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<HTML> <HEAD> <TITLE>Cephes long double precision special functions suite</TITLE> <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1"> <META NAME="description" CONTENT="Long double precision special functions suite of <META NAME="keywords" CONTENT="numerical analysis, special functions, Cephes"> <META NAME="GENERATOR" CONTENT="Mozilla/4.04 [en] (X11; I; Linux 2.0.32 i486) [Nets </HEAD> <BODY> <H3>Cephes Mathematical Library</H3> <H3>Source code archives</H3> <BR><A HREF="singldoc.html">Documentation for single precision library.</A> <BR><A HREF="doubldoc.html">Documentation for double precision library.</A> <BR><A HREF="ldoubdoc.html">Documentation for 80-bit long double library.</A> <BR><A HREF="128bdoc.html">Documentation for 128-bit long double library.</A> <BR><A HREF="qlibdoc.html">Documentation for extended precision library.</A> <H3>Long Double Precision Special Functions</H3> Select function name for additional information. For other precisions, see the archives and descriptions listed above. <DIR> <LI><A HREF="#acosh">acoshl, Inverse hyperbolic cosine</A> <LI><A HREF="#arcdot">arcdotl, Angle between two vectors</A> <LI><A HREF="#asinh">asinh, Inverse hyperbolic sine</A> <LI><A HREF="#asin">asin, Inverse circular sine</A> <LI><A HREF="#acos">acos, Inverse circular cosine</A> <LI><A HREF="#atanh">atanh, Inverse hyperbolic tangent</A> <LI><A HREF="#atan">atan, Inverse circular tangent</A> <LI><A HREF="#atan2">atan2, Quadrant correct inverse circular tangent</A> <LI><A HREF="#bdtr">bdtr, Binomial distribution</A> <LI><A HREF="#bdtrc">bdtrc, Complemented binomial distribution</A> <LI><A HREF="#bdtri">bdtri, Inverse binomial distribution</A> <LI><A HREF="#btdtr">btdtr, Beta distribution</A> <LI><A HREF="#cbrt">cbrt, Cube root</A> <LI><A HREF="#chdtr">chdtr, Chi-square distribution</A> <LI><A HREF="#chdtrc">chdtrc, Complemented Chi-square distribution</A> <LI><A HREF="#chdtri">chdtri, Inverse of complemented Chi-square distribution</A> <LI><A HREF="#clog">clog, Complex natural logarithm</A> <LI><A HREF="#cexp">cexp, Complex exponential function</A> <LI><A HREF="#csin">csin, Complex circular sine</A> <LI><A HREF="#ccos">ccos, Complex circular cosine</A> <LI><A HREF="#ctan">ctan, Complex circular tangent</A> <LI><A HREF="#ccot">ccot, Complex circular cotangent</A> <LI><A HREF="#casin">casin, Complex circular arc sine</A> <LI><A HREF="#cacos">cacos, Complex circular arc cosine</A> <LI><A HREF="#catan">catan, Complex circular arc tangent</A> <LI><A HREF="#cmplx">cmplx, Complex number arithmetic</A> <LI><A HREF="#cosh">cosh, Hyperbolic cosine</A> <LI><A HREF="#ellie">ellie, Incomplete elliptic integral of the second kind</A> <LI><A HREF="#ellik">ellik, Incomplete elliptic integral of the first kind</A> <LI><A HREF="#ellpe">ellpe, Complete elliptic integral of the second kind</A> <LI><A HREF="#ellpj">ellpj, Jacobian elliptic functions</A> <LI><A HREF="#ellpk">ellpk, Complete elliptic integral of the first kind</A> <LI><A HREF="#exp10">exp10, Base 10 exponential function</A> <LI><A HREF="#exp2">exp2, Base 2 exponential function</A> <LI><A HREF="#exp">exp, Exponential function</A> <LI><A HREF="#expm1">expm1, Exponential function, minus 1</A> <LI><A HREF="#expx2">expx2, Exponential function</A> <LI><A HREF="#fdtr">fdtr, F distribution</A> <LI><A HREF="#fdtrc">fdtrc, Complemented F distribution</A> <LI><A HREF="#fdtri">fdtri, Inverse of complemented F distribution</A> <LI><A HREF="#floor">floor, Floor function</A> <LI><A HREF="#ceil">ceil, Ceil function</A> <LI><A HREF="#frexp">frexp, Extract exponent</A> <LI><A HREF="#ldexp">ldexp, Apply exponent</A> <LI><A HREF="#fabs">fabs, Absolute value</A> <LI><A HREF="#gamma">gamma, Gamma function</A> <LI><A HREF="#lgam">lgam, Natural logarithm of gamma function</A> <LI><A HREF="#gdtr">gdtr, Gamma distribution function</A> <LI><A HREF="#gdtrc">gdtrc, Complemented gamma distribution function</A> <LI><A HREF="#gels">gels, Linear system with symmetric coefficient matrix</A> <LI><A HREF="#hyperg">hyperg, Confluent hypergeometric function</A> <LI><A HREF="#ieee">ieee, Extended precision arithmetic</A> <LI><A HREF="#igami">igami, Inverse of complemented imcomplete gamma integral</A> <LI><A HREF="#igam">igam, Incomplete gamma integral</A> <LI><A HREF="#igamc">igamc, Complemented incomplete gamma integral</A> <LI><A HREF="#incbet">incbet, Incomplete beta integral</A> <LI><A HREF="#incbi">incbi, Inverse of imcomplete beta integral</A> <LI><A HREF="#isnan">isnan, Test for not a number</A> <LI><A HREF="#isfinite">isfinite, Test for infinity</A> <LI><A HREF="#signbit">signbit, Extract sign</A> <LI><A HREF="#j0">j0, Bessel function of order zero</A> <LI><A HREF="#y0">y0, Bessel function of the second kind, order zero</A> <LI><A HREF="#j1">j1, Bessel function of order one</A> <LI><A HREF="#y1">y1, Bessel function of the second kind, order one</A> <LI><A HREF="#jn">jn, Bessel function of integer order</A> <LI><A HREF="#ldrand">ldrand, Pseudorandom number generator</A> <LI><A HREF="#log10">log10, Common logarithm</A> <LI><A HREF="#log1p">log1p, Relative error logarithm</A> <LI><A HREF="#log2">log2, Base 2 logarithm</A> <LI><A HREF="#log">log, Natural logarithm</A> <LI><A HREF="#mtherr">mtherr, Library common error handling routine</A> <LI><A HREF="#nbdtr">nbdtr, Negative binomial distribution</A> <LI><A HREF="#nbdtrc">nbdtrc, Complemented negative binomial distribution</A> <LI><A HREF="#nbdtri">nbdtri, Functional inverse of negative binomial distribution</A> <LI><A HREF="#ndtri">ndtri, Inverse of normal distribution function</A> <LI><A HREF="#ndtr">ndtr, Normal distribution function</A> <LI><A HREF="#erf">erf, Error function</A> <LI><A HREF="#erfc">erfc, Complementary error function</A> <LI><A HREF="#pdtr">pdtr, Poisson distribution function</A> <LI><A HREF="#pdtrc">pdtrc, Complemented Poisson distribution function</A> <LI><A HREF="#pdtri">pdtri, Inverse of Poisson distribution function</A> <LI><A HREF="#polevl">polevl, Evaluate polynomial</A> <LI><A HREF="#p1evl">p1evl, Evaluate polynomial</A> <LI><A HREF="#powi">powi, Integer power function</A> <LI><A HREF="#pow">pow, Power function</A> <LI><A HREF="#sinh">sinh, Hyperbolic sine</A> <LI><A HREF="#sin">sin, Circular sine</A> <LI><A HREF="#cos">cos, Circular cosine</A> <LI><A HREF="#sqrt">sqrt, Square root</A> <LI><A HREF="#stdtr">stdtr, Student's t distribution <LI><A HREF="#stdtri">stdtri, Functional inverse of Student's t distribution <LI><A HREF="#tanh">tanh, Hyperbolic tangent</A> <LI><A HREF="#tan">tan, Circular tangent</A> <LI><A HREF="#cot">cot, Circular cotangent</A> <LI><A HREF="#cosm1">cosm1, Relative error cosine</A> <LI><A HREF="#yn">yn, Bessel function of second kind of integer order</A> </DIR> <A NAME="acosh"> </A> <PRE> /* acoshl.c * * Inverse hyperbolic cosine, long double precision * * * * SYNOPSIS: * * long double x, y, acoshl(); * * y = acoshl( x ); * * * * DESCRIPTION: * * Returns inverse hyperbolic cosine of argument. * * If 1 <= x < 1.5, a rational approximation * * sqrt(2z) * P(z)/Q(z) * * where z = x-1, is used. Otherwise, * * acosh(x) = log( x + sqrt( (x-1)(x+1) ). * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE 1,3 30000 2.0e-19 3.9e-20 * * * ERROR MESSAGES: * * message condition value returned * acoshl domain |x| < 1 0.0 * */ </PRE> <A NAME="arcdot"> </A> <PRE> /* arcdot.c * * Angle between two vectors * * * * * SYNOPSIS: * * long double p[3], q[3], arcdotl(); * * y = arcdotl( p, q ); * * * * DESCRIPTION: * * For two vectors p, q, the angle A between them is given by * * p.q / (|p| |q|) = cos A . * * where "." represents inner product, "|x|" the length of vector x. * If the angle is small, an expression in sin A is preferred. * Set r = q - p. Then * * p.q = p.p + p.r , * * |p|^2 = p.p , * * |q|^2 = p.p + 2 p.r + r.r , * * p.p^2 + 2 p.p p.r + p.r^2 * cos^2 A = ---------------------------- * p.p (p.p + 2 p.r + r.r) * * p.p + 2 p.r + p.r^2 / p.p * = --------------------------- , * p.p + 2 p.r + r.r * * sin^2 A = 1 - cos^2 A * * r.r - p.r^2 / p.p * = -------------------- * p.p + 2 p.r + r.r * * = (r.r - p.r^2 / p.p) / q.q . * * ACCURACY: * * About 1 ULP. See arcdot.c. * */ </PRE> <A NAME="asinh"> </A> <PRE> /* asinhl.c * * Inverse hyperbolic sine, long double precision * * * * SYNOPSIS: * * long double x, y, asinhl(); * * y = asinhl( x ); * * * * DESCRIPTION: * * Returns inverse hyperbolic sine of argument. * * If |x| < 0.5, the function is approximated by a rational * form x + x**3 P(x)/Q(x). Otherwise, * * asinh(x) = log( x + sqrt(1 + x*x) ). * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -3,3 30000 1.7e-19 3.5e-20 * */ </PRE> <A NAME="asin"> </A> <PRE> /* asinl.c * * Inverse circular sine, long double precision * * * * SYNOPSIS: * * double x, y, asinl(); * * y = asinl( x ); * * * * DESCRIPTION: * * Returns radian angle between -pi/2 and +pi/2 whose sine is x. * * A rational function of the form x + x**3 P(x**2)/Q(x**2) * is used for |x| in the interval [0, 0.5]. If |x| > 0.5 it is * transformed by the identity * * asin(x) = pi/2 - 2 asin( sqrt( (1-x)/2 ) ). * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -1, 1 30000 2.7e-19 4.8e-20 * * * ERROR MESSAGES: * * message condition value returned * asinl domain |x| > 1 NANL * */ </PRE> <A NAME="acos"> </A> <PRE> /* acosl() * * Inverse circular cosine, long double precision * * * * SYNOPSIS: * * double x, y, acosl(); * * y = acosl( x ); * * * * DESCRIPTION: * * Returns radian angle between -pi/2 and +pi/2 whose cosine * is x. * * Analytically, acos(x) = pi/2 - asin(x). However if |x| is * near 1, there is cancellation error in subtracting asin(x) * from pi/2. Hence if x < -0.5, * * acos(x) = pi - 2.0 * asin( sqrt((1+x)/2) ); * * or if x > +0.5, * * acos(x) = 2.0 * asin( sqrt((1-x)/2) ). * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -1, 1 30000 1.4e-19 3.5e-20 * * * ERROR MESSAGES: * * message condition value returned * acosl domain |x| > 1 NANL */ </PRE> <A NAME="atanh"> </A> <PRE> /* atanhl.c * * Inverse hyperbolic tangent, long double precision * * * * SYNOPSIS: * * long double x, y, atanhl(); * * y = atanhl( x ); * * * * DESCRIPTION: * * Returns inverse hyperbolic tangent of argument in the range * MINLOGL to MAXLOGL. * * If |x| < 0.5, the rational form x + x**3 P(x)/Q(x) is * employed. Otherwise, * atanh(x) = 0.5 * log( (1+x)/(1-x) ). * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -1,1 30000 1.1e-19 3.3e-20 * */ </PRE> <A NAME="atan"> </A> <PRE> /* atanl.c * * Inverse circular tangent, long double precision * (arctangent) * * * * SYNOPSIS: * * long double x, y, atanl(); * * y = atanl( x ); * * * * DESCRIPTION: * * Returns radian angle between -pi/2 and +pi/2 whose tangent * is x. * * Range reduction is from four intervals into the interval * from zero to tan( pi/8 ). The approximant uses a rational * function of degree 3/4 of the form x + x**3 P(x)/Q(x). * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -10, 10 150000 1.3e-19 3.0e-20 * */ </PRE> <A NAME="atan2"> </A> <PRE> /* atan2l() * * Quadrant correct inverse circular tangent, * long double precision * * * * SYNOPSIS: * * long double x, y, z, atan2l(); * * z = atan2l( y, x ); * * * * DESCRIPTION: * * Returns radian angle whose tangent is y/x. * Define compile time symbol ANSIC = 1 for ANSI standard, * range -PI < z <= +PI, args (y,x); else ANSIC = 0 for range * 0 to 2PI, args (x,y). * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -10, 10 60000 1.7e-19 3.2e-20 * See atan.c. * */ </PRE> <A NAME="bdtr"> </A> <PRE> /* bdtrl.c * * Binomial distribution * * * * SYNOPSIS: * * int k, n; * long double p, y, bdtrl(); * * y = bdtrl( k, n, p ); * * * * DESCRIPTION: * * Returns the sum of the terms 0 through k of the Binomial * probability density: * * k * -- ( n ) j n-j * > ( ) p (1-p) * -- ( j ) * j=0 * * The terms are not summed directly; instead the incomplete * beta integral is employed, according to the formula * * y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ). * * The arguments must be positive, with p ranging from 0 to 1. * * * * ACCURACY: * * Tested at random points (k,n,p) with a and b between 0 * and 10000 and p between 0 and 1. * Relative error: * arithmetic domain # trials peak rms * IEEE 0,10000 3000 1.6e-14 2.2e-15 * * ERROR MESSAGES: * * message condition value returned * bdtrl domain k < 0 0.0 * n < k * x < 0, x > 1 * */ </PRE> <A NAME="bdtrc"> </A> <PRE> /* bdtrcl() * * Complemented binomial distribution * * * * SYNOPSIS: * * int k, n; * long double p, y, bdtrcl(); * * y = bdtrcl( k, n, p ); * * * * DESCRIPTION: * * Returns the sum of the terms k+1 through n of the Binomial * probability density: * * n * -- ( n ) j n-j * > ( ) p (1-p) * -- ( j ) * j=k+1 * * The terms are not summed directly; instead the incomplete * beta integral is employed, according to the formula * * y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ). * * The arguments must be positive, with p ranging from 0 to 1. * * * * ACCURACY: * * See incbet.c. * * ERROR MESSAGES: * * message condition value returned * bdtrcl domain x<0, x>1, n<k 0.0 */ </PRE> <A NAME="bdtri"> </A> <PRE> /* bdtril() * * Inverse binomial distribution * * * * SYNOPSIS: * * int k, n; * long double p, y, bdtril(); * * p = bdtril( k, n, y ); * * * * DESCRIPTION: * * Finds the event probability p such that the sum of the * terms 0 through k of the Binomial probability density * is equal to the given cumulative probability y. * * This is accomplished using the inverse beta integral * function and the relation * * 1 - p = incbi( n-k, k+1, y ). * * ACCURACY: * * See incbi.c. * Tested at random k, n between 1 and 10000. The "domain" refers to p: * Relative error: * arithmetic domain # trials peak rms * IEEE 0,1 3500 2.0e-15 8.2e-17 * * ERROR MESSAGES: * * message condition value returned * bdtril domain k < 0, n <= k 0.0 * x < 0, x > 1 */ </PRE> <A NAME="btdtr"> </A> <PRE> /* btdtrl.c * * Beta distribution * * * * SYNOPSIS: * * long double a, b, x, y, btdtrl(); * * y = btdtrl( a, b, x ); * * * * DESCRIPTION: * * Returns the area from zero to x under the beta density * function: * * * x * - - * | (a+b) | | a-1 b-1 * P(x) = ---------- | t (1-t) dt * - - | | * | (a) | (b) - * 0 * * * The mean value of this distribution is a/(a+b). The variance * is ab/[(a+b)^2 (a+b+1)]. * * This function is identical to the incomplete beta integral * function, incbetl(a, b, x). * * The complemented function is * * 1 - P(1-x) = incbetl( b, a, x ); * * * ACCURACY: * * See incbetl.c. * */ </PRE> <A NAME="cbrt"> </A> <PRE> /* cbrtl.c * * Cube root, long double precision * * * * SYNOPSIS: * * long double x, y, cbrtl(); * * y = cbrtl( x ); * * * * DESCRIPTION: * * Returns the cube root of the argument, which may be negative. * * Range reduction involves determining the power of 2 of * the argument. A polynomial of degree 2 applied to the * mantissa, and multiplication by the cube root of 1, 2, or 4 * approximates the root to within about 0.1%. Then Newton's * iteration is used three times to converge to an accurate * result. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE .125,8 80000 7.0e-20 2.2e-20 * IEEE exp(+-707) 100000 7.0e-20 2.4e-20 * */ </PRE> <A NAME="chdtr"> </A> <PRE> /* chdtrl.c * * Chi-square distribution * * * * SYNOPSIS: * * long double df, x, y, chdtrl(); * * y = chdtrl( df, x ); * * * * DESCRIPTION: * * Returns the area under the left hand tail (from 0 to x) * of the Chi square probability density function with * v degrees of freedom. * * * inf. * - * 1 | | v/2-1 -t/2 * P( x | v ) = ----------- | t e dt * v/2 - | | * 2 | (v/2) - * x * * where x is the Chi-square variable. * * The incomplete gamma integral is used, according to the * formula * * y = chdtr( v, x ) = igam( v/2.0, x/2.0 ). * * * The arguments must both be positive. * * * * ACCURACY: * * See igam(). * * ERROR MESSAGES: * * message condition value returned * chdtr domain x < 0 or v < 1 0.0 */ </PRE> <A NAME="chdtrc"> </A> <PRE> /* chdtrcl() * * Complemented Chi-square distribution * * * * SYNOPSIS: * * long double v, x, y, chdtrcl(); * * y = chdtrcl( v, x ); * * * * DESCRIPTION: * * Returns the area under the right hand tail (from x to * infinity) of the Chi square probability density function * with v degrees of freedom: * * * inf. * - * 1 | | v/2-1 -t/2 * P( x | v ) = ----------- | t e dt * v/2 - | | * 2 | (v/2) - * x * * where x is the Chi-square variable. * * The incomplete gamma integral is used, according to the * formula * * y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ). * * * The arguments must both be positive. * * * * ACCURACY: * * See igamc(). * * ERROR MESSAGES: * * message condition value returned * chdtrc domain x < 0 or v < 1 0.0 */ </PRE> <A NAME="chdtri"> </A> <PRE> /* chdtril() * * Inverse of complemented Chi-square distribution * * * * SYNOPSIS: * * long double df, x, y, chdtril(); * * x = chdtril( df, y ); * * * * * DESCRIPTION: * * Finds the Chi-square argument x such that the integral * from x to infinity of the Chi-square density is equal * to the given cumulative probability y. * * This is accomplished using the inverse gamma integral * function and the relation * * x/2 = igami( df/2, y ); * * * * * ACCURACY: * * See igami.c. * * ERROR MESSAGES: * * message condition value returned * chdtri domain y < 0 or y > 1 0.0 * v < 1 * */ </PRE> <A NAME="clog"> </A> <PRE> /* clogl.c * * Complex natural logarithm * * * * SYNOPSIS: * * void clogl(); * cmplxl z, w; * * clogl( &z, &w ); * * * * DESCRIPTION: * * Returns complex logarithm to the base e (2.718...) of * the complex argument x. * * If z = x + iy, r = sqrt( x**2 + y**2 ), * then * w = log(r) + i arctan(y/x). * * The arctangent ranges from -PI to +PI. * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC -10,+10 7000 8.5e-17 1.9e-17 * IEEE -10,+10 30000 5.0e-15 1.1e-16 * * Larger relative error can be observed for z near 1 +i0. * In IEEE arithmetic the peak absolute error is 5.2e-16, rms * absolute error 1.0e-16. */ </PRE> <A NAME="cexp"> </A> <PRE> /* cexpl() * * Complex exponential function * * * * SYNOPSIS: * * void cexpl(); * cmplxl z, w; * * cexpl( &z, &w ); * * * * DESCRIPTION: * * Returns the exponential of the complex argument z * into the complex result w. * * If * z = x + iy, * r = exp(x), * * then * * w = r cos y + i r sin y. * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC -10,+10 8700 3.7e-17 1.1e-17 * IEEE -10,+10 30000 3.0e-16 8.7e-17 * */ </PRE> <A NAME="csin"> </A> <PRE> /* csinl() * * Complex circular sine * * * * SYNOPSIS: * * void csinl(); * cmplxl z, w; * * csinl( &z, &w ); * * * * DESCRIPTION: * * If * z = x + iy, * * then * * w = sin x cosh y + i cos x sinh y. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC -10,+10 8400 5.3e-17 1.3e-17 * IEEE -10,+10 30000 3.8e-16 1.0e-16 * Also tested by csin(casin(z)) = z. * */ </PRE> <A NAME="ccos"> </A> <PRE> /* ccosl() * * Complex circular cosine * * * * SYNOPSIS: * * void ccosl(); * cmplxl z, w; * * ccosl( &z, &w ); * * * * DESCRIPTION: * * If * z = x + iy, * * then * * w = cos x cosh y - i sin x sinh y. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC -10,+10 8400 4.5e-17 1.3e-17 * IEEE -10,+10 30000 3.8e-16 1.0e-16 */ </PRE> <A NAME="ctan"> </A> <PRE> /* ctanl() * * Complex circular tangent * * * * SYNOPSIS: * * void ctanl(); * cmplxl z, w; * * ctanl( &z, &w ); * * * * DESCRIPTION: * * If * z = x + iy, * * then * * sin 2x + i sinh 2y * w = --------------------. * cos 2x + cosh 2y * * On the real axis the denominator is zero at odd multiples * of PI/2. The denominator is evaluated by its Taylor * series near these points. * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC -10,+10 5200 7.1e-17 1.6e-17 * IEEE -10,+10 30000 7.2e-16 1.2e-16 * Also tested by ctan * ccot = 1 and catan(ctan(z)) = z. */ </PRE> <A NAME="ccot"> </A> <PRE> /* ccotl() * * Complex circular cotangent * * * * SYNOPSIS: * * void ccotl(); * cmplxl z, w; * * ccotl( &z, &w ); * * * * DESCRIPTION: * * If * z = x + iy, * * then * * sin 2x - i sinh 2y * w = --------------------. * cosh 2y - cos 2x * * On the real axis, the denominator has zeros at even * multiples of PI/2. Near these points it is evaluated * by a Taylor series. * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC -10,+10 3000 6.5e-17 1.6e-17 * IEEE -10,+10 30000 9.2e-16 1.2e-16 * Also tested by ctan * ccot = 1 + i0. */ </PRE> <A NAME="casin"> </A> <PRE> /* casinl() * * Complex circular arc sine * * * * SYNOPSIS: * * void casinl(); * cmplxl z, w; * * casinl( &z, &w ); * * * * DESCRIPTION: * * Inverse complex sine: * * 2 * w = -i clog( iz + csqrt( 1 - z ) ). * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC -10,+10 10100 2.1e-15 3.4e-16 * IEEE -10,+10 30000 2.2e-14 2.7e-15 * Larger relative error can be observed for z near zero. * Also tested by csin(casin(z)) = z. */ </PRE> <A NAME="cacos"> </A> <PRE> /* cacosl() * * Complex circular arc cosine * * * * SYNOPSIS: * * void cacosl(); * cmplxl z, w; * * cacosl( &z, &w ); * * * * DESCRIPTION: * * * w = arccos z = PI/2 - arcsin z. * * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC -10,+10 5200 1.6e-15 2.8e-16 * IEEE -10,+10 30000 1.8e-14 2.2e-15 */ </PRE> <A NAME="catan"> </A> <PRE> /* catanl() * * Complex circular arc tangent * * * * SYNOPSIS: * * void catanl(); * cmplxl z, w; * * catanl( &z, &w ); * * * * DESCRIPTION: * * If * z = x + iy, * * then * 1 ( 2x ) * Re w = - arctan(-----------) + k PI * 2 ( 2 2) * (1 - x - y ) * * ( 2 2) * 1 (x + (y+1) ) * Im w = - log(------------) * 4 ( 2 2) * (x + (y-1) ) * * Where k is an arbitrary integer. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC -10,+10 5900 1.3e-16 7.8e-18 * IEEE -10,+10 30000 2.3e-15 8.5e-17 * The check catan( ctan(z) ) = z, with |x| and |y| < PI/2, * had peak relative error 1.5e-16, rms relative error * 2.9e-17. See also clog(). */ </PRE> <A NAME="cmplx"> </A> <PRE> /* cmplxl.c * * Complex number arithmetic * * * * SYNOPSIS: * * typedef struct { * long double r; real part * long double i; imaginary part * }cmplxl; * * cmplxl *a, *b, *c; * * caddl( a, b, c ); c = b + a * csubl( a, b, c ); c = b - a * cmull( a, b, c ); c = b * a * cdivl( a, b, c ); c = b / a * cnegl( c ); c = -c * cmovl( b, c ); c = b * * * * DESCRIPTION: * * Addition: * c.r = b.r + a.r * c.i = b.i + a.i * * Subtraction: * c.r = b.r - a.r * c.i = b.i - a.i * * Multiplication: * c.r = b.r * a.r - b.i * a.i * c.i = b.r * a.i + b.i * a.r * * Division: * d = a.r * a.r + a.i * a.i * c.r = (b.r * a.r + b.i * a.i)/d * c.i = (b.i * a.r - b.r * a.i)/d * ACCURACY: * * In DEC arithmetic, the test (1/z) * z = 1 had peak relative * error 3.1e-17, rms 1.2e-17. The test (y/z) * (z/y) = 1 had * peak relative error 8.3e-17, rms 2.1e-17. * * Tests in the rectangle {-10,+10}: * Relative error: * arithmetic function # trials peak rms * DEC cadd 10000 1.4e-17 3.4e-18 * IEEE cadd 100000 1.1e-16 2.7e-17 * DEC csub 10000 1.4e-17 4.5e-18 * IEEE csub 100000 1.1e-16 3.4e-17 * DEC cmul 3000 2.3e-17 8.7e-18 * IEEE cmul 100000 2.1e-16 6.9e-17 * DEC cdiv 18000 4.9e-17 1.3e-17 * IEEE cdiv 100000 3.7e-16 1.1e-16 */ </PRE> <A NAME="cosh"> </A> <PRE> /* coshl.c * * Hyperbolic cosine, long double precision * * * * SYNOPSIS: * * long double x, y, coshl(); * * y = coshl( x ); * * * * DESCRIPTION: * * Returns hyperbolic cosine of argument in the range MINLOGL to * MAXLOGL. * * cosh(x) = ( exp(x) + exp(-x) )/2. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE +-10000 30000 1.1e-19 2.8e-20 * * * ERROR MESSAGES: * * message condition value returned * cosh overflow |x| > MAXLOGL+LOGE2L INFINITYL * * */ </PRE> <A NAME="ellie"> </A> <PRE> /* elliel.c * * Incomplete elliptic integral of the second kind * * * * SYNOPSIS: * * long double phi, m, y, elliel(); * * y = elliel( phi, m ); * * * * DESCRIPTION: * * Approximates the integral * * * phi * - * | | * | 2 * E(phi_\m) = | sqrt( 1 - m sin t ) dt * | * | | * - * 0 * * of amplitude phi and modulus m, using the arithmetic - * geometric mean algorithm. * * * * ACCURACY: * * Tested at random arguments with phi in [-10, 10] and m in * [0, 1]. * Relative error: * arithmetic domain # trials peak rms * IEEE -10,10 50000 2.7e-18 2.3e-19 * * */ </PRE> <A NAME="ellik"> </A> <PRE> /* ellikl.c * * Incomplete elliptic integral of the first kind * * * * SYNOPSIS: * * long double phi, m, y, ellikl(); * * y = ellikl( phi, m ); * * * * DESCRIPTION: * * Approximates the integral * * * * phi * - * | | * | dt * F(phi_\m) = | ------------------ * | 2 * | | sqrt( 1 - m sin t ) * - * 0 * * of amplitude phi and modulus m, using the arithmetic - * geometric mean algorithm. * * * * * ACCURACY: * * Tested at random points with m in [0, 1] and phi as indicated. * * Relative error: * arithmetic domain # trials peak rms * IEEE -10,10 30000 3.6e-18 4.1e-19 * * */ </PRE> <A NAME="ellpe"> </A> <PRE> /* ellpel.c * * Complete elliptic integral of the second kind * * * * SYNOPSIS: * * long double m1, y, ellpel(); * * y = ellpel( m1 ); * * * * DESCRIPTION: * * Approximates the integral * * * pi/2 * - * | | 2 * E(m) = | sqrt( 1 - m sin t ) dt * | | * - * 0 * * 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 ellpk(). * * E(1) = 1; E(0) = pi/2. * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE 0, 1 10000 1.1e-19 3.5e-20 * * * ERROR MESSAGES: * * message condition value returned * ellpel domain x<0, x>1 0.0 * */ </PRE> <A NAME="ellpj"> </A> <PRE> /* ellpjl.c * * Jacobian Elliptic Functions * * * * SYNOPSIS: * * long double u, m, sn, cn, dn, phi; * int ellpjl(); * * ellpjl( u, m, &sn, &cn, &dn, &phi ); * * * * DESCRIPTION: * * * Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m), * and dn(u|m) of parameter m between 0 and 1, and real * argument u. * * These functions are periodic, with quarter-period on the * real axis equal to the complete elliptic integral * ellpk(1.0-m). * * Relation to incomplete elliptic integral: * If u = ellik(phi,m), then sn(u|m) = sin(phi), * and cn(u|m) = cos(phi). Phi is called the amplitude of u. * * Computation is by means of the arithmetic-geometric mean * algorithm, except when m is within 1e-12 of 0 or 1. In the * latter case with m close to 1, the approximation applies * only for phi < pi/2. * * ACCURACY: * * Tested at random points with u between 0 and 10, m between * 0 and 1. * * Absolute error (* = relative error): * arithmetic function # trials peak rms * IEEE sn 10000 1.7e-18 2.3e-19 * IEEE cn 20000 1.6e-18 2.2e-19 * IEEE dn 10000 4.7e-15 2.7e-17 * IEEE phi 10000 4.0e-19* 6.6e-20* * * Accuracy deteriorates when u is large. * */ </PRE> <A NAME="ellpk"> </A> <PRE> /* ellpkl.c * * Complete elliptic integral of the first kind * * * * SYNOPSIS: * * long double m1, y, ellpkl(); * * y = ellpkl( m1 ); * * * * DESCRIPTION: * * Approximates the integral * * * * pi/2 * - * | | * | dt * K(m) = | ------------------ * | 2 * | | sqrt( 1 - m sin t ) * - * 0 * * where m = 1 - m1, using the approximation * * P(x) - log x Q(x). * * The argument m1 is used rather than m so that the logarithmic * singularity at m = 1 will be shifted to the origin; this * preserves maximum accuracy. * * K(0) = pi/2. * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE 0,1 10000 1.1e-19 3.3e-20 * * ERROR MESSAGES: * * message condition value returned * ellpkl domain x<0, x>1 0.0 * */ </PRE> <A NAME="exp10"> </A> <PRE> /* exp10l.c * * Base 10 exponential function, long double precision * (Common antilogarithm) * * * * SYNOPSIS: * * long double x, y, exp10l() * * y = exp10l( x ); * * * * DESCRIPTION: * * Returns 10 raised to the x power. * * Range reduction is accomplished by expressing the argument * as 10**x = 2**n 10**f, with |f| < 0.5 log10(2). * The Pade' form * * 1 + 2x P(x**2)/( Q(x**2) - P(x**2) ) * * is used to approximate 10**f. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE +-4900 30000 1.0e-19 2.7e-20 * * ERROR MESSAGES: * * message condition value returned * exp10l underflow x < -MAXL10 0.0 * exp10l overflow x > MAXL10 MAXNUM * * IEEE arithmetic: MAXL10 = 4932.0754489586679023819 * */ </PRE> <A NAME="exp2"> </A> <PRE> /* exp2l.c * * Base 2 exponential function, long double precision * * * * SYNOPSIS: * * long double x, y, exp2l(); * * y = exp2l( x ); * * * * DESCRIPTION: * * Returns 2 raised to the x power. * * Range reduction is accomplished by separating the argument * into an integer k and fraction f such that * x k f * 2 = 2 2. * * A Pade' form * * 1 + 2x P(x**2) / (Q(x**2) - x P(x**2) ) * * approximates 2**x in the basic range [-0.5, 0.5]. * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE +-16300 300000 9.1e-20 2.6e-20 * * * See exp.c for comments on error amplification. * * * ERROR MESSAGES: * * message condition value returned * exp2l underflow x < -16382 0.0 * exp2l overflow x >= 16384 MAXNUM * */ </PRE> <A NAME="exp"> </A> <PRE> /* expl.c * * Exponential function, long double precision * * * * SYNOPSIS: * * long double x, y, expl(); * * y = expl( x ); * * * * DESCRIPTION: * * Returns e (2.71828...) raised to the x power. * * Range reduction is accomplished by separating the argument * into an integer k and fraction f such that * * x k f * e = 2 e. * * A Pade' form of degree 2/3 is used to approximate exp(f) - 1 * in the basic range [-0.5 ln 2, 0.5 ln 2]. * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE +-10000 50000 1.12e-19 2.81e-20 * * * Error amplification in the exponential function can be * a serious matter. The error propagation involves * exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ), * which shows that a 1 lsb error in representing X produces * a relative error of X times 1 lsb in the function. * While the routine gives an accurate result for arguments * that are exactly represented by a long double precision * computer number, the result contains amplified roundoff * error for large arguments not exactly represented. * * * ERROR MESSAGES: * * message condition value returned * exp underflow x < MINLOG 0.0 * exp overflow x > MAXLOG MAXNUM * */ </PRE> <A NAME="expm1"> </A> <PRE> /* expm1l.c * * Exponential function, minus 1 * Long double precision * * * SYNOPSIS: * * long double x, y, expm1l(); * * y = expm1l( x ); * * * * DESCRIPTION: * * Returns e (2.71828...) raised to the x power, minus 1. * * Range reduction is accomplished by separating the argument * into an integer k and fraction f such that * * x k f * e = 2 e. * * An expansion x + .5 x^2 + x^3 R(x) approximates exp(f) - 1 * in the basic range [-0.5 ln 2, 0.5 ln 2]. * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -45,+MAXLOG 200,000 1.2e-19 2.5e-20 * * ERROR MESSAGES: * * message condition value returned * expm1l overflow x > MAXLOG MAXNUM * */ </PRE> <A NAME="expx2"> </A> <PRE> /* expx2l.c * * Exponential of squared argument * * * * SYNOPSIS: * * long double x, y, expx2l(); * int sign; * * y = expx2l( x, sign ); * * * * DESCRIPTION: * * Computes y = exp(x*x) while suppressing error amplification * that would ordinarily arise from the inexactness of the * exponential argument x*x. * * If sign < 0, the result is inverted; i.e., y = exp(-x*x) . * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -106.566, 106.566 10^5 1.6e-19 4.4e-20 * */ </PRE> <A NAME="fdtr"> </A> <PRE> /* fdtrl.c * * F distribution, long double precision * * * * SYNOPSIS: * * int df1, df2; * long double x, y, fdtrl(); * * y = fdtrl( df1, df2, x ); * * * * DESCRIPTION: * * Returns the area from zero to x under the F density * function (also known as Snedcor's density or the * variance ratio density). This is the density * of x = (u1/df1)/(u2/df2), where u1 and u2 are random * variables having Chi square distributions with df1 * and df2 degrees of freedom, respectively. * * The incomplete beta integral is used, according to the * formula * * P(x) = incbetl( df1/2, df2/2, (df1*x/(df2 + df1*x) ). * * * The arguments a and b are greater than zero, and x * x is nonnegative. * * ACCURACY: * * Tested at random points (a,b,x) in the indicated intervals. * x a,b Relative error: * arithmetic domain domain # trials peak rms * IEEE 0,1 1,100 10000 9.3e-18 2.9e-19 * IEEE 0,1 1,10000 10000 1.9e-14 2.9e-15 * IEEE 1,5 1,10000 10000 5.8e-15 1.4e-16 * * ERROR MESSAGES: * * message condition value returned * fdtrl domain a<0, b<0, x<0 0.0 * */ </PRE> <A NAME="fdtrc"> </A> <PRE> /* fdtrcl() * * Complemented F distribution * * * * SYNOPSIS: * * int df1, df2; * long double x, y, fdtrcl(); * * y = fdtrcl( df1, df2, x ); * * * * DESCRIPTION: * * Returns the area from x to infinity under the F density * function (also known as Snedcor's density or the * variance ratio density). * * * inf. * - * 1 | | a-1 b-1 * 1-P(x) = ------ | t (1-t) dt * B(a,b) | | * - * x * * (See fdtr.c.) * * The incomplete beta integral is used, according to the * formula * * P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ). * * * ACCURACY: * * See incbet.c. * Tested at random points (a,b,x). * * x a,b Relative error: * arithmetic domain domain # trials peak rms * IEEE 0,1 0,100 10000 4.2e-18 3.3e-19 * IEEE 0,1 1,10000 10000 7.2e-15 2.6e-16 * IEEE 1,5 1,10000 10000 1.7e-14 3.0e-15 * * ERROR MESSAGES: * * message condition value returned * fdtrcl domain a<0, b<0, x<0 0.0 * */ </PRE> <A NAME="fdtri"> </A> <PRE> /* fdtril() * * Inverse of complemented F distribution * * * * SYNOPSIS: * * int df1, df2; * long double x, p, fdtril(); * * x = fdtril( df1, df2, p ); * * DESCRIPTION: * * Finds the F density argument x such that the integral * from x to infinity of the F density is equal to the * given probability p. * * This is accomplished using the inverse beta integral * function and the relations * * z = incbi( df2/2, df1/2, p ) * x = df2 (1-z) / (df1 z). * * Note: the following relations hold for the inverse of * the uncomplemented F distribution: * * z = incbi( df1/2, df2/2, p ) * x = df2 z / (df1 (1-z)). * * ACCURACY: * * See incbi.c. * Tested at random points (a,b,p). * * a,b Relative error: * arithmetic domain # trials peak rms * For p between .001 and 1: * IEEE 1,100 40000 4.6e-18 2.7e-19 * IEEE 1,10000 30000 1.7e-14 1.4e-16 * For p between 10^-6 and .001: * IEEE 1,100 20000 1.9e-15 3.9e-17 * IEEE 1,10000 30000 2.7e-15 4.0e-17 * * ERROR MESSAGES: * * message condition value returned * fdtril domain p <= 0 or p > 1 0.0 * v < 1 */ </PRE> <A NAME="ceil"> </A> <A NAME="floor"> </A> <A NAME="frexp"> </A> <A NAME="ldexp"> </A> <A NAME="fabs"> </A> <PRE> /* ceill() * floorl() * frexpl() * ldexpl() * fabsl() * signbitl() * isnanl() * isfinitel() * * Floating point numeric utilities * * * * SYNOPSIS: * * long double ceill(), floorl(), frexpl(), ldexpl(), fabsl(); * int signbitl(), isnanl(), isfinitel(); * long double x, y; * int expnt, n; * * y = floorl(x); * y = ceill(x); * y = frexpl( x, &expnt ); * y = ldexpl( x, n ); * y = fabsl( x ); * n = signbitl(x); * n = isnanl(x); * n = isfinitel(x); * * * * DESCRIPTION: * * The following routines return a long double precision floating point * result: * * floorl() returns the largest integer less than or equal to x. * It truncates toward minus infinity. * * ceill() returns the smallest integer greater than or equal * to x. It truncates toward plus infinity. * * frexpl() extracts the exponent from x. It returns an integer * power of two to expnt and the significand between 0.5 and 1 * to y. Thus x = y * 2**expn. * * ldexpl() multiplies x by 2**n. * * fabsl() returns the absolute value of its argument. * * These functions are part of the standard C run time library * for some but not all C compilers. The ones supplied are * written in C for IEEE arithmetic. They should * be used only if your compiler library does not already have * them. * * The IEEE versions assume that denormal numbers are implemented * in the arithmetic. Some modifications will be required if * the arithmetic has abrupt rather than gradual underflow. */ </PRE> <A NAME="gamma"> </A> <PRE> /* gammal.c * * Gamma function * * * * SYNOPSIS: * * long double x, y, gammal(); * extern int sgngam; * * y = gammal( x ); * * * * DESCRIPTION: * * Returns gamma function of the argument. The result is * correctly signed, and the sign (+1 or -1) is also * returned in a global (extern) variable named sgngam. * This variable is also filled in by the logarithmic gamma * function lgam(). * * 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. * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -40,+40 10000 3.6e-19 7.9e-20 * IEEE -1755,+1755 10000 4.8e-18 6.5e-19 * * Accuracy for large arguments is dominated by error in powl(). * */ </PRE> <A NAME="lgam"> </A> <PRE> /* lgaml() * * Natural logarithm of gamma function * * * * SYNOPSIS: * * long double x, y, lgaml(); * extern int sgngam; * * y = lgaml( x ); * * * * DESCRIPTION: * * Returns the base e (2.718...) logarithm of the absolute * value of the gamma function of the argument. * The sign (+1 or -1) of the gamma function is returned in a * global (extern) variable named sgngam. * * 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. * * Arguments greater than MAXLGML (10^4928) return MAXNUML. * * * * ACCURACY: * * * arithmetic domain # trials peak rms * IEEE -40, 40 100000 2.2e-19 4.6e-20 * IEEE 10^-2000,10^+2000 20000 1.6e-19 3.3e-20 * The error criterion was relative when the function magnitude * was greater than one but absolute when it was less than one. * */ </PRE> <A NAME="gdtr"> </A> <PRE> /* gdtrl.c * * Gamma distribution function * * * * SYNOPSIS: * * long double a, b, x, y, gdtrl(); * * y = gdtrl( a, b, x ); * * * * DESCRIPTION: * * Returns the integral from zero to x of the gamma probability * density function: * * * x * b - * a | | b-1 -at * y = ----- | t e dt * - | | * | (b) - * 0 * * The incomplete gamma integral is used, according to the * relation * * y = igam( b, ax ). * * * ACCURACY: * * See igam(). * * ERROR MESSAGES: * * message condition value returned * gdtrl domain x < 0 0.0 * */ </PRE> <A NAME="gdtrc"> </A> <PRE> /* gdtrcl.c * * Complemented gamma distribution function * * * * SYNOPSIS: * * long double a, b, x, y, gdtrcl(); * * y = gdtrcl( a, b, x ); * * * * DESCRIPTION: * * Returns the integral from x to infinity of the gamma * probability density function: * * * inf. * b - * a | | b-1 -at * y = ----- | t e dt * - | | * | (b) - * x * * The incomplete gamma integral is used, according to the * relation * * y = igamc( b, ax ). * * * ACCURACY: * * See igamc(). * * ERROR MESSAGES: * * message condition value returned * gdtrcl domain x < 0 0.0 * */ </PRE> <A NAME="gels"> </A> <PRE> /* C C .................................................................. C C SUBROUTINE GELS C C PURPOSE C TO SOLVE A SYSTEM OF SIMULTANEOUS LINEAR EQUATIONS WITH C SYMMETRIC COEFFICIENT MATRIX UPPER TRIANGULAR PART OF WHICH C IS ASSUMED TO BE STORED COLUMNWISE. C C USAGE C CALL GELS(R,A,M,N,EPS,IER,AUX) C C DESCRIPTION OF PARAMETERS C R - M BY N RIGHT HAND SIDE MATRIX. (DESTROYED) C ON RETURN R CONTAINS THE SOLUTION OF THE EQUATIONS. C A - UPPER TRIANGULAR PART OF THE SYMMETRIC C M BY M COEFFICIENT MATRIX. (DESTROYED) C M - THE NUMBER OF EQUATIONS IN THE SYSTEM. C N - THE NUMBER OF RIGHT HAND SIDE VECTORS. C EPS - AN INPUT CONSTANT WHICH IS USED AS RELATIVE C TOLERANCE FOR TEST ON LOSS OF SIGNIFICANCE. C IER - RESULTING ERROR PARAMETER CODED AS FOLLOWS C IER=0 - NO ERROR, C IER=-1 - NO RESULT BECAUSE OF M LESS THAN 1 OR C PIVOT ELEMENT AT ANY ELIMINATION STEP C EQUAL TO 0, C IER=K - WARNING DUE TO POSSIBLE LOSS OF SIGNIFI- C CANCE INDICATED AT ELIMINATION STEP K+1, C WHERE PIVOT ELEMENT WAS LESS THAN OR C EQUAL TO THE INTERNAL TOLERANCE EPS TIMES C ABSOLUTELY GREATEST MAIN DIAGONAL C ELEMENT OF MATRIX A. C AUX - AN AUXILIARY STORAGE ARRAY WITH DIMENSION M-1. C C REMARKS C UPPER TRIANGULAR PART OF MATRIX A IS ASSUMED TO BE STORED C COLUMNWISE IN M*(M+1)/2 SUCCESSIVE STORAGE LOCATIONS, RIGHT C HAND SIDE MATRIX R COLUMNWISE IN N*M SUCCESSIVE STORAGE C LOCATIONS. ON RETURN SOLUTION MATRIX R IS STORED COLUMNWISE C TOO. C THE PROCEDURE GIVES RESULTS IF THE NUMBER OF EQUATIONS M IS C GREATER THAN 0 AND PIVOT ELEMENTS AT ALL ELIMINATION STEPS C ARE DIFFERENT FROM 0. HOWEVER WARNING IER=K - IF GIVEN - C INDICATES POSSIBLE LOSS OF SIGNIFICANCE. IN CASE OF A WELL C SCALED MATRIX A AND APPROPRIATE TOLERANCE EPS, IER=K MAY BE C INTERPRETED THAT MATRIX A HAS THE RANK K. NO WARNING IS C GIVEN IN CASE M=1. C ERROR PARAMETER IER=-1 DOES NOT NECESSARILY MEAN THAT C MATRIX A IS SINGULAR, AS ONLY MAIN DIAGONAL ELEMENTS C ARE USED AS PIVOT ELEMENTS. POSSIBLY SUBROUTINE GELG (WHICH C WORKS WITH TOTAL PIVOTING) WOULD BE ABLE TO FIND A SOLUTION. C C SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED C NONE C C METHOD C SOLUTION IS DONE BY MEANS OF GAUSS-ELIMINATION WITH C PIVOTING IN MAIN DIAGONAL, IN ORDER TO PRESERVE C SYMMETRY IN REMAINING COEFFICIENT MATRICES. C C .................................................................. C */ </PRE> <A NAME="hyperg"> </A> <PRE> /* hypergl.c * * Confluent hypergeometric function * * * * SYNOPSIS: * * long double a, b, x, y, hypergl(); * * y = hypergl( a, b, x ); * * * * DESCRIPTION: * * Computes the confluent hypergeometric function * * 1 2 * a x a(a+1) x * F ( a,b;x ) = 1 + ---- + --------- + ... * 1 1 b 1! b(b+1) 2! * * Many higher transcendental functions are special cases of * this power series. * * As is evident from the formula, b must not be a negative * integer or zero unless a is an integer with 0 >= a > b. * * The routine attempts both a direct summation of the series * and an asymptotic expansion. In each case error due to * roundoff, cancellation, and nonconvergence is estimated. * The result with smaller estimated error is returned. * * * * ACCURACY: * * Tested at random points (a, b, x), all three variables * ranging from 0 to 30. * Relative error: * arithmetic domain # trials peak rms * IEEE 0,30 100000 3.3e-18 5.0e-19 * * Larger errors can be observed when b is near a negative * integer or zero. Certain combinations of arguments yield * serious cancellation error in the power series summation * and also are not in the region of near convergence of the * asymptotic series. An error message is printed if the * self-estimated relative error is greater than 1.0e-12. * */ </PRE> <A NAME="ieee"> </A> <PRE> /* ieee.c * * Extended precision IEEE binary floating point arithmetic routines * * Numbers are stored in C language as arrays of 16-bit unsigned * short integers. The arguments of the routines are pointers to * the arrays. * * * External e type data structure, simulates Intel 8087 chip * temporary real format but possibly with a larger significand: * * NE-1 significand words (least significant word first, * most significant bit is normally set) * exponent (value = EXONE for 1.0, * top bit is the sign) * * * Internal data structure of a number (a "word" is 16 bits): * * ei[0] sign word (0 for positive, 0xffff for negative) * ei[1] biased exponent (value = EXONE for the number 1.0) * ei[2] high guard word (always zero after normalization) * ei[3] * to ei[NI-2] significand (NI-4 significand words, * most significant word first, * most significant bit is set) * ei[NI-1] low guard word (0x8000 bit is rounding place) * * * * Routines for external format numbers * * asctoe( string, e ) ASCII string to extended double e type * asctoe64( string, &d ) ASCII string to long double * asctoe53( string, &d ) ASCII string to double * asctoe24( string, &f ) ASCII string to single * asctoeg( string, e, prec ) ASCII string to specified precision * e24toe( &f, e ) IEEE single precision to e type * e53toe( &d, e ) IEEE double precision to e type * e64toe( &d, e ) IEEE long double precision to e type * eabs(e) absolute value * eadd( a, b, c ) c = b + a * eclear(e) e = 0 * ecmp (a, b) Returns 1 if a > b, 0 if a == b, * -1 if a < b, -2 if either a or b is a NaN. * ediv( a, b, c ) c = b / a * efloor( a, b ) truncate to integer, toward -infinity * efrexp( a, exp, s ) extract exponent and significand * eifrac( e, &l, frac ) e to long integer and e type fraction * euifrac( e, &l, frac ) e to unsigned long integer and e type fraction * einfin( e ) set e to infinity, leaving its sign alone * eldexp( a, n, b ) multiply by 2**n * emov( a, b ) b = a * emul( a, b, c ) c = b * a * eneg(e) e = -e * eround( a, b ) b = nearest integer value to a * esub( a, b, c ) c = b - a * e24toasc( &f, str, n ) single to ASCII string, n digits after decimal * e53toasc( &d, str, n ) double to ASCII string, n digits after decimal * e64toasc( &d, str, n ) long double to ASCII string * etoasc( e, str, n ) e to ASCII string, n digits after decimal * etoe24( e, &f ) convert e type to IEEE single precision * etoe53( e, &d ) convert e type to IEEE double precision * etoe64( e, &d ) convert e type to IEEE long double precision * ltoe( &l, e ) long (32 bit) integer to e type * ultoe( &l, e ) unsigned long (32 bit) integer to e type * eisneg( e ) 1 if sign bit of e != 0, else 0 * eisinf( e ) 1 if e has maximum exponent (non-IEEE) * or is infinite (IEEE) * eisnan( e ) 1 if e is a NaN * esqrt( a, b ) b = square root of a * * * Routines for internal format numbers * * eaddm( ai, bi ) add significands, bi = bi + ai * ecleaz(ei) ei = 0 * ecleazs(ei) set ei = 0 but leave its sign alone * ecmpm( ai, bi ) compare significands, return 1, 0, or -1 * edivm( ai, bi ) divide significands, bi = bi / ai * emdnorm(ai,l,s,exp) normalize and round off * emovi( a, ai ) convert external a to internal ai * emovo( ai, a ) convert internal ai to external a * emovz( ai, bi ) bi = ai, low guard word of bi = 0 * emulm( ai, bi ) multiply significands, bi = bi * ai * enormlz(ei) left-justify the significand * eshdn1( ai ) shift significand and guards down 1 bit * eshdn8( ai ) shift down 8 bits * eshdn6( ai ) shift down 16 bits * eshift( ai, n ) shift ai n bits up (or down if n < 0) * eshup1( ai ) shift significand and guards up 1 bit * eshup8( ai ) shift up 8 bits * eshup6( ai ) shift up 16 bits * esubm( ai, bi ) subtract significands, bi = bi - ai * * * The result is always normalized and rounded to NI-4 word precision * after each arithmetic operation. * * Exception flags are NOT fully supported. * * Define INFINITY in mconf.h for support of infinity; otherwise a * saturation arithmetic is implemented. * * Define NANS for support of Not-a-Number items; otherwise the * arithmetic will never produce a NaN output, and might be confused * by a NaN input. * If NaN's are supported, the output of ecmp(a,b) is -2 if * either a or b is a NaN. This means asking if(ecmp(a,b) < 0) * may not be legitimate. Use if(ecmp(a,b) == -1) for less-than * if in doubt. * Signaling NaN's are NOT supported; they are treated the same * as quiet NaN's. * * Denormals are always supported here where appropriate (e.g., not * for conversion to DEC numbers). */ /* * Revision history: * * 5 Jan 84 PDP-11 assembly language version * 2 Mar 86 fixed bug in asctoq() * 6 Dec 86 C language version * 30 Aug 88 100 digit version, improved rounding * 15 May 92 80-bit long double support * * Author: S. L. Moshier. */ </PRE> <A NAME="igami"> </A> <PRE> /* igamil() * * Inverse of complemented imcomplete gamma integral * * * * SYNOPSIS: * * long double a, x, y, igamil(); * * x = igamil( a, y ); * * * * DESCRIPTION: * * Given y, the function finds x such that * * igamc( a, x ) = y. * * Starting with the approximate value * * 3 * x = a t * * where * * t = 1 - d - ndtri(y) sqrt(d) * * and * * d = 1/9a, * * the routine performs up to 10 Newton iterations to find the * root of igamc(a,x) - y = 0. * * * ACCURACY: * * Tested for a ranging from 0.5 to 30 and x from 0 to 0.5. * * Relative error: * arithmetic domain # trials peak rms * DEC 0,0.5 3400 8.8e-16 1.3e-16 * IEEE 0,0.5 10000 1.1e-14 1.0e-15 * */ </PRE> <A NAME="igam"> </A> <PRE> /* igaml.c * * Incomplete gamma integral * * * * SYNOPSIS: * * long double a, x, y, igaml(); * * y = igaml( a, x ); * * * * DESCRIPTION: * * The function is defined by * * x * - * 1 | | -t a-1 * igam(a,x) = ----- | e t dt. * - | | * | (a) - * 0 * * * 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. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC 0,30 4000 4.4e-15 6.3e-16 * IEEE 0,30 10000 3.6e-14 5.1e-15 * */ </PRE> <A NAME="igamc"> </A> <PRE> /* igamcl() * * Complemented incomplete gamma integral * * * * SYNOPSIS: * * long double a, x, y, igamcl(); * * y = igamcl( a, x ); * * * * DESCRIPTION: * * The function is defined by * * * igamc(a,x) = 1 - igam(a,x) * * inf. * - * 1 | | -t a-1 * = ----- | e t dt. * - | | * | (a) - * x * * * 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. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC 0,30 2000 2.7e-15 4.0e-16 * IEEE 0,30 60000 1.4e-12 6.3e-15 * */ </PRE> <A NAME="incbet"> </A> <PRE> /* incbetl.c * * Incomplete beta integral * * * SYNOPSIS: * * long double a, b, x, y, incbetl(); * * y = incbetl( a, b, x ); * * * DESCRIPTION: * * Returns incomplete beta integral of the arguments, evaluated * from zero to x. The function is defined as * * x * - - * | (a+b) | | a-1 b-1 * ----------- | t (1-t) dt. * - - | | * | (a) | (b) - * 0 * * 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 * * 1 - incbet( a, b, x ) = incbet( b, a, 1-x ). * * The integral is evaluated by a continued fraction expansion * or, when b*x is small, by a power series. * * ACCURACY: * * Tested at random points (a,b,x) with x between 0 and 1. * arithmetic domain # trials peak rms * IEEE 0,5 20000 4.5e-18 2.4e-19 * IEEE 0,100 100000 3.9e-17 1.0e-17 * Half-integer a, b: * IEEE .5,10000 100000 3.9e-14 4.4e-15 * Outputs smaller than the IEEE gradual underflow threshold * were excluded from these statistics. * * ERROR MESSAGES: * * message condition value returned * incbetl domain x<0, x>1 0.0 */ </PRE> <A NAME="incbi"> </A> <PRE> /* incbil() * * Inverse of imcomplete beta integral * * * * SYNOPSIS: * * long double a, b, x, y, incbil(); * * x = incbil( a, b, y ); * * * * DESCRIPTION: * * Given y, the function finds x such that * * incbet( a, b, x ) = y. * * the routine performs up to 10 Newton iterations to find the * root of incbet(a,b,x) - y = 0. * * * ACCURACY: * * Relative error: * x a,b * arithmetic domain domain # trials peak rms * IEEE 0,1 .5,10000 10000 1.1e-14 1.4e-16 */ </PRE> <A NAME="isnan"> </A> <A NAME="isfinite"> </A> <A NAME="signbit"> </A> <PRE> /* isnanl() * isfinitel() * signbitl() * * Floating point IEEE special number tests * * * * SYNOPSIS: * * int signbitl(), isnanl(), isfinitel(); * long double x, y; * * n = signbitl(x); * n = isnanl(x); * n = isfinitel(x); * * * * DESCRIPTION: * * These functions are part of the standard C run time library * for some but not all C compilers. The ones supplied are * written in C for IEEE arithmetic. They should * be used only if your compiler library does not already have * them. * */ </PRE> <A NAME="j0"> </A> <PRE> /* j0l.c * * Bessel function of order zero * * * * SYNOPSIS: * * long double x, y, j0l(); * * y = j0l( x ); * * * * DESCRIPTION: * * Returns Bessel function of first kind, order zero of the argument. * * The domain is divided into the intervals [0, 9] and * (9, infinity). In the first interval the rational approximation * is (x^2 - r^2) (x^2 - s^2) (x^2 - t^2) P7(x^2) / Q8(x^2), * where r, s, t are the first three zeros of the function. * In the second interval the expansion is in terms of the * modulus M0(x) = sqrt(J0(x)^2 + Y0(x)^2) and phase P0(x) * = atan(Y0(x)/J0(x)). M0 is approximated by sqrt(1/x)P7(1/x)/Q7(1/x). * The approximation to J0 is M0 * cos(x - pi/4 + 1/x P5(1/x^2)/Q6(1/x^2)). * * * ACCURACY: * * Absolute error: * arithmetic domain # trials peak rms * IEEE 0, 30 100000 2.8e-19 7.4e-20 * * */ </PRE> <A NAME="y0"> </A> <PRE> /* y0l.c * * Bessel function of the second kind, order zero * * * * SYNOPSIS: * * double x, y, y0l(); * * y = y0l( x ); * * * * DESCRIPTION: * * Returns Bessel function of the second kind, of order * zero, of the argument. * * The domain is divided into the intervals [0, 5>, [5,9> and * [9, infinity). In the first interval a rational approximation * R(x) is employed to compute y0(x) = R(x) + 2/pi * log(x) * j0(x). * * In the second interval, the approximation is * (x - p)(x - q)(x - r)(x - s)P7(x)/Q7(x) * where p, q, r, s are zeros of y0(x). * * The third interval uses the same approximations to modulus * and phase as j0(x), whence y0(x) = modulus * sin(phase). * * ACCURACY: * * Absolute error, when y0(x) < 1; else relative error: * * arithmetic domain # trials peak rms * IEEE 0, 30 100000 3.4e-19 7.6e-20 * */ </PRE> <A NAME="j1"> </A> <PRE> /* j1l.c * * Bessel function of order one * * * * SYNOPSIS: * * long double x, y, j1l(); * --> -------------------- --> maximum size reached --> -------------------- ¤ Dauer der Verarbeitung: 0.146 Sekunden (vorverarbeitet) ¤ |
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