/* qacosh.c * * Inverse hyperbolic cosine * * * * SYNOPSIS: * * int qacosh( x, y ) * QELT *x, *y; * * qacosh( x, y ); * * * * DESCRIPTION: * * acosh(x) = log( x + sqrt( (x-1)(x+1) ). * */
/* qairy.c * * Airy functions * * * * SYNOPSIS: * * int qairy( x, ai, aip, bi, bip ); * QELT *x, *ai, *aip, *bi, *bip; * * qairy( x, ai, aip, bi, bip ); * * * * DESCRIPTION: * * Solution of the differential equation * * y"(x) = xy. * * The function returns the two independent solutions Ai, Bi * and their first derivatives Ai'(x), Bi'(x). * * Evaluation is by power series summation for small x, * by asymptotic expansion for large x. * * * ACCURACY: * * The asymptotic expansion is truncated at less than full working precision. * */
/* qasin.c * * Inverse circular sine * * * * SYNOPSIS: * * int qasin( x, y ); * QELT *x, *y; * * qasin( x, y ); * * * * DESCRIPTION: * * Returns radian angle between -pi/2 and +pi/2 whose sine is x. * * asin(x) = arctan (x / sqrt(1 - x^2)) * * If |x| > 0.5 it is transformed by the identity * * asin(x) = pi/2 - 2 asin( sqrt( (1-x)/2 ) ). * */
/* qacos * * Inverse circular cosine * * * * SYNOPSIS: * * int qacos( x, y ); * QELT x[], y[]; * * qacos( x, y ); * * * * DESCRIPTION: * * Returns radian angle between 0 and pi whose cosine * is x. * * acos(x) = pi/2 - asin(x) * */
/* qasinh.c * * Inverse hyperbolic sine * * * * SYNOPSIS: * * int qasinh( x, y ); * QELT *x, *y; * * qasinh( x, y ); * * * * DESCRIPTION: * * Returns inverse hyperbolic sine of argument. * * asinh(x) = log( x + sqrt(1 + x*x) ). * * For very large x, asinh(x) = log x + log 2. * */
/* qatanh.c * * Inverse hyperbolic tangent * * * * SYNOPSIS: * * int qatanh( x, y ); * QELT x[], y[]; * * qatanh( x, y ); * * * * DESCRIPTION: * * Returns inverse hyperbolic tangent of argument. * * atanh(x) = 0.5 * log( (1+x)/(1-x) ). * * For very small x, the first few terms of the Taylor series * are summed. * */
/* qatn * * Inverse circular tangent * (arctangent) * * * * SYNOPSIS: * * int qatn( x, y ); * QELT *x, *y; * * qatn( x, y ); * * * * DESCRIPTION: * * Returns radian angle between -pi/2 and +pi/2 whose tangent * is x. * * Range reduction is from three intervals into the interval * from zero to pi/8. * * 2 2 2 * x x 4 x 9 x * arctan(x) = --- --- ---- ---- ... * 1 - 3 - 5 - 7 - * */
/* qatn2 * * Quadrant correct inverse circular tangent * * * * SYNOPSIS: * * int qatn2( y, x, z ); * QELT *x, *y, *z; * * qatn2( y, x, z ); * * * * DESCRIPTION: * * Returns radian angle -PI < z < PI whose tangent is y/x. * */
/* qbeta.c * * Beta function * * * * SYNOPSIS: * * int qbeta( a, b, y ); * QELT *a, *b, *y; * * qbeta( a, b, y ); * * * * DESCRIPTION: * * - - * | (a) | (b) * beta( a, b ) = -----------. * - * | (a+b) * */
/* qcbrt.c * * Cube root * * * * SYNOPSIS: * * int qcbrt( x, y ); * QELT *x, *y; * * qcbrt( x, y ); * * * * DESCRIPTION: * * Returns the cube root of the argument, which may be negative. * */
/* qcgamma * * Complex gamma function * * * * SYNOPSIS: * * int qcgamma( x, y ); * qcmplx *x, *y; * * qcgamma( x, y ); * * * * DESCRIPTION: * * Returns complex-valued gamma function of the complex argument. * * gamma(x) = exp (log(gamma(x))) * */
/* qclgam * * Natural logarithm of complex gamma function * * * * SYNOPSIS: * * int qclgam( x, y ); * qcmplx *x, *y; * * qclgam( x, y ); * * * * DESCRIPTION: * * Returns the base e (2.718...) logarithm of the complex gamma * function of the argument. * * The logarithm of the gamma function is approximated by the * logarithmic version of Stirling's asymptotic formula. * Arguments of real part less than +32 are increased by recurrence. * The cosecant reflection formula is employed for arguments * having real part less than -34. * */
/* qchyp1f1.c * * confluent hypergeometric function * * 1 2 * a x a(a+1) x * F ( a,b;x ) = 1 + ---- + --------- + ... * 1 1 b 1! b(b+1) 2! * * * Series summation terminates at 70 bits accuracy. * */
/* qcmplx.c * Q type complex number arithmetic * * The syntax of arguments in * * cfunc( a, b, c ) * * is * c = b + a * c = b - a * c = b * a * c = b / a. */
/* qcos.c * * Circular cosine * * * * SYNOPSIS: * * int qcos( x, y ); * QELT *x, *y; * * qcos( x, y ); * * * * DESCRIPTION: * * cos(x) = sin(pi/2 - x) * */
/* qcosh.c * * Hyperbolic cosine * * * * SYNOPSIS: * * int qcosh(x, y); * QELT *x, *y; * * qcosh(x, y); * * * * DESCRIPTION: * * cosh(x) = ( exp(x) + exp(-x) )/2. * */
/*
qcpolylog.c
Complex polylogarithms.
inf k
- x
Li (x) = > ---
n - n
k=1 k
x
-
| | -ln(1-t)
Li (x) = | -------- dt
2 | | t
-
0
1-x
-
| | ln t
= | ------ dt = spence(1-x)
| | 1 - t
-
1
2 3
x x
= x + --- + --- + ...
4 9
d 1
-- Li (x) = --- Li (x)
dx n x n-1
*/
/* qdawsn.c * * Dawson's Integral * * * * SYNOPSIS: * * int qdawsn( x, y ); * QELT *x, *y; * * qdawsn( x, y ); * * * * DESCRIPTION: * * Approximates the integral * * x * - * 2 | | 2 * dawsn(x) = exp( -x ) | exp( t ) dt * | | * - * 0 * * * * ACCURACY: * * Series expansions are truncated at NBITS/2. * */
/* qei.c * * Exponential integral * * * SYNOPSIS: * * QELT *x, *y; * * qei( x, y ); * * * * DESCRIPTION: * * x * - t * | | e * Ei(x) = -|- --- dt . * | | t * - * -inf * * Not defined for x <= 0. * See also qexpn.c. * * ACCURACY: * * Series truncated at NBITS/2. * */
/* qellie.c * * Incomplete elliptic integral of the second kind * * * * SYNOPSIS: * * int qellie( phi, m, y ); * QELT *phi, *m, *y; * * qellie( phi, m, y ); * * * * 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: * * Sequence terminates at NBITS/2. * */
/* qellik.c * * Incomplete elliptic integral of the first kind * * * * SYNOPSIS: * * int qellik( phi, m, y ); * QELT *phi, *m, *y; * * qellik( phi, m, y ); * * * * 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: * * Sequence terminates at NBITS/2. * */
/* qellpe.c * * Complete elliptic integral of the second kind * * * * SYNOPSIS: * * int qellpe(x, y); * QELT *x, *y; * * qellpe(x, y); * * * * DESCRIPTION: * * Approximates the integral * * * pi/2 * - * | | 2 * E(m) = | sqrt( 1 - m sin t ) dt * | | * - * 0 * * Where m = 1 - m1, using the arithmetic-geometric mean method. * * * ACCURACY: * * Method terminates at NBITS/2. * */
/* qellpj.c * * Jacobian Elliptic Functions * * * * SYNOPSIS: * * int qellpj( u, m, sn, cn, dn, ph ); * QELT *u, *m; * QELT *sn, *cn, *dn, *ph; * * qellpj( u, m, sn, cn, dn, ph ); * * * * 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-9 of 0 or 1. In the * latter case with m close to 1, the approximation applies * only for phi < pi/2. * * ACCURACY: * * Truncated at 70 bits. * */
/* qellpk.c * * Complete elliptic integral of the first kind * * * * SYNOPSIS: * * int qellpk(x, y); * QELT *x, *y; * * qellpk(x, y); * * * * DESCRIPTION: * * Approximates the integral * * * * pi/2 * - * | | * | dt * K(m) = | ------------------ * | 2 * | | sqrt( 1 - m sin t ) * - * 0 * * where m = 1 - m1, using the arithmetic-geometric mean method. * * 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: * * Truncated at NBITS/2. * */
/* qerf.c * * Error function * * * * SYNOPSIS: * * int qerf( x, y ); * QELT *x, *y; * * qerf( x, y ); * * * * DESCRIPTION: * * The integral is * * x * - * 2 | | 2 * erf(x) = -------- | exp( - t ) dt. * sqrt(pi) | | * - * 0 * * */
/* qerfc.c * * Complementary error function * * * * SYNOPSIS: * * int qerfc( x, y ); * QELT *x, *y; * * qerfc( x, y ); * * * * DESCRIPTION: * * * 1 - erf(x) = * * inf. * - * 2 | | 2 * erfc(x) = -------- | exp( - t ) dt * sqrt(pi) | | * - * x * */
/* qeuclid.c * * Rational arithmetic routines * * radd( a, b, c ) c = b + a * rsub( a, b, c ) c = b - a * rmul( a, b, c ) c = b * a * rdiv( a, b, c ) c = b / a * euclid( n, d ) Reduce n/d to lowest terms, return g.c.d. * * Note: arguments are assumed, * without checking, * to be integer valued. */
/* qexp.c * * Exponential function check routine * * * * SYNOPSIS: * * int qexp( x, y ); * QELT *x, *y; * * qexp( x, y ); * * * * DESCRIPTION: * * Returns e (2.71828...) raised to the x power. * */
/* exp10.c * * Base 10 exponential function * (Common antilogarithm) * * * * SYNOPSIS: * * int qexp10( x, y ); * QELT *x, *y; * * qexp10( x, y ); * * * * DESCRIPTION: * * Returns 10 raised to the x power. * * x x ln 10 * 10 = e * */
/* qexp2.c * * Check routine for base 2 exponential function * * * * SYNOPSIS: * * int qexp2( x, y ); * QELT *x, *y; * * qexp2( x, y ); * * * * DESCRIPTION: * * Returns 2 raised to the x power. * * x ln 2 x x ln 2 * y = 2 = ( e ) = e * */
/* qexpn.c * * Exponential integral En * * * * SYNOPSIS: * * int qexpn( n, x, y ); * int n; * QELT *x, *y; * * qexpn( n, x, y ); * * * * DESCRIPTION: * * Evaluates the exponential integral * * inf. * - * | | -xt * | e * E (x) = | ---- dt. * n | n * | | t * - * 1 * * * Both n and x must be nonnegative. * * * ACCURACY: * * Series expansions are truncated at less than full working precision. * */
/* qfloor.c * qfloor - largest integer not greater than x * qround - nearest integer to x */
/* qflt.c * QFLOAT * * Extended precision floating point routines * * asctoq( string, q ) ascii string to q type * dtoq( &d, q ) DEC double precision to q type * etoq( &d, q ) IEEE double precision to q type * e24toq( &d, q ) IEEE single precision to q type * e113toq( &d, q ) 128-bit long double precision to q type * ltoq( &l, q ) long integer to q type * qabs(q) absolute value * qadd( a, b, c ) c = b + a * qclear(q) q = 0 * qcmp( a, b ) compare a to b * qdiv( a, b, c ) c = b / a * qifrac( x, &l, frac ) x to integer part l and q type fraction * qfrexp( x, l, y ) find exponent l and fraction y between .5 and 1 * qldexp( x, l, y ) multiply x by 2^l * qinfin( x ) set x to infinity, leaving its sign alone * qmov( a, b ) b = a * qmul( a, b, c ) c = b * a * qmuli( a, b, c ) c = b * a, a has only 16 significant bits * qisneg(q) returns sign of q * qneg(q) q = -q * qnrmlz(q) adjust exponent and mantissa * qsub( a, b, c ) c = b - a * qtoasc( a, s, n ) q to ASCII string, n digits after decimal * qtod( q, &d ) convert q type to DEC double precision * qtoe( q, &d ) convert q type to IEEE double precision * qtoe24( q, &d ) convert q type to IEEE single precision * qtoe113( q, &d ) convert q type to 128-bit long double precision * * Data structure of the number (a "word" is 16 bits) * * sign word (0 for positive, -1 for negative) * exponent (EXPONE for 1.0) * high guard word (always zero after normalization) * N-1 mantissa words (most significant word first, * most significant bit is set) * * Numbers are stored in C language as arrays. All routines * use pointers to the arrays as arguments. * * The result is always normalized after each arithmetic operation. * All arithmetic results are chopped. No rounding is performed except * on conversion to double precision. */
/* qflta.c * Utilities for extended precision arithmetic, called by qflt.c. * These should all be written in machine language for speed. * * addm( x, y ) add significand of x to that of y * shdn1( x ) shift significand of x down 1 bit * shdn8( x ) shift significand of x down 8 bits * shdn16( x ) shift significand of x down 16 bits * shup1( x ) shift significand of x up 1 bit * shup8( x ) shift significand of x up 8 bits * shup16( x ) shift significand of x up 16 bits * divm( a, b ) divide significand of a into b * mulm( a, b ) multiply significands, result in b * mdnorm( x ) normalize and round off * * Copyright (c) 1984 - 1988 by Stephen L. Moshier. All rights reserved. */
/* qfresnl * * Fresnel integral * * * * SYNOPSIS: * * int qfresnl( x, s, c ); * QELT *x, *s, *c; * * qfresnl( x, s, c ); * * * DESCRIPTION: * * Evaluates the Fresnel integrals * * x * - * | | * C(x) = | cos(pi/2 t**2) dt, * | | * - * 0 * * x * - * | | * S(x) = | sin(pi/2 t**2) dt. * | | * - * 0 * * * The integrals are evaluated by a power series for x < 1. * For large x auxiliary functions f(x) and g(x) are employed * such that * * C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 ) * S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 ) * * Routine qfresfg computes f and g. * * * ACCURACY: * * Series expansions are truncated at less than full working precision. */
/* qlgam * * Natural logarithm of gamma function * * * * SYNOPSIS: * * int qlgam( x, y ); * QELT *x, *y; * * qlgam( x, y ); * * * * DESCRIPTION: * * Returns the base e (2.718...) logarithm of the absolute * value of the gamma function of the argument. * */
/* qgamma * * Gamma function * * * * SYNOPSIS: * * int qgamma( x, y ); * QELT *x, *y; * * qgamma( x, y ); * * * * DESCRIPTION: * * Returns gamma function of the argument. * * qgamma(x) = exp(qlgam(x)) * */
/* hyp2f1.c * * Gauss hypergeometric function F * 2 1 * * * SYNOPSIS: * * int qhy2f1( a, b, c, x, y ); * QELT *a, *b, *c, *x, *y; * * qhy2f1( a, b, c, x, y ); * * * DESCRIPTION: * * * hyp2f1( a, b, c, x ) = F ( a, b; c; x ) * 2 1 * * inf. * - a(a+1)...(a+k) b(b+1)...(b+k) k+1 * = 1 + > ----------------------------- x . * - c(c+1)...(c+k) (k+1)! * k = 0 * * * ACCURACY: * * Expansions are set to terminate at less than full working precision. * */
/* qhyp.c * * Confluent hypergeometric function * * * * SYNOPSIS: * * int qhyp( a, b, x, y ); * QELT *a, *b, *x, *y; * * qhyp( a, b, x, y ); * * * * 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! * * * ACCURACY: * * Series expansion is truncated at less than full working precision. * */
/* qigam.c * Check routine for incomplete gamma integral * * * * SYNOPSIS: * * For the left tail: * int qigam( a, x, y ); * QELT *a, *x, *y; * qigam( a, x, y ); * * For the right tail: * int qigamc( a, x, y ); * QELT *a, *x, *y; * qigamc( a, x, y ); * * * 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: * * Expansions terminate at less than full working precision. * */
/* qigami() * * Inverse of complemented imcomplete gamma integral * * * * SYNOPSIS: * * int qigami( a, p, x ); * QELT *a, *p, *x; * * qigami( a, p, x ); * * DESCRIPTION: * * The program refines an initial estimate generated by the * double precision routine igami to find the root of * * igamc(a,x) - p = 0. * * * ACCURACY: * * Set to do just one Newton-Raphson iteration. * */
/* qin.c
*
* Modified Bessel function I of noninteger order
*
*
* SYNOPSIS:
*
* int qin( v, x, y );
* QELT *v, *x, *y;
*
* qin( v, x, y );
*
*
*
* DESCRIPTION:
*
* Returns modified Bessel function of order v of the
* argument.
*
* The power series is
*
* inf 2 k
* v - (z /4)
* I (z) = (z/2) > --------------
* v - -
* k=0 k! | (v+k+1)
*
*
* For large x,
* 2 2 2
* exp(z) u - 1 (u - 1 )(u - 3 )
* I (z) = ------------ { 1 - -------- + ---------------- + ...}
* v sqrt(2 pi z) 1 2
* 1! (8z) 2! (8z)
*
* asymptotically, where
*
* 2
* u = 4 v .
*
*
* x <= 0 is not supported.
*
* Series expansion is truncated at less than full working precision.
*
*/
/* qincb.c * * Incomplete beta integral * * * SYNOPSIS: * * int qincb( a, b, x, y ); * QELT *a, *b, *x, *y; * * qincb( a, b, x, y ); * * * DESCRIPTION: * * Returns incomplete beta integral of the arguments, evaluated * from zero to x. * * x * - - * | (a+b) | | a-1 b-1 * ----------- | t (1-t) dt. * - - | | * | (a) | (b) - * 0 * * * ACCURACY: * * Series expansions terminate at less than full working precision. * */
/* qincbi() * * Inverse of imcomplete beta integral * * * * SYNOPSIS: * * double a, b, x, y, incbi(); * * x = incbi( 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. * */
/* qine.c
*
* Modified Bessel function I of noninteger order
* Exponentially scaled
*
*
* SYNOPSIS:
*
* int qine( v, x, y );
* QELT *v, *x, *y;
*
* qine( v, x, y );
*
*
*
* DESCRIPTION:
*
* Returns modified Bessel function of order v of the
* argument.
*
* The power series is
*
* inf 2 k
* v - (z /4)
* I (z) = (z/2) > --------------
* v - -
* k=0 k! | (v+k+1)
*
*
* For large x,
* 2 2 2
* exp(z) u - 1 (u - 1 )(u - 3 )
* I (z) = ------------ { 1 - -------- + ---------------- + ...}
* v sqrt(2 pi z) 1 2
* 1! (8z) 2! (8z)
*
* asymptotically, where
*
* 2
* u = 4 v .
*
*
* The routine returns
*
* sqrt(x) exp(-x) I (x)
* v
*
* x <= 0 is not supported.
*
* Series expansion is truncated at less than full working precision.
*
*/
/* qjn.c * * Bessel function of noninteger order * * * * SYNOPSIS: * * int qjn( v, x, y ); * QELT *v, *x, *y; * * qjn( v, x, y ); * * * * DESCRIPTION: * * Returns Bessel function of order v of the argument, * where v is real. Negative x is allowed if v is an integer. * * Two expansions are used: the ascending power series and the * Hankel expansion for large v. If v is not too large, it * is reduced by recurrence to a region of better accuracy. * */
/* kn.c * * Modified Bessel function, third kind, integer order * * * * SYNOPSIS: * * int qkn( n, x, y ); * int n; * QELT *x, *y; * * qkn( n, x, y ); * * * * DESCRIPTION: * * Returns modified Bessel function of the third kind * of order n of the argument. * * The range is partitioned into the two intervals [0,9.55] and * (9.55, infinity). An ascending power series is used in the * low range, and an asymptotic expansion in the high range. * * ACCURACY: * * Series expansions are set to terminate at less than full * working precision. * */
/* qkne.c * * exp(x) sqrt(x) Kn(x) */
/* qlog.c * * Natural logarithm * * * * SYNOPSIS: * * int qlog( x, y ); * QELT *x, *y; * * qlog( x, y ); * * * * DESCRIPTION: * * Returns the base e (2.718...) logarithm of x. * * After reducing the argument into the interval [1/sqrt(2), sqrt(2)], * the logarithm is calculated by * * x-1 * w = --- * x+1 * 3 5 * w w * ln(x) / 2 = w + --- + --- + ... * 3 5 */
/* qlog1.c * * Relative error logarithm * * * * SYNOPSIS: * * int qlog1( x, y ); * QELT *x, *y; * * qlog1( x, y ); * * * * DESCRIPTION: * * Returns the base e (2.718...) logarithm of 1 + x. * * For small x, this continued fraction is used: * * 1+z * w = --- * 1-z * * 2 2 2 * 2z z 4z 9z * ln(w) = --- --- --- --- ... * 1 - 3 - 5 - 7 - * * after setting z = x/(x+2). * */
/* qlog10.c * * Common logarithm * * * * SYNOPSIS: * * int qlog10( x, y ); * QELT *x, *y; * * qlog10( x, y ); * * * * DESCRIPTION: * * Returns base 10, or common, logarithm of x. * * log (x) = log (e) log (x) * 10 10 e * */
/* qndtr.c * * Normal distribution function * * * * SYNOPSIS: * * int qndtr( x, y ); * QELT *x, *y; * * qndtr( x, y ); * * * * DESCRIPTION: * * Returns the area under the Gaussian probability density * function, integrated from minus infinity to x: * * x * - * 1 | | 2 * ndtr(x) = --------- | exp( - t /2 ) dt * sqrt(2pi) | | * - * -inf. * * = ( 1 + erf(z) ) / 2 * = erfc(z) / 2 * * where z = x/sqrt(2). * */
/* qndtri.c * * Inverse of Normal distribution function * * * * SYNOPSIS: * * int qndtri(y, x); * QELT *y, *x; * * qndtri(y, x); * * * * DESCRIPTION: * * Returns the argument, x, for which the area under the * Gaussian probability density function (integrated from * minus infinity to x) is equal to y. * * The routine refines a trial solution computed by the double * precision function ndtri. * */
/* qplanck.c * Integral of Planck's radiation formula. * * 1 * ------------------ * 5 * t (exp(1/bw) - 1) * * Set * b = T/c2 * u = exp(1/bw) * * In terms of polylogarithms Li_n(u)¸ the integral is * * ( Li (u) Li (u) ) * 1 4 ( 3 2 log(1-u) ) * ---- - 6 b ( Li (u) - ------ + -------- + ---------- ) * 4 ( 4 bw 2 3 ) * 4 w ( 2 (bw) 6 (bw) ) * * Since u > 1, the Li_n are complex valued. This is not * the best way to calculate the result, which is real, but it * is adopted as a the priori formula against which other formulas * can be verified. */
/* qpolylog.c
*
Polylogarithms.
inf k
- x
Li (x) = > ---
n - n
k=1 k
x
-
| | -ln(1-t)
Li (x) = | -------- dt
2 | | t
-
0
1-x
-
| | ln t
= | ------ dt = spence(1-x)
| | 1 - t
-
1
2 3
x x
= x + --- + --- + ...
4 9
d 1
-- Li (x) = --- Li (x)
dx n x n-1
Series expansions are set to terminate at less than full
working precision.
*/
/* qpolyr.c * * Arithmetic operations on polynomials with rational coefficients * * In the following descriptions a, b, c are polynomials of degree * na, nb, nc respectively. The degree of a polynomial cannot * exceed a run-time value MAXPOL. An operation that attempts * to use or generate a polynomial of higher degree may produce a * result that suffers truncation at degree MAXPOL. The value of * MAXPOL is set by calling the function * * polini( maxpol ); * * where maxpol is the desired maximum degree. This must be * done prior to calling any of the other functions in this module. * Memory for internal temporary polynomial storage is allocated * by polini(). * * Each polynomial is represented by an array containing its * coefficients, together with a separately declared integer equal * to the degree of the polynomial. The coefficients appear in * ascending order; that is, * * 2 na * a(x) = a[0] + a[1] * x + a[2] * x + ... + a[na] * x . * * * * sum = poleva( a, na, x ); Evaluate polynomial a(t) at t = x. * polprt( a, na, D ); Print the coefficients of a to D digits. * polclr( a, na ); Set a identically equal to zero, up to a[na]. * polmov( a, na, b ); Set b = a. * poladd( a, na, b, nb, c ); c = b + a, nc = max(na,nb) * polsub( a, na, b, nb, c ); c = b - a, nc = max(na,nb) * polmul( a, na, b, nb, c ); c = b * a, nc = na+nb * * * Division: * * i = poldiv( a, na, b, nb, c ); c = b / a, nc = MAXPOL * * returns i = the degree of the first nonzero coefficient of a. * The computed quotient c must be divided by x^i. An error message * is printed if a is identically zero. * * * Change of variables: * If a and b are polynomials, and t = a(x), then * c(t) = b(a(x)) * is a polynomial found by substituting a(x) for t. The * subroutine call for this is * * polsbt( a, na, b, nb, c ); * * * Notes: * poldiv() is an integer routine; poleva() is double. * Any of the arguments a, b, c may refer to the same array. * */
/* qpow * * Power function check routine * * * * SYNOPSIS: * * int qpow( x, y, z ); * QELT *x, *y, *z; * * qpow( x, y, z ); * * * * DESCRIPTION: * * Computes x raised to the yth power. * * y * x = exp( y log(x) ). * */
/* qprob.c */ /* various probability integrals * computed via incomplete beta and gamma integrals */
/* qbdtr * * Binomial distribution * * * * SYNOPSIS: * * int qbdtr( k, n, p, y ); * int k, n; * QELT *p, *y; * * qbdtr( k, n, p, y ); * * DESCRIPTION: * * Returns (in y) 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. * */
/* qbdtrc * * Complemented binomial distribution * * * * SYNOPSIS: * * int qbdtrc( k, n, p, y ); * int k, n; * QELT *p, *y; * * y = qbdtrc( k, n, p, y ); * * 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. * */
/* qbdtri * * Inverse binomial distribution * * * * SYNOPSIS: * * int qbdtri( k, n, y, p ); * int k, n; * QELT *p, *y; * * qbdtri( k, n, y, p ); * * 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 ). * */
/* qchdtr * * Chi-square distribution * * * * SYNOPSIS: * * int qchdtr( df, x, y ); * QELT *df, *x, *y; * * qchdtr( df, x, y ); * * * * 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. * */
/* qchdtc * * Complemented Chi-square distribution * * * * SYNOPSIS: * * int qchdtc( df, x, y ); * QELT df[], x[], y[]; * * qchdtc( df, x, y ); * * * * 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. * */
/* qchdti * * Inverse of complemented Chi-square distribution * * * * SYNOPSIS: * * int qchdti( df, y, x ); * QELT *df, *x, *y; * * qchdti( df, y, x ); * * * * * 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 * */
/* qfdtr * * F distribution * * * * SYNOPSIS: * * int qfdtr( ia, ib, x, y ); * int ia, ib; * QELT *x, *y; * * qfdtr( ia, ib, x, y ); * * 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) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ). * * * The arguments a and b are greater than zero, and x is * nonnegative. * */
/* qfdtrc * * Complemented F distribution * * * * SYNOPSIS: * * int qfdtrc( ia, ib, x, y ); * int ia, ib; * QELT x[], y[]; * * qfdtrc( ia, ib, x, y ); * * 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 * * * The incomplete beta integral is used, according to the * formula * * P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ). * */
/* qfdtri * * Inverse of complemented F distribution * * * * SYNOPSIS: * * int qfdtri( ia, ib, y, x ); * int ia, ib; * QELT x[], y[]; * * qfdtri( ia, ib, y, x ); * * 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)). * */
/* qgdtr * * Gamma distribution function * * * * SYNOPSIS: * * int qgdtr( a, b, x, y ); * QELT *a, *b, *x, *y; * * qgdtr( a, b, x, y ); * * * * 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 ). * */
/* qgdtrc * * Complemented gamma distribution function * * * * SYNOPSIS: * * int qgdtrc( a, b, x, y ); * QELT *a, *b, *x, *y; * * qgdtrc( a, b, x, y ); * * * * 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 ). * */
/* qnbdtr * * Negative binomial distribution * * * * SYNOPSIS: * * int qnbdtr( k, n, p, y ); * int k, n; * QELT *p, *y; * * qnbdtr( k, n, p, y ); * * DESCRIPTION: * * Returns the sum of the terms 0 through k of the negative * binomial distribution: * * k * -- ( n+j-1 ) n j * > ( ) p (1-p) * -- ( j ) * j=0 * * In a sequence of Bernoulli trials, this is the probability * that k or fewer failures precede the nth success. * * The terms are not computed individually; instead the incomplete * beta integral is employed, according to the formula * * y = nbdtr( k, n, p ) = incbet( n, k+1, p ). * * The arguments must be positive, with p ranging from 0 to 1. * */
/* qnbdtc * * Complemented negative binomial distribution * * * * SYNOPSIS: * * int qnbdtc( k, n, p, y ); * int k, n; * QELT *p, *y; * * qnbdtc( k, n, p, y ); * * DESCRIPTION: * * Returns the sum of the terms k+1 to infinity of the negative * binomial distribution: * * inf * -- ( n+j-1 ) n j * > ( ) p (1-p) * -- ( j ) * j=k+1 * * The terms are not computed individually; instead the incomplete * beta integral is employed, according to the formula * * y = nbdtrc( k, n, p ) = incbet( k+1, n, 1-p ). * * The arguments must be positive, with p ranging from 0 to 1. * */
/* qpdtr * * Poisson distribution * * * * SYNOPSIS: * * int qpdtr( k, m, y ); * int k; * QELT *m, *y; * * qpdtr( k, m, y ); * * * * DESCRIPTION: * * Returns the sum of the first k terms of the Poisson * distribution: * * k j * -- -m m * > e -- * -- j! * j=0 * * The terms are not summed directly; instead the incomplete * gamma integral is employed, according to the relation * * y = pdtr( k, m ) = igamc( k+1, m ). * * The arguments must both be positive. * */
/* qpdtrc * * Complemented poisson distribution * * * * SYNOPSIS: * * int qpdtrc( k, m, y ); * int k; * QELT *m, *y; * * qpdtrc( k, m, y ); * * * * DESCRIPTION: * * Returns the sum of the terms k+1 to infinity of the Poisson * distribution: * * inf. j * -- -m m * > e -- * -- j! * j=k+1 * * The terms are not summed directly; instead the incomplete * gamma integral is employed, according to the formula * * y = pdtrc( k, m ) = igam( k+1, m ). * * The arguments must both be positive. * */
/* qpdtri * * Inverse Poisson distribution * * * * SYNOPSIS: * * int qpdtri( k, y, m ); * int k; * QELT *m, *y; * * qpdtri( k, y, m ); * * * * * DESCRIPTION: * * Finds the Poisson variable x such that the integral * from 0 to x of the Poisson density is equal to the * given probability y. * * This is accomplished using the inverse gamma integral * function and the relation * * m = igami( k+1, y ). * */
/* qpsi.c * Psi (digamma) function check routine * * * SYNOPSIS: * * int qpsi( x, y ); * QELT *x, *y; * * qpsi( x, y ); * * * DESCRIPTION: * * d - * psi(x) = -- ln | (x) * dx * * is the logarithmic derivative of the gamma function. * For general positive x, the argument is made greater than 16 * using the recurrence psi(x+1) = psi(x) + 1/x. * Then the following asymptotic expansion is applied: * * inf. B * - 2k * psi(x) = log(x) - 1/2x - > ------- * - 2k * k=1 2k x * * where the B2k are Bernoulli numbers. * * psi(-x) = psi(x+1) + pi/tan(pi(x+1)) */
/* qrand.c * * Pseudorandom number generator * * * * SYNOPSIS: * * int qrand( q ); * QELT q[NQ]; * * qrand( q ); * * * * DESCRIPTION: * * Yields a random number 1.0 <= q < 2.0. * * A three-generator congruential algorithm adapted from Brian * Wichmann and David Hill (BYTE magazine, March, 1987, * pp 127-8) is used to generate random 16-bit integers. * These are copied into the significand area to produce * a pseudorandom bit pattern. */
/* qshici.c * * Hyperbolic sine and cosine integrals * * * * SYNOPSIS: * * int qshici( x, si, ci ); * QELT *x, *si, *ci; * * qshici( x, si, ci ); * * * DESCRIPTION: * * * x * - * | | cosh t - 1 * Chi(x) = eul + ln x + | ----------- dt * | | t * - * 0 * * x * - * | | sinh t * Shi(x) = | ------ dt * | | t * - * 0 * * where eul = 0.57721566490153286061 is Euler's constant. * * The power series are * * inf 2n+1 * - z * Shi(z) = > -------------- * - (2n+1) (2n+1)! * n=0 * * inf 2n * - z * Chi(z) = eul + ln(z) + > ----------- * - 2n (2n)! * n=1 * * Asymptotically, * * * -x 1 2! 3! * 2x e Shi(x) = 1 + - + -- + -- + ... * x 2 3 * x x * * ACCURACY: * * Series expansions are set to terminate at less than full * working precision. * */
/* qsici.c * Sine and cosine integrals * * * * SYNOPSIS: * * int qsici( x, si, ci ); * QELT *x, *si, *ci; * * qsici( x, si, ci ); * * * DESCRIPTION: * * Evaluates the integrals * * x * - * | cos t - 1 * Ci(x) = eul + ln x + | --------- dt, * | t * - * 0 * x * - * | sin t * Si(x) = | ----- dt * | t * - * 0 * * where eul = 0.57721566490153286061 is Euler's constant. * * The power series are * * inf n 2n+1 * - (-1) z * Si(z) = > -------------- * - (2n+1) (2n+1)! * n=0 * * inf n 2n * - (-1) z * Ci(z) = eul + ln(z) + > ----------- * - 2n (2n)! * n=1 * * ACCURACY: * * Series expansions are set to terminate at less than full * working precision. * */
/* qsimq.c * * Solution of simultaneous linear equations AX = B * by Gaussian elimination with partial pivoting * * * * SYNOPSIS: * * double A[n*n], B[n], X[n]; * int n, flag; * int IPS[]; * int simq(); * * ercode = simq( A, B, X, n, flag, IPS ); * * * * DESCRIPTION: * * B, X, IPS are vectors of length n. * A is an n x n matrix (i.e., a vector of length n*n), * stored row-wise: that is, A(i,j) = A[ij], * where ij = i*n + j, which is the transpose of the normal * column-wise storage. * * The contents of matrix A are destroyed. * * Set flag=0 to solve. * Set flag=-1 to do a new back substitution for different B vector * using the same A matrix previously reduced when flag=0. * * The routine returns nonzero on error; messages are printed. * * * ACCURACY: * * Depends on the conditioning (range of eigenvalues) of matrix A. * * * REFERENCE: * * Computer Solution of Linear Algebraic Systems, * by George E. Forsythe and Cleve B. Moler; Prentice-Hall, 1967. * */
/* qsin.c * Circular sine check routine * * * * SYNOPSIS: * * int qsin( x, y ); * QELT *x, *y; * * qsin( x, y ); * * * * DESCRIPTION: * * Range reduction is into intervals of pi/2. * Then * * 3 5 7 * z z z * sin(z) = z - -- + -- - -- + ... * 3! 5! 7! * */
/* qsindg.c * * sin, cos, tan in degrees */
/* qsinh.c * * Hyperbolic sine check routine * * * * SYNOPSIS: * * int qsinh( x, y ); * QELT *x, *y; * * qsinh( x, y ); * * * * DESCRIPTION: * * The range is partitioned into two segments. If |x| <= 1/4, * * 3 5 7 * x x x * sinh(x) = x + -- + -- + -- + ... * 3! 5! 7! * * Otherwise the calculation is sinh(x) = ( exp(x) - exp(-x) )/2. * */
/* qspenc.c * * Dilogarithm * * * * SYNOPSIS: * * int qspenc( x, y ); * QELT *x, *y; * * qspenc( x, y ); * * * * DESCRIPTION: * * Computes the integral * * x * - * | | log t * spence(x) = - | ----- dt * | | t - 1 * - * 1 * * for x >= 0. A power series gives the integral in * the interval (0.5, 1.5). Transformation formulas for 1/x * and 1-x are employed outside the basic expansion range. * * * */
/* qsqrt.c * Square root check routine * * * * SYNOPSIS: * * int qsqrt( x, y ); * QELT *x, *y; * * qsqrt( x, y ); * * * * DESCRIPTION: * * Returns the square root of x. * * Range reduction involves isolating the power of two of the * argument and using a polynomial approximation to obtain * a rough value for the square root. Then Heron's iteration * is used to converge to an accurate value. * */
/* qsqrta.c */ /* Square root check routine, done by long division. */ /* Copyright (C) 1984-1988 by Stephen L. Moshier. */
/* qstdtr.c * * Student's t distribution * * * * SYNOPSIS: * * int qstudt( k, t, y ); * int k; * QELT *t, *y; * * qstudt( k, t, y ); * * * DESCRIPTION: * * Computes the integral from minus infinity to t of the Student * t distribution with integer k > 0 degrees of freedom: * * t * - * | | * - | 2 -(k+1)/2 * | ( (k+1)/2 ) | ( x ) * ---------------------- | ( 1 + --- ) dx * - | ( k ) * sqrt( k pi ) | ( k/2 ) | * | | * - * -inf. * * Relation to incomplete beta integral: * * 1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z ) * where * z = k/(k + t**2). * * For t < -2, this is the method of computation. For higher t, * a direct method is derived from integration by parts. * Since the function is symmetric about t=0, the area under the * right tail of the density is found by calling the function * with -t instead of t. * * ACCURACY: * */
/* qtan.c * Circular tangent check routine * * * * SYNOPSIS: * * int qtan( x, y ); * QELT *x, *y; * * qtan( x, y ); * * * * DESCRIPTION: * * Domain of approximation is reduced by the transformation * * x -> x - pi floor((x + pi/2)/pi) * * * then tan(x) is the continued fraction * * 2 2 2 * x x x x * tan(x) = --- --- --- --- ... * 1 - 3 - 5 - 7 - * */
/* qcot * * Circular cotangent check routine * * * * SYNOPSIS: * * int qcot( x, y ); * QELT *x, *y; * * qcot( x, y ); * * * * DESCRIPTION: * * cot (x) = 1 / tan (x). * */
/* qtanh.c * Hyperbolic tangent check routine * * * * * SYNOPSIS: * * int qtanh( x, y ); * QELT *x, *y; * * qtanh( x, y ); * * * * DESCRIPTION: * * For x >= 1 the program uses the definition * * exp(x) - exp(-x) * tanh(x) = ---------------- * exp(x) + exp(-x) * * * For x < 1 the method is a continued fraction * * 2 2 2 * x x x x * tanh(x) = --- --- --- --- ... * 1+ 3+ 5+ 7+ * */
/* qyn.c * * Real bessel function of second kind and general order. * * * * SYNOPSIS: * * int qyn( v, x, y ); * QELT *v, *x, *y; * * qyn( v, x, y ); * * * * DESCRIPTION: * * Returns Bessel function of order v. * If v is not an integer, the result is * * Y (z) = ( cos(pi v) * J (x) - J (x) )/sin(pi v) * v v -v * * Hankel's expansion is used for large x: * * Y (z) = sqrt(2/(pi z)) (P sin w + Q cos w) * v * * w = z - (.5 v + .25) pi * * * (u-1)(u-9) (u-1)(u-9)(u-25)(u-49) * P = 1 - ---------- + ---------------------- - ... * 2 4 * 2! (8z) 4! (8z) * * * (u-1) (u-1)(u-9)(u-25) * Q = ----- - ---------------- + ... * 8z 3 * 3! (8z) * * 2 * u = 4 v * * (AMS55 #9.2.6). * * * -n n-1 * -(z/2) - (n-k-1)! 2 k * Y (z) = ------- > -------- (z / 4) + (2/pi) ln (z/2) J (z) * n pi - k! n * k=0 * * * n inf 2 k * (z/2) - (- z / 4) * - ------ - > (psi(k+1) + psi(n+k+1)) ---------- * pi - k!(n+k)! * k=0 * * (AMS55 #9.1.11). * * ACCURACY: * * Series expansions are set to terminate at less than full working * precision. * */
/* qzetac.c * * Riemann zeta function * * * * SYNOPSIS: * * int qzetac( x, y ); * QELT *x, *y; * * qzetac( x, y ); * * * * DESCRIPTION: * * * * inf. * - -x * zetac(x) = > k , x > 1, * - * k=2 * * is related to the Riemann zeta function by * * Riemann zeta(x) = zetac(x) + 1. * * Extension of the function definition for x < 1 is implemented. * * * ACCURACY: * * Series summation terminates at NBITS/2. * */
To Cephes home page www.moshier.net:
Last update: 31 July 2000