Mercurial > projects > ldc
diff tango/tango/math/GammaFunction.d @ 132:1700239cab2e trunk
[svn r136] MAJOR UNSTABLE UPDATE!!!
Initial commit after moving to Tango instead of Phobos.
Lots of bugfixes...
This build is not suitable for most things.
author | lindquist |
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date | Fri, 11 Jan 2008 17:57:40 +0100 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/tango/tango/math/GammaFunction.d Fri Jan 11 17:57:40 2008 +0100 @@ -0,0 +1,1463 @@ +/** + * Implementation of the gamma and beta functions, and their integrals. + * + * 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 + * + * +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) + */ +module tango.math.GammaFunction; +private import tango.math.Math; +private import tango.math.IEEE; +private import tango.math.ErrorFunction; + +version(Windows) { // Some tests only pass on DMD Windows + version(DigitalMars) { + version = FailsOnLinux; +} +} + +//------------------------------------------------------------------ + +/// The maximum value of x for which gamma(x) < real.infinity. +const real MAXGAMMA = 1755.5483429L; + +private { + +const real SQRT2PI = 2.50662827463100050242E0L; // sqrt(2pi) + +// Polynomial approximations for gamma and loggamma. + +const real GammaNumeratorCoeffs[] = [ 1.0, + 0x1.acf42d903366539ep-1, 0x1.73a991c8475f1aeap-2, 0x1.c7e918751d6b2a92p-4, + 0x1.86d162cca32cfe86p-6, 0x1.0c378e2e6eaf7cd8p-8, 0x1.dc5c66b7d05feb54p-12, + 0x1.616457b47e448694p-15 +]; + +const real GammaDenominatorCoeffs[] = [ 1.0, + 0x1.a8f9faae5d8fc8bp-2, -0x1.cb7895a6756eebdep-3, -0x1.7b9bab006d30652ap-5, + 0x1.c671af78f312082ep-6, -0x1.a11ebbfaf96252dcp-11, -0x1.447b4d2230a77ddap-10, + 0x1.ec1d45bb85e06696p-13,-0x1.d4ce24d05bd0a8e6p-17 +]; + +const real GammaSmallCoeffs[] = [ 1.0, + 0x1.2788cfc6fb618f52p-1, -0x1.4fcf4026afa2f7ecp-1, -0x1.5815e8fa24d7e306p-5, + 0x1.5512320aea2ad71ap-3, -0x1.59af0fb9d82e216p-5, -0x1.3b4b61d3bfdf244ap-7, + 0x1.d9358e9d9d69fd34p-8, -0x1.38fc4bcbada775d6p-10 +]; + +const real GammaSmallNegCoeffs[] = [ -1.0, + 0x1.2788cfc6fb618f54p-1, 0x1.4fcf4026afa2bc4cp-1, -0x1.5815e8fa2468fec8p-5, + -0x1.5512320baedaf4b6p-3, -0x1.59af0fa283baf07ep-5, 0x1.3b4a70de31e05942p-7, + 0x1.d9398be3bad13136p-8, 0x1.291b73ee05bcbba2p-10 +]; + +const real logGammaStirlingCoeffs[] = [ + 0x1.5555555555553f98p-4, -0x1.6c16c16c07509b1p-9, 0x1.a01a012461cbf1e4p-11, + -0x1.3813089d3f9d164p-11, 0x1.b911a92555a277b8p-11, -0x1.ed0a7b4206087b22p-10, + 0x1.402523859811b308p-8 +]; + +const real logGammaNumerator[] = [ + -0x1.0edd25913aaa40a2p+23, -0x1.31c6ce2e58842d1ep+24, -0x1.f015814039477c3p+23, + -0x1.74ffe40c4b184b34p+22, -0x1.0d9c6d08f9eab55p+20, -0x1.54c6b71935f1fc88p+16, + -0x1.0e761b42932b2aaep+11 +]; + +const real logGammaDenominator[] = [ + -0x1.4055572d75d08c56p+24, -0x1.deeb6013998e4d76p+24, -0x1.106f7cded5dcc79ep+24, + -0x1.25e17184848c66d2p+22, -0x1.301303b99a614a0ap+19, -0x1.09e76ab41ae965p+15, + -0x1.00f95ced9e5f54eep+9, 1.0 +]; + +/* + * Helper function: Gamma function computed by Stirling's formula. + * + * Stirling's formula for the gamma function is: + * + * $(GAMMA)(x) = sqrt(2 π) x<sup>x-0.5</sup> exp(-x) (1 + 1/x P(1/x)) + * + */ +real gammaStirling(real x) +{ + // CEPHES code Copyright 1994 by Stephen L. Moshier + + const real SmallStirlingCoeffs[] = [ + 0x1.55555555555543aap-4, 0x1.c71c71c720dd8792p-9, -0x1.5f7268f0b5907438p-9, + -0x1.e13cd410e0477de6p-13, 0x1.9b0f31643442616ep-11, 0x1.2527623a3472ae08p-14, + -0x1.37f6bc8ef8b374dep-11,-0x1.8c968886052b872ap-16, 0x1.76baa9c6d3eeddbcp-11 + ]; + + const real LargeStirlingCoeffs[] = [ 1.0L, + 8.33333333333333333333E-2L, 3.47222222222222222222E-3L, + -2.68132716049382716049E-3L, -2.29472093621399176955E-4L, + 7.84039221720066627474E-4L, 6.97281375836585777429E-5L + ]; + + real w = 1.0L/x; + real y = exp(x); + if ( x > 1024.0L ) { + // For large x, use rational coefficients from the analytical expansion. + w = poly(w, LargeStirlingCoeffs); + // Avoid overflow in pow() + real v = pow( x, 0.5L * x - 0.25L ); + y = v * (v / y); + } + else { + w = 1.0L + w * poly( w, SmallStirlingCoeffs); + y = pow( x, x - 0.5L ) / y; + } + y = SQRT2PI * y * w; + return y; +} + +} // private + +/**************** + * The sign of $(GAMMA)(x). + * + * Returns -1 if $(GAMMA)(x) < 0, +1 if $(GAMMA)(x) > 0, + * $(NAN) if sign is indeterminate. + */ +real sgnGamma(real x) +{ + /* Author: Don Clugston. */ + if (isNaN(x)) return x; + if (x > 0) return 1.0; + if (x < -1/real.epsilon) { + // Large negatives lose all precision + return NaN(TANGO_NAN.SGNGAMMA); + } +// if (remquo(x, -1.0, n) == 0) { + long n = rndlong(x); + if (x == n) { + return x == 0 ? copysign(1, x) : NaN(TANGO_NAN.SGNGAMMA); + } + return n & 1 ? 1.0 : -1.0; +} + +debug(UnitTest) { +unittest { + assert(sgnGamma(5.0) == 1.0); + assert(isNaN(sgnGamma(-3.0))); + assert(sgnGamma(-0.1) == -1.0); + assert(sgnGamma(-55.1) == 1.0); + assert(isNaN(sgnGamma(-real.infinity))); + assert(isIdentical(sgnGamma(NaN(0xABC)), NaN(0xABC))); +} +} + +/***************************************************** + * The Gamma function, $(GAMMA)(x) + * + * $(GAMMA)(x) is a generalisation of the factorial function + * to real and complex numbers. + * Like x!, $(GAMMA)(x+1) = x*$(GAMMA)(x). + * + * Mathematically, if z.re > 0 then + * $(GAMMA)(z) = $(INTEGRATE 0, ∞) $(POWER t, z-1)$(POWER e, -t) dt + * + * $(TABLE_SV + * $(SVH x, $(GAMMA)(x) ) + * $(SV $(NAN), $(NAN) ) + * $(SV ±0.0, ±∞) + * $(SV integer > 0, (x-1)! ) + * $(SV integer < 0, $(NAN) ) + * $(SV +∞, +∞ ) + * $(SV -∞, $(NAN) ) + * ) + */ +real gamma(real x) +{ +/* Based on code from the CEPHES library. + * CEPHES code Copyright 1994 by Stephen L. Moshier + * + * Arguments |x| <= 13 are reduced by recurrence and the function + * approximated by a rational function of degree 7/8 in the + * interval (2,3). Large arguments are handled by Stirling's + * formula. Large negative arguments are made positive using + * a reflection formula. + */ + + real q, z; + if (isNaN(x)) return x; + if (x == -x.infinity) return NaN(TANGO_NAN.GAMMA_DOMAIN); + if ( fabs(x) > MAXGAMMA ) return real.infinity; + if (x==0) return 1.0/x; // +- infinity depending on sign of x, create an exception. + + q = fabs(x); + + if ( q > 13.0L ) { + // Large arguments are handled by Stirling's + // formula. Large negative arguments are made positive using + // the reflection formula. + + if ( x < 0.0L ) { + int sgngam = 1; // sign of gamma. + real p = floor(q); + if (p == q) + return NaN(TANGO_NAN.GAMMA_DOMAIN); // poles for all integers <0. + int intpart = cast(int)(p); + if ( (intpart & 1) == 0 ) + sgngam = -1; + z = q - p; + if ( z > 0.5L ) { + p += 1.0L; + z = q - p; + } + z = q * sin( PI * z ); + z = fabs(z) * gammaStirling(q); + if ( z <= PI/real.max ) return sgngam * real.infinity; + return sgngam * PI/z; + } else { + return gammaStirling(x); + } + } + + // Arguments |x| <= 13 are reduced by recurrence and the function + // approximated by a rational function of degree 7/8 in the + // interval (2,3). + + z = 1.0L; + while ( x >= 3.0L ) { + x -= 1.0L; + z *= x; + } + + while ( x < -0.03125L ) { + z /= x; + x += 1.0L; + } + + if ( x <= 0.03125L ) { + if ( x == 0.0L ) + return NaN(TANGO_NAN.GAMMA_POLE); + else { + if ( x < 0.0L ) { + x = -x; + return z / (x * poly( x, GammaSmallNegCoeffs )); + } else { + return z / (x * poly( x, GammaSmallCoeffs )); + } + } + } + + while ( x < 2.0L ) { + z /= x; + x += 1.0L; + } + if ( x == 2.0L ) return z; + + x -= 2.0L; + return z * poly( x, GammaNumeratorCoeffs ) / poly( x, GammaDenominatorCoeffs ); +} + +debug(UnitTest) { +unittest { + // gamma(n) = factorial(n-1) if n is an integer. + real fact = 1.0L; + for (int i=1; fact<real.max; ++i) { + // Require exact equality for small factorials + if (i<14) assert(gamma(i*1.0L) == fact); + version(FailsOnLinux) assert(feqrel(gamma(i*1.0L), fact) > real.mant_dig-15); + fact *= (i*1.0L); + } + assert(gamma(0.0) == real.infinity); + assert(gamma(-0.0) == -real.infinity); + assert(isNaN(gamma(-1.0))); + assert(isNaN(gamma(-15.0))); + assert(isIdentical(gamma(NaN(0xABC)), NaN(0xABC))); + assert(gamma(real.infinity) == real.infinity); + assert(gamma(real.max) == real.infinity); + assert(isNaN(gamma(-real.infinity))); + assert(gamma(real.min*real.epsilon) == real.infinity); + assert(gamma(MAXGAMMA)< real.infinity); + assert(gamma(MAXGAMMA*2) == real.infinity); + + // Test some high-precision values (50 decimal digits) + const real SQRT_PI = 1.77245385090551602729816748334114518279754945612238L; + + version(FailsOnLinux) assert(feqrel(gamma(0.5L), SQRT_PI) == real.mant_dig); + + assert(feqrel(gamma(1.0/3.L), 2.67893853470774763365569294097467764412868937795730L) >= real.mant_dig-2); + assert(feqrel(gamma(0.25L), + 3.62560990822190831193068515586767200299516768288006) >= real.mant_dig-1); + assert(feqrel(gamma(1.0/5.0L), + 4.59084371199880305320475827592915200343410999829340L) >= real.mant_dig-1); +} +} + +/***************************************************** + * Natural logarithm of gamma function. + * + * Returns the base e (2.718...) logarithm of the absolute + * value of the gamma function of the argument. + * + * For reals, logGamma is equivalent to log(fabs(gamma(x))). + * + * $(TABLE_SV + * $(SVH x, logGamma(x) ) + * $(SV $(NAN), $(NAN) ) + * $(SV integer <= 0, +∞ ) + * $(SV ±∞, +∞ ) + * ) + */ +real logGamma(real x) +{ + /* Author: Don Clugston. Based on code from the CEPHES library. + * CEPHES code Copyright 1994 by Stephen L. Moshier + * + * For arguments greater than 33, the logarithm of the gamma + * function is approximated by the logarithmic version of + * Stirling's formula using a polynomial approximation of + * degree 4. Arguments between -33 and +33 are reduced by + * recurrence to the interval [2,3] of a rational approximation. + * The cosecant reflection formula is employed for arguments + * less than -33. + */ + real q, w, z, f, nx; + + if (isNaN(x)) return x; + if (fabs(x) == x.infinity) return x.infinity; + + if( x < -34.0L ) { + q = -x; + w = logGamma(q); + real p = floor(q); + if ( p == q ) return real.infinity; + int intpart = cast(int)(p); + real sgngam = 1; + if ( (intpart & 1) == 0 ) + sgngam = -1; + z = q - p; + if ( z > 0.5L ) { + p += 1.0L; + z = p - q; + } + z = q * sin( PI * z ); + if ( z == 0.0L ) return sgngam * real.infinity; + /* z = LOGPI - logl( z ) - w; */ + z = log( PI/z ) - w; + return z; + } + + if( x < 13.0L ) { + z = 1.0L; + nx = floor( x + 0.5L ); + f = x - nx; + while ( x >= 3.0L ) { + nx -= 1.0L; + x = nx + f; + z *= x; + } + while ( x < 2.0L ) { + if( fabs(x) <= 0.03125 ) { + if ( x == 0.0L ) return real.infinity; + if ( x < 0.0L ) { + x = -x; + q = z / (x * poly( x, GammaSmallNegCoeffs)); + } else + q = z / (x * poly( x, GammaSmallCoeffs)); + return log( fabs(q) ); + } + z /= nx + f; + nx += 1.0L; + x = nx + f; + } + z = fabs(z); + if ( x == 2.0L ) + return log(z); + x = (nx - 2.0L) + f; + real p = x * rationalPoly( x, logGammaNumerator, logGammaDenominator); + return log(z) + p; + } + + // const real MAXLGM = 1.04848146839019521116e+4928L; + // if( x > MAXLGM ) return sgngaml * real.infinity; + + const real LOGSQRT2PI = 0.91893853320467274178L; // log( sqrt( 2*pi ) ) + + q = ( x - 0.5L ) * log(x) - x + LOGSQRT2PI; + if (x > 1.0e10L) return q; + real p = 1.0L / (x*x); + q += poly( p, logGammaStirlingCoeffs ) / x; + return q ; +} + +debug(UnitTest) { +unittest { + assert(isIdentical(logGamma(NaN(0xDEF)), NaN(0xDEF))); + assert(logGamma(real.infinity) == real.infinity); + assert(logGamma(-1.0) == real.infinity); + assert(logGamma(0.0) == real.infinity); + assert(logGamma(-50.0) == real.infinity); + assert(isIdentical(0.0L, logGamma(1.0L))); + assert(isIdentical(0.0L, logGamma(2.0L))); + assert(logGamma(real.min*real.epsilon) == real.infinity); + assert(logGamma(-real.min*real.epsilon) == real.infinity); + + // x, correct loggamma(x), correct d/dx loggamma(x). + static real[] testpoints = [ + 8.0L, 8.525146484375L + 1.48766904143001655310E-5, 2.01564147795560999654E0L, + 8.99993896484375e-1L, 6.6375732421875e-2L + 5.11505711292524166220E-6L, -7.54938684259372234258E-1, + 7.31597900390625e-1L, 2.2369384765625e-1 + 5.21506341809849792422E-6L, -1.13355566660398608343E0L, + 2.31639862060546875e-1L, 1.3686676025390625L + 1.12609441752996145670E-5L, -4.56670961813812679012E0, + 1.73162841796875L, -8.88214111328125e-2L + 3.36207740803753034508E-6L, 2.33339034686200586920E-1L, + 1.23162841796875L, -9.3902587890625e-2L + 1.28765089229009648104E-5L, -2.49677345775751390414E-1L, + 7.3786976294838206464e19L, 3.301798506038663053312e21L - 1.656137564136932662487046269677E5L, + 4.57477139169563904215E1L, + 1.08420217248550443401E-19L, 4.36682586669921875e1L + 1.37082843669932230418E-5L, + -9.22337203685477580858E18L, + 1.0L, 0.0L, -5.77215664901532860607E-1L, + 2.0L, 0.0L, 4.22784335098467139393E-1L, + -0.5L, 1.2655029296875L + 9.19379714539648894580E-6L, 3.64899739785765205590E-2L, + -1.5L, 8.6004638671875e-1L + 6.28657731014510932682E-7L, 7.03156640645243187226E-1L, + -2.5L, -5.6243896484375E-2L + 1.79986700949327405470E-7, 1.10315664064524318723E0L, + -3.5L, -1.30902099609375L + 1.43111007079536392848E-5L, 1.38887092635952890151E0L + ]; + // TODO: test derivatives as well. + for (int i=0; i<testpoints.length; i+=3) { + assert( feqrel(logGamma(testpoints[i]), testpoints[i+1]) > real.mant_dig-5); + if (testpoints[i]<MAXGAMMA) { + assert( feqrel(log(fabs(gamma(testpoints[i]))), testpoints[i+1]) > real.mant_dig-5); + } + } + assert(logGamma(-50.2) == log(fabs(gamma(-50.2)))); + assert(logGamma(-0.008) == log(fabs(gamma(-0.008)))); + assert(feqrel(logGamma(-38.8),log(fabs(gamma(-38.8)))) > real.mant_dig-4); + assert(feqrel(logGamma(1500.0L),log(gamma(1500.0L))) > real.mant_dig-2); +} +} + +private { +const real MAXLOG = 0x1.62e42fefa39ef358p+13L; // log(real.max) +const real MINLOG = -0x1.6436716d5406e6d8p+13L; // log(real.min*real.epsilon) = log(smallest denormal) +const real BETA_BIG = 9.223372036854775808e18L; +const real BETA_BIGINV = 1.084202172485504434007e-19L; +} + +/** Beta function + * + * The beta function is defined as + * + * beta(x, y) = (Γ(x) Γ(y))/Γ(x + y) + */ +real beta(real x, real y) +{ + if ((x+y)> MAXGAMMA) { + return exp(logGamma(x) + logGamma(y) - logGamma(x+y)); + } else return gamma(x)*gamma(y)/gamma(x+y); +} + +debug(UnitTest) { +unittest { + assert(isIdentical(beta(NaN(0xABC), 4), NaN(0xABC))); + assert(isIdentical(beta(2, NaN(0xABC)), NaN(0xABC))); +} +} + +/** Incomplete beta integral + * + * Returns incomplete beta integral of the arguments, evaluated + * from zero to x. The regularized incomplete beta function is defined as + * + * betaIncomplete(a, b, x) = Γ(a+b)/(Γ(a) Γ(b)) * + * $(INTEGRATE 0, x) $(POWER t, a-1)$(POWER (1-t),b-1) dt + * + * and is the same as the the cumulative distribution function. + * + * The domain of definition is 0 <= x <= 1. In this + * implementation a and b are restricted to positive values. + * The integral from x to 1 may be obtained by the symmetry + * relation + * + * betaIncompleteCompl(a, b, x ) = betaIncomplete( b, a, 1-x ) + * + * The integral is evaluated by a continued fraction expansion + * or, when b*x is small, by a power series. + */ +real betaIncomplete(real aa, real bb, real xx ) +{ + if (!(aa>0 && bb>0)) { + if (isNaN(aa)) return aa; + if (isNaN(bb)) return bb; + return NaN(TANGO_NAN.BETA_DOMAIN); // domain error + } + if (!(xx>0 && xx<1.0)) { + if (isNaN(xx)) return xx; + if ( xx == 0.0L ) return 0.0; + if ( xx == 1.0L ) return 1.0; + return NaN(TANGO_NAN.BETA_DOMAIN); // domain error + } + if ( (bb * xx) <= 1.0L && xx <= 0.95L) { + return betaDistPowerSeries(aa, bb, xx); + } + real x; + real xc; // = 1 - x + + real a, b; + int flag = 0; + + /* Reverse a and b if x is greater than the mean. */ + if( xx > (aa/(aa+bb)) ) { + // here x > aa/(aa+bb) and (bb*x>1 or x>0.95) + flag = 1; + a = bb; + b = aa; + xc = xx; + x = 1.0L - xx; + } else { + a = aa; + b = bb; + xc = 1.0L - xx; + x = xx; + } + + if( flag == 1 && (b * x) <= 1.0L && x <= 0.95L) { + // here xx > aa/(aa+bb) and ((bb*xx>1) or xx>0.95) and (aa*(1-xx)<=1) and xx > 0.05 + return 1.0 - betaDistPowerSeries(a, b, x); // note loss of precision + } + + real w; + // Choose expansion for optimal convergence + // One is for x * (a+b+2) < (a+1), + // the other is for x * (a+b+2) > (a+1). + real y = x * (a+b-2.0L) - (a-1.0L); + if( y < 0.0L ) { + w = betaDistExpansion1( a, b, x ); + } else { + w = betaDistExpansion2( a, b, x ) / xc; + } + + /* Multiply w by the factor + a b + x (1-x) Gamma(a+b) / ( a Gamma(a) Gamma(b) ) . */ + + y = a * log(x); + real t = b * log(xc); + if ( (a+b) < MAXGAMMA && fabs(y) < MAXLOG && fabs(t) < MAXLOG ) { + t = pow(xc,b); + t *= pow(x,a); + t /= a; + t *= w; + t *= gamma(a+b) / (gamma(a) * gamma(b)); + } else { + /* Resort to logarithms. */ + y += t + logGamma(a+b) - logGamma(a) - logGamma(b); + y += log(w/a); + + t = exp(y); +/+ + // There seems to be a bug in Cephes at this point. + // Problems occur for y > MAXLOG, not y < MINLOG. + if( y < MINLOG ) { + t = 0.0L; + } else { + t = exp(y); + } ++/ + } + if( flag == 1 ) { +/+ // CEPHES includes this code, but I think it is erroneous. + if( t <= real.epsilon ) { + t = 1.0L - real.epsilon; + } else ++/ + t = 1.0L - t; + } + return t; +} + +/** Inverse of incomplete beta integral + * + * Given y, the function finds x such that + * + * betaIncomplete(a, b, x) == y + * + * Newton iterations or interval halving is used. + */ +real betaIncompleteInv(real aa, real bb, real yy0 ) +{ + real a, b, y0, d, y, x, x0, x1, lgm, yp, di, dithresh, yl, yh, xt; + int i, rflg, dir, nflg; + + if (isNaN(yy0)) return yy0; + if (isNaN(aa)) return aa; + if (isNaN(bb)) return bb; + if( yy0 <= 0.0L ) + return 0.0L; + if( yy0 >= 1.0L ) + return 1.0L; + x0 = 0.0L; + yl = 0.0L; + x1 = 1.0L; + yh = 1.0L; + if( aa <= 1.0L || bb <= 1.0L ) { + dithresh = 1.0e-7L; + rflg = 0; + a = aa; + b = bb; + y0 = yy0; + x = a/(a+b); + y = betaIncomplete( a, b, x ); + nflg = 0; + goto ihalve; + } else { + nflg = 0; + dithresh = 1.0e-4L; + } + + /* approximation to inverse function */ + + yp = -normalDistributionInvImpl( yy0 ); + + if( yy0 > 0.5L ) { + rflg = 1; + a = bb; + b = aa; + y0 = 1.0L - yy0; + yp = -yp; + } else { + rflg = 0; + a = aa; + b = bb; + y0 = yy0; + } + + lgm = (yp * yp - 3.0L)/6.0L; + x = 2.0L/( 1.0L/(2.0L * a-1.0L) + 1.0L/(2.0L * b - 1.0L) ); + d = yp * sqrt( x + lgm ) / x + - ( 1.0L/(2.0L * b - 1.0L) - 1.0L/(2.0L * a - 1.0L) ) + * (lgm + (5.0L/6.0L) - 2.0L/(3.0L * x)); + d = 2.0L * d; + if( d < MINLOG ) { + x = 1.0L; + goto under; + } + x = a/( a + b * exp(d) ); + y = betaIncomplete( a, b, x ); + yp = (y - y0)/y0; + if( fabs(yp) < 0.2 ) + goto newt; + + /* Resort to interval halving if not close enough. */ +ihalve: + + dir = 0; + di = 0.5L; + for( i=0; i<400; i++ ) { + if( i != 0 ) { + x = x0 + di * (x1 - x0); + if( x == 1.0L ) { + x = 1.0L - real.epsilon; + } + if( x == 0.0L ) { + di = 0.5; + x = x0 + di * (x1 - x0); + if( x == 0.0 ) + goto under; + } + y = betaIncomplete( a, b, x ); + yp = (x1 - x0)/(x1 + x0); + if( fabs(yp) < dithresh ) + goto newt; + yp = (y-y0)/y0; + if( fabs(yp) < dithresh ) + goto newt; + } + if( y < y0 ) { + x0 = x; + yl = y; + if( dir < 0 ) { + dir = 0; + di = 0.5L; + } else if( dir > 3 ) + di = 1.0L - (1.0L - di) * (1.0L - di); + else if( dir > 1 ) + di = 0.5L * di + 0.5L; + else + di = (y0 - y)/(yh - yl); + dir += 1; + if( x0 > 0.95L ) { + if( rflg == 1 ) { + rflg = 0; + a = aa; + b = bb; + y0 = yy0; + } else { + rflg = 1; + a = bb; + b = aa; + y0 = 1.0 - yy0; + } + x = 1.0L - x; + y = betaIncomplete( a, b, x ); + x0 = 0.0; + yl = 0.0; + x1 = 1.0; + yh = 1.0; + goto ihalve; + } + } else { + x1 = x; + if( rflg == 1 && x1 < real.epsilon ) { + x = 0.0L; + goto done; + } + yh = y; + if( dir > 0 ) { + dir = 0; + di = 0.5L; + } + else if( dir < -3 ) + di = di * di; + else if( dir < -1 ) + di = 0.5L * di; + else + di = (y - y0)/(yh - yl); + dir -= 1; + } + } + // loss of precision has occurred + + //mtherr( "incbil", PLOSS ); + if( x0 >= 1.0L ) { + x = 1.0L - real.epsilon; + goto done; + } + if( x <= 0.0L ) { +under: + // underflow has occurred + //mtherr( "incbil", UNDERFLOW ); + x = 0.0L; + goto done; + } + +newt: + + if ( nflg ) { + goto done; + } + nflg = 1; + lgm = logGamma(a+b) - logGamma(a) - logGamma(b); + + for( i=0; i<15; i++ ) { + /* Compute the function at this point. */ + if ( i != 0 ) + y = betaIncomplete(a,b,x); + if ( y < yl ) { + x = x0; + y = yl; + } else if( y > yh ) { + x = x1; + y = yh; + } else if( y < y0 ) { + x0 = x; + yl = y; + } else { + x1 = x; + yh = y; + } + if( x == 1.0L || x == 0.0L ) + break; + /* Compute the derivative of the function at this point. */ + d = (a - 1.0L) * log(x) + (b - 1.0L) * log(1.0L - x) + lgm; + if ( d < MINLOG ) { + goto done; + } + if ( d > MAXLOG ) { + break; + } + d = exp(d); + /* Compute the step to the next approximation of x. */ + d = (y - y0)/d; + xt = x - d; + if ( xt <= x0 ) { + y = (x - x0) / (x1 - x0); + xt = x0 + 0.5L * y * (x - x0); + if( xt <= 0.0L ) + break; + } + if ( xt >= x1 ) { + y = (x1 - x) / (x1 - x0); + xt = x1 - 0.5L * y * (x1 - x); + if ( xt >= 1.0L ) + break; + } + x = xt; + if ( fabs(d/x) < (128.0L * real.epsilon) ) + goto done; + } + /* Did not converge. */ + dithresh = 256.0L * real.epsilon; + goto ihalve; + +done: + if ( rflg ) { + if( x <= real.epsilon ) + x = 1.0L - real.epsilon; + else + x = 1.0L - x; + } + return x; +} + +debug(UnitTest) { +unittest { // also tested by the normal distribution + // check NaN propagation + assert(isIdentical(betaIncomplete(NaN(0xABC),2,3), NaN(0xABC))); + assert(isIdentical(betaIncomplete(7,NaN(0xABC),3), NaN(0xABC))); + assert(isIdentical(betaIncomplete(7,15,NaN(0xABC)), NaN(0xABC))); + assert(isIdentical(betaIncompleteInv(NaN(0xABC),1,17), NaN(0xABC))); + assert(isIdentical(betaIncompleteInv(2,NaN(0xABC),8), NaN(0xABC))); + assert(isIdentical(betaIncompleteInv(2,3, NaN(0xABC)), NaN(0xABC))); + + assert(isNaN(betaIncomplete(-1, 2, 3))); + + assert(betaIncomplete(1, 2, 0)==0); + assert(betaIncomplete(1, 2, 1)==1); + assert(isNaN(betaIncomplete(1, 2, 3))); + assert(betaIncompleteInv(1, 1, 0)==0); + assert(betaIncompleteInv(1, 1, 1)==1); + + // Test some values against Microsoft Excel 2003. + + assert(fabs(betaIncomplete(8, 10, 0.2) - 0.010_934_315_236_957_2L) < 0.000_000_000_5); + assert(fabs(betaIncomplete(2, 2.5, 0.9) - 0.989_722_597_604_107L) < 0.000_000_000_000_5); + assert(fabs(betaIncomplete(1000, 800, 0.5) - 1.17914088832798E-06L) < 0.000_000_05e-6); + + assert(fabs(betaIncomplete(0.0001, 10000, 0.0001) - 0.999978059369989L) < 0.000_000_000_05); + + assert(fabs(betaIncompleteInv(5, 10, 0.2) - 0.229121208190918L) < 0.000_000_5L); + assert(fabs(betaIncompleteInv(4, 7, 0.8) - 0.483657360076904L) < 0.000_000_5L); + + // Coverage tests. I don't have correct values for these tests, but + // these values cover most of the code, so they are useful for + // regression testing. + // Extensive testing failed to increase the coverage. It seems likely that about + // half the code in this function is unnecessary; there is potential for + // significant improvement over the original CEPHES code. + +// Excel 2003 gives clearly erroneous results (betadist>1) when a and x are tiny and b is huge. +// The correct results are for these next tests are unknown. + +// real testpoint1 = betaIncomplete(1e-10, 5e20, 8e-21); +// assert(testpoint1 == 0x1.ffff_ffff_c906_404cp-1L); + + assert(betaIncomplete(0.01, 327726.7, 0.545113) == 1.0); + assert(betaIncompleteInv(0.01, 8e-48, 5.45464e-20)==1-real.epsilon); + assert(betaIncompleteInv(0.01, 8e-48, 9e-26)==1-real.epsilon); + + assert(betaIncomplete(0.01, 498.437, 0.0121433) == 0x1.ffff_8f72_19197402p-1); + assert(1- betaIncomplete(0.01, 328222, 4.0375e-5) == 0x1.5f62926b4p-30); + version(FailsOnLinux) assert(betaIncompleteInv(0x1.b3d151fbba0eb18p+1, 1.2265e-19, 2.44859e-18)==0x1.c0110c8531d0952cp-1); + version(FailsOnLinux) assert(betaIncompleteInv(0x1.ff1275ae5b939bcap-41, 4.6713e18, 0.0813601)==0x1.f97749d90c7adba8p-63); + real a1; + a1 = 3.40483; + version(FailsOnLinux) assert(betaIncompleteInv(a1, 4.0640301659679627772e19L, 0.545113)== 0x1.ba8c08108aaf5d14p-109); + real b1; + b1= 2.82847e-25; + version(FailsOnLinux) assert(betaIncompleteInv(0.01, b1, 9e-26) == 0x1.549696104490aa9p-830); + + // --- Problematic cases --- + // This is a situation where the series expansion fails to converge + assert( isNaN(betaIncompleteInv(0.12167, 4.0640301659679627772e19L, 0.0813601))); + // This next result is almost certainly erroneous. + assert(betaIncomplete(1.16251e20, 2.18e39, 5.45e-20)==-real.infinity); +} +} + +private { +// Implementation functions + +// Continued fraction expansion #1 for incomplete beta integral +// Use when x < (a+1)/(a+b+2) +real betaDistExpansion1(real a, real b, real x ) +{ + real xk, pk, pkm1, pkm2, qk, qkm1, qkm2; + real k1, k2, k3, k4, k5, k6, k7, k8; + real r, t, ans; + int n; + + k1 = a; + k2 = a + b; + k3 = a; + k4 = a + 1.0L; + k5 = 1.0L; + k6 = b - 1.0L; + k7 = k4; + k8 = a + 2.0L; + + pkm2 = 0.0L; + qkm2 = 1.0L; + pkm1 = 1.0L; + qkm1 = 1.0L; + ans = 1.0L; + r = 1.0L; + n = 0; + const real thresh = 3.0L * real.epsilon; + do { + xk = -( x * k1 * k2 )/( k3 * k4 ); + pk = pkm1 + pkm2 * xk; + qk = qkm1 + qkm2 * xk; + pkm2 = pkm1; + pkm1 = pk; + qkm2 = qkm1; + qkm1 = qk; + + xk = ( x * k5 * k6 )/( k7 * k8 ); + pk = pkm1 + pkm2 * xk; + qk = qkm1 + qkm2 * xk; + pkm2 = pkm1; + pkm1 = pk; + qkm2 = qkm1; + qkm1 = qk; + + if( qk != 0.0L ) + r = pk/qk; + if( r != 0.0L ) { + t = fabs( (ans - r)/r ); + ans = r; + } else { + t = 1.0L; + } + + if( t < thresh ) + return ans; + + k1 += 1.0L; + k2 += 1.0L; + k3 += 2.0L; + k4 += 2.0L; + k5 += 1.0L; + k6 -= 1.0L; + k7 += 2.0L; + k8 += 2.0L; + + if( (fabs(qk) + fabs(pk)) > BETA_BIG ) { + pkm2 *= BETA_BIGINV; + pkm1 *= BETA_BIGINV; + qkm2 *= BETA_BIGINV; + qkm1 *= BETA_BIGINV; + } + if( (fabs(qk) < BETA_BIGINV) || (fabs(pk) < BETA_BIGINV) ) { + pkm2 *= BETA_BIG; + pkm1 *= BETA_BIG; + qkm2 *= BETA_BIG; + qkm1 *= BETA_BIG; + } + } + while( ++n < 400 ); +// loss of precision has occurred +// mtherr( "incbetl", PLOSS ); + return ans; +} + +// Continued fraction expansion #2 for incomplete beta integral +// Use when x > (a+1)/(a+b+2) +real betaDistExpansion2(real a, real b, real x ) +{ + real xk, pk, pkm1, pkm2, qk, qkm1, qkm2; + real k1, k2, k3, k4, k5, k6, k7, k8; + real r, t, ans, z; + + k1 = a; + k2 = b - 1.0L; + k3 = a; + k4 = a + 1.0L; + k5 = 1.0L; + k6 = a + b; + k7 = a + 1.0L; + k8 = a + 2.0L; + + pkm2 = 0.0L; + qkm2 = 1.0L; + pkm1 = 1.0L; + qkm1 = 1.0L; + z = x / (1.0L-x); + ans = 1.0L; + r = 1.0L; + int n = 0; + const real thresh = 3.0L * real.epsilon; + do { + + xk = -( z * k1 * k2 )/( k3 * k4 ); + pk = pkm1 + pkm2 * xk; + qk = qkm1 + qkm2 * xk; + pkm2 = pkm1; + pkm1 = pk; + qkm2 = qkm1; + qkm1 = qk; + + xk = ( z * k5 * k6 )/( k7 * k8 ); + pk = pkm1 + pkm2 * xk; + qk = qkm1 + qkm2 * xk; + pkm2 = pkm1; + pkm1 = pk; + qkm2 = qkm1; + qkm1 = qk; + + if( qk != 0.0L ) + r = pk/qk; + if( r != 0.0L ) { + t = fabs( (ans - r)/r ); + ans = r; + } else + t = 1.0L; + + if( t < thresh ) + return ans; + k1 += 1.0L; + k2 -= 1.0L; + k3 += 2.0L; + k4 += 2.0L; + k5 += 1.0L; + k6 += 1.0L; + k7 += 2.0L; + k8 += 2.0L; + + if( (fabs(qk) + fabs(pk)) > BETA_BIG ) { + pkm2 *= BETA_BIGINV; + pkm1 *= BETA_BIGINV; + qkm2 *= BETA_BIGINV; + qkm1 *= BETA_BIGINV; + } + if( (fabs(qk) < BETA_BIGINV) || (fabs(pk) < BETA_BIGINV) ) { + pkm2 *= BETA_BIG; + pkm1 *= BETA_BIG; + qkm2 *= BETA_BIG; + qkm1 *= BETA_BIG; + } + } while( ++n < 400 ); +// loss of precision has occurred +//mtherr( "incbetl", PLOSS ); + return ans; +} + +/* Power series for incomplete gamma integral. + Use when b*x is small. */ +real betaDistPowerSeries(real a, real b, real x ) +{ + real ai = 1.0L / a; + real u = (1.0L - b) * x; + real v = u / (a + 1.0L); + real t1 = v; + real t = u; + real n = 2.0L; + real s = 0.0L; + real z = real.epsilon * ai; + while( fabs(v) > z ) { + u = (n - b) * x / n; + t *= u; + v = t / (a + n); + s += v; + n += 1.0L; + } + s += t1; + s += ai; + + u = a * log(x); + if ( (a+b) < MAXGAMMA && fabs(u) < MAXLOG ) { + t = gamma(a+b)/(gamma(a)*gamma(b)); + s = s * t * pow(x,a); + } else { + t = logGamma(a+b) - logGamma(a) - logGamma(b) + u + log(s); + + if( t < MINLOG ) { + s = 0.0L; + } else + s = exp(t); + } + return s; +} + +} + +/*************************************** + * Incomplete gamma integral and its complement + * + * These functions are defined by + * + * gammaIncomplete = ( $(INTEGRATE 0, x) $(POWER e, -t) $(POWER t, a-1) dt )/ $(GAMMA)(a) + * + * gammaIncompleteCompl(a,x) = 1 - gammaIncomplete(a,x) + * = ($(INTEGRATE x, ∞) $(POWER e, -t) $(POWER t, a-1) dt )/ $(GAMMA)(a) + * + * In this implementation both arguments must be positive. + * The integral is evaluated by either a power series or + * continued fraction expansion, depending on the relative + * values of a and x. + */ +real gammaIncomplete(real a, real x ) +in { + assert(x >= 0); + assert(a > 0); +} +body { + /* left tail of incomplete gamma function: + * + * inf. k + * a -x - x + * x e > ---------- + * - - + * k=0 | (a+k+1) + * + */ + if (x==0) + return 0.0L; + + if ( (x > 1.0L) && (x > a ) ) + return 1.0L - gammaIncompleteCompl(a,x); + + real ax = a * log(x) - x - logGamma(a); +/+ + if( ax < MINLOGL ) return 0; // underflow + // { mtherr( "igaml", UNDERFLOW ); return( 0.0L ); } ++/ + ax = exp(ax); + + /* power series */ + real r = a; + real c = 1.0L; + real ans = 1.0L; + + do { + r += 1.0L; + c *= x/r; + ans += c; + } while( c/ans > real.epsilon ); + + return ans * ax/a; +} + +/** ditto */ +real gammaIncompleteCompl(real a, real x ) +in { + assert(x >= 0); + assert(a > 0); +} +body { + if (x==0) + return 1.0L; + if ( (x < 1.0L) || (x < a) ) + return 1.0L - gammaIncomplete(a,x); + + // DAC (Cephes bug fix): This is necessary to avoid + // spurious nans, eg + // log(x)-x = NaN when x = real.infinity + const real MAXLOGL = 1.1356523406294143949492E4L; + if (x > MAXLOGL) return 0; // underflow + + real ax = a * log(x) - x - logGamma(a); +//const real MINLOGL = -1.1355137111933024058873E4L; +// if ( ax < MINLOGL ) return 0; // underflow; + ax = exp(ax); + + + /* continued fraction */ + real y = 1.0L - a; + real z = x + y + 1.0L; + real c = 0.0L; + + real pk, qk, t; + + real pkm2 = 1.0L; + real qkm2 = x; + real pkm1 = x + 1.0L; + real qkm1 = z * x; + real ans = pkm1/qkm1; + + do { + c += 1.0L; + y += 1.0L; + z += 2.0L; + real yc = y * c; + pk = pkm1 * z - pkm2 * yc; + qk = qkm1 * z - qkm2 * yc; + if( qk != 0.0L ) { + real r = pk/qk; + t = fabs( (ans - r)/r ); + ans = r; + } else { + t = 1.0L; + } + pkm2 = pkm1; + pkm1 = pk; + qkm2 = qkm1; + qkm1 = qk; + + const real BIG = 9.223372036854775808e18L; + + if ( fabs(pk) > BIG ) { + pkm2 /= BIG; + pkm1 /= BIG; + qkm2 /= BIG; + qkm1 /= BIG; + } + } while ( t > real.epsilon ); + + return ans * ax; +} + +/** Inverse of complemented incomplete gamma integral + * + * Given a and y, the function finds x such that + * + * gammaIncompleteCompl( a, x ) = p. + * + * Starting with the approximate value x = a $(POWER t, 3), where + * t = 1 - d - normalDistributionInv(p) sqrt(d), + * and d = 1/9a, + * the routine performs up to 10 Newton iterations to find the + * root of incompleteGammaCompl(a,x) - p = 0. + */ +real gammaIncompleteComplInv(real a, real p) +in { + assert(p>=0 && p<= 1); + assert(a>0); +} +body { + if (p==0) return real.infinity; + + real y0 = p; + const real MAXLOGL = 1.1356523406294143949492E4L; + real x0, x1, x, yl, yh, y, d, lgm, dithresh; + int i, dir; + + /* bound the solution */ + x0 = real.max; + yl = 0.0L; + x1 = 0.0L; + yh = 1.0L; + dithresh = 4.0 * real.epsilon; + + /* approximation to inverse function */ + d = 1.0L/(9.0L*a); + y = 1.0L - d - normalDistributionInvImpl(y0) * sqrt(d); + x = a * y * y * y; + + lgm = logGamma(a); + + for( i=0; i<10; i++ ) { + if( x > x0 || x < x1 ) + goto ihalve; + y = gammaIncompleteCompl(a,x); + if ( y < yl || y > yh ) + goto ihalve; + if ( y < y0 ) { + x0 = x; + yl = y; + } else { + x1 = x; + yh = y; + } + /* compute the derivative of the function at this point */ + d = (a - 1.0L) * log(x0) - x0 - lgm; + if ( d < -MAXLOGL ) + goto ihalve; + d = -exp(d); + /* compute the step to the next approximation of x */ + d = (y - y0)/d; + x = x - d; + if ( i < 3 ) continue; + if ( fabs(d/x) < dithresh ) return x; + } + + /* Resort to interval halving if Newton iteration did not converge. */ +ihalve: + d = 0.0625L; + if ( x0 == real.max ) { + if( x <= 0.0L ) + x = 1.0L; + while( x0 == real.max ) { + x = (1.0L + d) * x; + y = gammaIncompleteCompl( a, x ); + if ( y < y0 ) { + x0 = x; + yl = y; + break; + } + d = d + d; + } + } + d = 0.5L; + dir = 0; + + for( i=0; i<400; i++ ) { + x = x1 + d * (x0 - x1); + y = gammaIncompleteCompl( a, x ); + lgm = (x0 - x1)/(x1 + x0); + if ( fabs(lgm) < dithresh ) + break; + lgm = (y - y0)/y0; + if ( fabs(lgm) < dithresh ) + break; + if ( x <= 0.0L ) + break; + if ( y > y0 ) { + x1 = x; + yh = y; + if ( dir < 0 ) { + dir = 0; + d = 0.5L; + } else if ( dir > 1 ) + d = 0.5L * d + 0.5L; + else + d = (y0 - yl)/(yh - yl); + dir += 1; + } else { + x0 = x; + yl = y; + if ( dir > 0 ) { + dir = 0; + d = 0.5L; + } else if ( dir < -1 ) + d = 0.5L * d; + else + d = (y0 - yl)/(yh - yl); + dir -= 1; + } + } + /+ + if( x == 0.0L ) + mtherr( "igamil", UNDERFLOW ); + +/ + return x; +} + +debug(UnitTest) { +unittest { +//Values from Excel's GammaInv(1-p, x, 1) +assert(fabs(gammaIncompleteComplInv(1, 0.5) - 0.693147188044814) < 0.00000005); +assert(fabs(gammaIncompleteComplInv(12, 0.99) - 5.42818075054289) < 0.00000005); +assert(fabs(gammaIncompleteComplInv(100, 0.8) - 91.5013985848288L) < 0.000005); + +assert(gammaIncomplete(1, 0)==0); +assert(gammaIncompleteCompl(1, 0)==1); +assert(gammaIncomplete(4545, real.infinity)==1); + +// Values from Excel's (1-GammaDist(x, alpha, 1, TRUE)) + +assert(fabs(1.0L-gammaIncompleteCompl(0.5, 2) - 0.954499729507309L) < 0.00000005); +assert(fabs(gammaIncomplete(0.5, 2) - 0.954499729507309L) < 0.00000005); +// Fixed Cephes bug: +assert(gammaIncompleteCompl(384, real.infinity)==0); +assert(gammaIncompleteComplInv(3, 0)==real.infinity); +} +} + +/** Digamma function +* +* The digamma function is the logarithmic derivative of the gamma function. +* +* digamma(x) = d/dx logGamma(x) +* +*/ +real digamma(real x) +{ + // Based on CEPHES, Stephen L. Moshier. + + // DAC: These values are Bn / n for n=2,4,6,8,10,12,14. + const real [] Bn_n = [ + 1.0L/(6*2), -1.0L/(30*4), 1.0L/(42*6), -1.0L/(30*8), + 5.0L/(66*10), -691.0L/(2730*12), 7.0L/(6*14) ]; + + real p, q, nz, s, w, y, z; + int i, n, negative; + + negative = 0; + nz = 0.0; + + if ( x <= 0.0 ) { + negative = 1; + q = x; + p = floor(q); + if( p == q ) { + return NaN(TANGO_NAN.GAMMA_POLE); // singularity. + } + /* Remove the zeros of tan(PI x) + * by subtracting the nearest integer from x + */ + nz = q - p; + if ( nz != 0.5 ) { + if ( nz > 0.5 ) { + p += 1.0; + nz = q - p; + } + nz = PI/tan(PI*nz); + } else { + nz = 0.0; + } + x = 1.0 - x; + } + + // check for small positive integer + if ((x <= 13.0) && (x == floor(x)) ) { + y = 0.0; + n = rndint(x); + // DAC: CEPHES bugfix. Cephes did this in reverse order, which + // created a larger roundoff error. + for (i=n-1; i>0; --i) { + y+=1.0L/i; + } + y -= EULERGAMMA; + goto done; + } + + s = x; + w = 0.0; + while ( s < 10.0 ) { + w += 1.0/s; + s += 1.0; + } + + if ( s < 1.0e17 ) { + z = 1.0/(s * s); + y = z * poly(z, Bn_n); + } else + y = 0.0; + + y = log(s) - 0.5L/s - y - w; + +done: + if ( negative ) { + y -= nz; + } + return y; +} + +import tango.stdc.stdio; +debug(UnitTest) { +unittest { + // Exact values + assert(digamma(1)== -EULERGAMMA); + assert(feqrel(digamma(0.25), -PI/2 - 3* LN2 - EULERGAMMA)>=real.mant_dig-6); + assert(feqrel(digamma(1.0L/6), -PI/2 *sqrt(3.0L) - 2* LN2 -1.5*log(3.0L) - EULERGAMMA)>=real.mant_dig-7); + assert(digamma(-5)!<>0); + assert(feqrel(digamma(2.5), -EULERGAMMA - 2*LN2 + 2.0 + 2.0L/3)>=real.mant_dig-9); + assert(isIdentical(digamma(NaN(0xABC)), NaN(0xABC))); + + for (int k=1; k<40; ++k) { + real y=0; + for (int u=k; u>=1; --u) { + y+= 1.0L/u; + } + assert(feqrel(digamma(k+1),-EULERGAMMA + y) >=real.mant_dig-2); + } + +// printf("%d %La %La %d %d\n", k+1, digamma(k+1), -EULERGAMMA + x, feqrel(digamma(k+1),-EULERGAMMA + y), feqrel(digamma(k+1), -EULERGAMMA + x)); +} +} +