590279993142323 0x1.fa05af797624c586p-6, // 0.030885144578720423267 0x1.e979cdfac7249746p-7, // 0.01493761594388688915 0x1.1f4cc8890cff803cp-7, // 0.0087676982094322259125 0x1.7befb3bb1fa978acp-8, // 0.0057973684116620276454 0x1.2c2566aa1d5fe6b8p-8, // 0.0045798659940508010431 0x1.7333514e7fe57c98p-8, // 0.0056640695097481470287 0x1.09292d1c8621348cp-7, // 0.0080920667906392630755 0x1.b89ab5fe793a6062p-8, // 0.0067230886765842542487 0x1.28e9c44dc5e26e66p-9, // 0.002265267575136470585 0x1.c2c43245d445addap-13, // 0.00021494216542320112406 0x1.4ee247035a03e13p-20 // 1.2475397291548388385e-06 ]; const real [] Q = [ 0x1p-1, // 0.5 0x1.fffffffffff635eap-4, // 0.12499999999999782631 0x1.1fffffff8a2bea1p-4, // 0.070312499993302227507 0x1.8ffffe6f40ec2078p-5, // 0.04882812208418620146 0x1.323f4dbf7f4d0c2ap-5, // 0.037383701182969303058 0x1.efe8a028541b50bp-6, // 0.030267864612427881354 0x1.9d58c49718d6617cp-6, // 0.025228683455123323041 0x1.4d1a8d2292ff6e2ep-6, // 0.020331037356569904872 0x1.b637687027d664aap-7, // 0.013373304362459048444 0x1.687a640ae5c71332p-8, // 0.0055004591221382442135 0x1.0f9c30a94a1dcb4ep-10, // 0.001036110372590318803 0x1.d321746708e92d48p-15 // 5.568631677757315399e-05 ]; const real LOG4 = 0x1.62e42fefa39ef358p+0; // log(4) if( x > real.epsilon ) return poly(x,P) - log(x) * poly(x,Q); if ( x == 0.0L ) return real.infinity; return LOG4 - 0.5L * log(x); } /** * Complete elliptic integral of the second kind * * Approximates the integral * * E(m) = $(INTEGRATE 0, &pi/2) sqrt( 1- m $(POWER sin, 2) t) dt * * Where m = 1 - m1, using the approximation * * P(x) - x log x Q(x). * * Though there are no singularities, the argument m1 is used * rather than m for compatibility with ellipticKComplete(). * * E(1) = 1; E(0) = &pi/2. * m must be in the range 0 <= m <= 1. */ real ellipticEComplete(real x) in { assert(x>=0 && x<=1.0); } body { const real [] P = [ 0x1.c5c85fdf473f78f2p-2, // 0.44314718055994670505 0x1.d1591f9e9a66477p-5, // 0.056805192715569305834 0x1.65af6a7a61f587cp-6, // 0.021831373198011179718 0x1.7a4d48ed00d5745ap-7, // 0.011544857605264509506 0x1.d4f5fe4f93b60688p-8, // 0.0071557756305783152481 0x1.4cb71c73bac8656ap-8, // 0.0050768322432573952962 0x1.4a9167859a1d0312p-8, // 0.0050440671671840438539 0x1.dd296daa7b1f5b7ap-8, // 0.0072809117068399675418 0x1.04f2c29224ba99b6p-7, // 0.0079635095646944542686 0x1.0f5820e2d80194d8p-8, // 0.0041403847015715420009 0x1.95ee634752ca69b6p-11, // 0.00077425232385887751162 0x1.0c58aa9ab404f4fp-15 // 3.1989378120323412946e-05 ]; const real [] Q = [ 0x1.ffffffffffffb1cep-3, // 0.24999999999999986434 0x1.7ffffffff29eaa0cp-4, // 0.093749999999239422678 0x1.dfffffbd51eb098p-5, // 0.058593749514839092674 0x1.5dffd791cb834c92p-5, // 0.04272453406734691973 0x1.1397b63c2f09a8ep-5, // 0.033641677787700181541 0x1.c567cde5931e75bcp-6, // 0.02767367465121309044 0x1.75e0cae852be9ddcp-6, // 0.022819708015315777007 0x1.12bb968236d4e434p-6, // 0.016768357258894633433 0x1.1f6572c1c402d07cp-7, // 0.0087706384979640787504 0x1.452c6909f88b8306p-9, // 0.0024808767529843311337 0x1.1f7504e72d664054p-12, // 0.00027414045912208516032 0x1.ad17054dc46913e2p-18 // 6.3939381343012054851e-06 ]; if (x==0) return 1.0L; return 1.0L + x * poly(x,P) - log(x) * (x * poly(x,Q) ); } unittest { assert( ellipticF(1, 0)==1); assert(ellipticEComplete(0)==1); assert(ellipticEComplete(1)==PI_2); assert(feqrel(ellipticKComplete(1),PI_2)>= real.mant_dig-1); assert(ellipticKComplete(0)==real.infinity); // assert(ellipticKComplete(1)==0); //-real.infinity); real x=0.5653L; assert(ellipticKComplete(1-x) == ellipticF(PI_2, x) ); assert(ellipticEComplete(1-x) == ellipticE(PI_2, x) ); }