From 7ce331c01ce6eb7b3f5c715a38a24359da9c6ee2 Mon Sep 17 00:00:00 2001 From: Eric Andersen Date: Thu, 22 Nov 2001 14:04:29 +0000 Subject: Totally rework the math library, this time based on the MacOs X math library (which is itself based on the math lib from FreeBSD). -Erik --- libm/ldouble/Makefile | 122 -- libm/ldouble/README.txt | 3502 --------------------------------------- libm/ldouble/acoshl.c | 167 -- libm/ldouble/arcdotl.c | 108 -- libm/ldouble/asinhl.c | 156 -- libm/ldouble/asinl.c | 249 --- libm/ldouble/atanhl.c | 163 -- libm/ldouble/atanl.c | 376 ----- libm/ldouble/bdtrl.c | 260 --- libm/ldouble/btdtrl.c | 68 - libm/ldouble/cbrtl.c | 143 -- libm/ldouble/chdtrl.c | 200 --- libm/ldouble/clogl.c | 720 -------- libm/ldouble/cmplxl.c | 461 ------ libm/ldouble/coshl.c | 89 - libm/ldouble/econst.c | 96 -- libm/ldouble/ehead.h | 45 - libm/ldouble/elliel.c | 146 -- libm/ldouble/ellikl.c | 148 -- libm/ldouble/ellpel.c | 173 -- libm/ldouble/ellpjl.c | 164 -- libm/ldouble/ellpkl.c | 203 --- libm/ldouble/exp10l.c | 192 --- libm/ldouble/exp2l.c | 166 -- libm/ldouble/expl.c | 183 --- libm/ldouble/fdtrl.c | 237 --- libm/ldouble/floorl.c | 432 ----- libm/ldouble/flrtstl.c | 104 -- libm/ldouble/fltestl.c | 265 --- libm/ldouble/gammal.c | 764 --------- libm/ldouble/gdtrl.c | 130 -- libm/ldouble/gelsl.c | 240 --- libm/ldouble/ieee.c | 4182 ----------------------------------------------- libm/ldouble/igamil.c | 193 --- libm/ldouble/igaml.c | 220 --- libm/ldouble/incbetl.c | 406 ----- libm/ldouble/incbil.c | 305 ---- libm/ldouble/isnanl.c | 186 --- libm/ldouble/j0l.c | 541 ------ libm/ldouble/j1l.c | 551 ------- libm/ldouble/jnl.c | 130 -- libm/ldouble/lcalc.c | 1484 ----------------- libm/ldouble/lcalc.h | 79 - libm/ldouble/ldrand.c | 175 -- libm/ldouble/log10l.c | 319 ---- libm/ldouble/log2l.c | 302 ---- libm/ldouble/logl.c | 292 ---- libm/ldouble/lparanoi.c | 2348 -------------------------- libm/ldouble/monotl.c | 307 ---- libm/ldouble/mtherr.c | 102 -- libm/ldouble/mtstl.c | 521 ------ libm/ldouble/nantst.c | 61 - libm/ldouble/nbdtrl.c | 197 --- libm/ldouble/ndtril.c | 416 ----- libm/ldouble/ndtrl.c | 473 ------ libm/ldouble/pdtrl.c | 184 --- libm/ldouble/polevll.c | 182 --- libm/ldouble/powil.c | 164 -- libm/ldouble/powl.c | 739 --------- libm/ldouble/sinhl.c | 150 -- libm/ldouble/sinl.c | 342 ---- libm/ldouble/sqrtl.c | 172 -- libm/ldouble/stdtrl.c | 225 --- libm/ldouble/tanhl.c | 129 -- libm/ldouble/tanl.c | 279 ---- libm/ldouble/testvect.c | 497 ------ libm/ldouble/unityl.c | 128 -- libm/ldouble/wronkl.c | 67 - libm/ldouble/ynl.c | 113 -- 69 files changed, 27633 deletions(-) delete mode 100644 libm/ldouble/Makefile delete mode 100644 libm/ldouble/README.txt delete mode 100644 libm/ldouble/acoshl.c delete mode 100644 libm/ldouble/arcdotl.c delete mode 100644 libm/ldouble/asinhl.c delete mode 100644 libm/ldouble/asinl.c delete mode 100644 libm/ldouble/atanhl.c delete mode 100644 libm/ldouble/atanl.c delete mode 100644 libm/ldouble/bdtrl.c delete mode 100644 libm/ldouble/btdtrl.c delete mode 100644 libm/ldouble/cbrtl.c delete mode 100644 libm/ldouble/chdtrl.c delete mode 100644 libm/ldouble/clogl.c delete mode 100644 libm/ldouble/cmplxl.c delete mode 100644 libm/ldouble/coshl.c delete mode 100644 libm/ldouble/econst.c delete mode 100644 libm/ldouble/ehead.h delete mode 100644 libm/ldouble/elliel.c delete mode 100644 libm/ldouble/ellikl.c delete mode 100644 libm/ldouble/ellpel.c delete mode 100644 libm/ldouble/ellpjl.c delete mode 100644 libm/ldouble/ellpkl.c delete mode 100644 libm/ldouble/exp10l.c delete mode 100644 libm/ldouble/exp2l.c delete mode 100644 libm/ldouble/expl.c delete mode 100644 libm/ldouble/fdtrl.c delete mode 100644 libm/ldouble/floorl.c delete mode 100644 libm/ldouble/flrtstl.c delete mode 100644 libm/ldouble/fltestl.c delete mode 100644 libm/ldouble/gammal.c delete mode 100644 libm/ldouble/gdtrl.c delete mode 100644 libm/ldouble/gelsl.c delete mode 100644 libm/ldouble/ieee.c delete mode 100644 libm/ldouble/igamil.c delete mode 100644 libm/ldouble/igaml.c delete mode 100644 libm/ldouble/incbetl.c delete mode 100644 libm/ldouble/incbil.c delete mode 100644 libm/ldouble/isnanl.c delete mode 100644 libm/ldouble/j0l.c delete mode 100644 libm/ldouble/j1l.c delete mode 100644 libm/ldouble/jnl.c delete mode 100644 libm/ldouble/lcalc.c delete mode 100644 libm/ldouble/lcalc.h delete mode 100644 libm/ldouble/ldrand.c delete mode 100644 libm/ldouble/log10l.c delete mode 100644 libm/ldouble/log2l.c delete mode 100644 libm/ldouble/logl.c delete mode 100644 libm/ldouble/lparanoi.c delete mode 100644 libm/ldouble/monotl.c delete mode 100644 libm/ldouble/mtherr.c delete mode 100644 libm/ldouble/mtstl.c delete mode 100644 libm/ldouble/nantst.c delete mode 100644 libm/ldouble/nbdtrl.c delete mode 100644 libm/ldouble/ndtril.c delete mode 100644 libm/ldouble/ndtrl.c delete mode 100644 libm/ldouble/pdtrl.c delete mode 100644 libm/ldouble/polevll.c delete mode 100644 libm/ldouble/powil.c delete mode 100644 libm/ldouble/powl.c delete mode 100644 libm/ldouble/sinhl.c delete mode 100644 libm/ldouble/sinl.c delete mode 100644 libm/ldouble/sqrtl.c delete mode 100644 libm/ldouble/stdtrl.c delete mode 100644 libm/ldouble/tanhl.c delete mode 100644 libm/ldouble/tanl.c delete mode 100644 libm/ldouble/testvect.c delete mode 100644 libm/ldouble/unityl.c delete mode 100644 libm/ldouble/wronkl.c delete mode 100644 libm/ldouble/ynl.c (limited to 'libm/ldouble') diff --git a/libm/ldouble/Makefile b/libm/ldouble/Makefile deleted file mode 100644 index dad448840..000000000 --- a/libm/ldouble/Makefile +++ /dev/null @@ -1,122 +0,0 @@ -# Makefile for uClibc's math library -# Copyright (C) 2001 by Lineo, inc. -# -# This math library is derived primarily from the Cephes Math Library, -# copyright by Stephen L. Moshier -# -# This program is free software; you can redistribute it and/or modify it under -# the terms of the GNU Library General Public License as published by the Free -# Software Foundation; either version 2 of the License, or (at your option) any -# later version. -# -# This program is distributed in the hope that it will be useful, but WITHOUT -# ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS -# FOR A PARTICULAR PURPOSE. See the GNU Library General Public License for more -# details. -# -# You should have received a copy of the GNU Library General Public License -# along with this program; if not, write to the Free Software Foundation, Inc., -# 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA -# - -TOPDIR=../../ -include $(TOPDIR)Rules.mak - -LIBM=../libm.a -TARGET_CC= $(TOPDIR)/extra/gcc-uClibc/$(TARGET_ARCH)-uclibc-gcc - -CSRC=acoshl.c asinhl.c asinl.c atanhl.c atanl.c bdtrl.c btdtrl.c cbrtl.c \ - chdtrl.c coshl.c ellpel.c ellpkl.c elliel.c ellikl.c ellpjl.c \ - exp10l.c exp2l.c expl.c fdtrl.c gammal.c gdtrl.c igamil.c igaml.c \ - incbetl.c incbil.c isnanl.c j0l.c j1l.c jnl.c ldrand.c log10l.c log2l.c \ - logl.c nbdtrl.c ndtril.c ndtrl.c pdtrl.c powl.c powil.c sinhl.c sinl.c \ - sqrtl.c stdtrl.c tanhl.c tanl.c unityl.c ynl.c \ - floorl.c polevll.c mtherr.c #cmplxl.c clogl.c -COBJS=$(patsubst %.c,%.o, $(CSRC)) - - -OBJS=$(COBJS) - -all: $(OBJS) $(LIBM) - -$(LIBM): ar-target - -ar-target: $(OBJS) - $(AR) $(ARFLAGS) $(LIBM) $(OBJS) - -$(COBJS): %.o : %.c - $(TARGET_CC) $(TARGET_CFLAGS) -c $< -o $@ - $(STRIPTOOL) -x -R .note -R .comment $*.o - -$(OBJ): Makefile - -clean: - rm -f *.[oa] *~ core - - - -#----------------------------------------- - - -#all: mtstl lparanoi lcalc fltestl nantst testvect monotl libml.a - -mtstl: libml.a mtstl.o $(OBJS) - $(TARGET_CC) $(TARGET_CFLAGS) -o mtstl mtstl.o libml.a $(LIBS) - -mtstl.o: mtstl.c - -lparanoi: libml.a lparanoi.o setprec.o ieee.o econst.o $(OBJS) - $(TARGET_CC) $(TARGET_CFLAGS) -o lparanoi lparanoi.o setprec.o ieee.o econst.o libml.a $(LIBS) - -lparanoi.o: lparanoi.c - $(TARGET_CC) $(TARGET_CFLAGS) -Wno-implicit -c lparanoi.c - -econst.o: econst.c ehead.h - -lcalc: libml.a lcalc.o ieee.o econst.o $(OBJS) - $(TARGET_CC) $(TARGET_CFLAGS) -o lcalc lcalc.o ieee.o econst.o libml.a $(LIBS) - -lcalc.o: lcalc.c lcalc.h ehead.h - -ieee.o: ieee.c ehead.h - -# Use $(OBJS) in ar command for libml.a if possible; else *.o -libml.a: $(OBJS) mconf.h - ar -rv libml.a $(OBJS) - ranlib libml.a - - -fltestl: fltestl.c libml.a - $(TARGET_CC) $(TARGET_CFLAGS) -o fltestl fltestl.c libml.a - -fltestl.o: fltestl.c - -flrtstl: flrtstl.c libml.a - $(TARGET_CC) $(TARGET_CFLAGS) -o flrtstl flrtstl.c libml.a - -flrtstl.o: flrtstl.c - -nantst: nantst.c libml.a - $(TARGET_CC) $(TARGET_CFLAGS) -o nantst nantst.c libml.a - -nantst.o: nantst.c - -testvect: testvect.o libml.a - $(TARGET_CC) $(TARGET_CFLAGS) -o testvect testvect.o libml.a - -testvect.o: testvect.c - $(TARGET_CC) -g -c -o testvect.o testvect.c - -monotl: monotl.o libml.a - $(TARGET_CC) $(TARGET_CFLAGS) -o monotl monotl.o libml.a - -monotl.o: monotl.c - $(TARGET_CC) -g -c -o monotl.o monotl.c - -# Run test programs -check: mtstl fltestl testvect monotl libml.a - -mtstl - -fltestl - -testvect - -monotl - diff --git a/libm/ldouble/README.txt b/libm/ldouble/README.txt deleted file mode 100644 index 30fcaad36..000000000 --- a/libm/ldouble/README.txt +++ /dev/null @@ -1,3502 +0,0 @@ -/* 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 - * - */ - -/* 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 - * - */ - -/* 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 - * asin domain |x| > 1 0.0 - * - */ - /* 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 - * asin domain |x| > 1 0.0 - */ - -/* 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 - * - */ - -/* 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 - * - */ - /* 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. - * - */ - -/* 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 - * - */ - /* 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 1 - */ - - -/* 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. - * - */ - -/* 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 - * - */ - -/* 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 - */ - /* 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 - */ - /* 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 - * - */ - -/* 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. - */ - - /* 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 - * - */ - /* 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. - * - */ - /* 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 - */ - /* 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. - */ - /* 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. - */ - - /* 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. - */ - /* 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 - */ - - /* 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(). - */ - -/* 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 - */ - -/* cabsl() - * - * Complex absolute value - * - * - * - * SYNOPSIS: - * - * long double cabsl(); - * cmplxl z; - * long double a; - * - * a = cabs( &z ); - * - * - * - * DESCRIPTION: - * - * - * If z = x + iy - * - * then - * - * a = sqrt( x**2 + y**2 ). - * - * Overflow and underflow are avoided by testing the magnitudes - * of x and y before squaring. If either is outside half of - * the floating point full scale range, both are rescaled. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -30,+30 30000 3.2e-17 9.2e-18 - * IEEE -10,+10 100000 2.7e-16 6.9e-17 - */ - /* csqrtl() - * - * Complex square root - * - * - * - * SYNOPSIS: - * - * void csqrtl(); - * cmplxl z, w; - * - * csqrtl( &z, &w ); - * - * - * - * DESCRIPTION: - * - * - * If z = x + iy, r = |z|, then - * - * 1/2 - * Im w = [ (r - x)/2 ] , - * - * Re w = y / 2 Im w. - * - * - * Note that -w is also a square root of z. The root chosen - * is always in the upper half plane. - * - * Because of the potential for cancellation error in r - x, - * the result is sharpened by doing a Heron iteration - * (see sqrt.c) in complex arithmetic. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * DEC -10,+10 25000 3.2e-17 9.6e-18 - * IEEE -10,+10 100000 3.2e-16 7.7e-17 - * - * 2 - * Also tested by csqrt( z ) = z, and tested by arguments - * close to the real axis. - */ - -/* 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 MAXNUML - * - * - */ - -/* 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 - * - * - */ - -/* 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 - * - * - */ - -/* 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 - * - */ - -/* 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. - * - */ - -/* 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 - * - */ - -/* 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 - * - */ - -/* 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 - * - */ - -/* 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 - * - */ - -/* fabsl.c - * - * Absolute value - * - * - * - * SYNOPSIS: - * - * long double x, y; - * - * y = fabsl( x ); - * - * - * - * DESCRIPTION: - * - * Returns the absolute value of the argument. - * - */ - -/* 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 - * - */ - /* 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 - * - */ - /* 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 - */ - -/* ceill() - * floorl() - * frexpl() - * ldexpl() - * fabsl() - * - * Floating point numeric utilities - * - * - * - * SYNOPSIS: - * - * long double x, y; - * long double ceill(), floorl(), frexpl(), ldexpl(), fabsl(); - * int expnt, n; - * - * y = floorl(x); - * y = ceill(x); - * y = frexpl( x, &expnt ); - * y = ldexpl( x, n ); - * y = fabsl( x ); - * - * - * - * DESCRIPTION: - * - * All four 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. - */ - -/* 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(). - * - */ -/* 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. - * - */ - -/* 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 - * - */ - /* 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 - * - */ - -/* -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 -*/ - -/* 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 - * - */ - -/* 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 - * - */ - /* 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 - * - */ - -/* 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 - */ - -/* 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 - */ - -/* 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 - * - * - */ - /* 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 - * - */ - -/* j1l.c - * - * Bessel function of order one - * - * - * - * SYNOPSIS: - * - * long double x, y, j1l(); - * - * y = j1l( x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order one 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) x P8(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 M1(x) = sqrt(J1(x)^2 + Y1(x)^2) and phase P1(x) - * = atan(Y1(x)/J1(x)). M1 is approximated by sqrt(1/x)P7(1/x)/Q8(1/x). - * The approximation to j1 is M1 * cos(x - 3 pi/4 + 1/x P5(1/x^2)/Q6(1/x^2)). - * - * - * ACCURACY: - * - * Absolute error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 40000 1.8e-19 5.0e-20 - * - * - */ - /* y1l.c - * - * Bessel function of the second kind, order zero - * - * - * - * SYNOPSIS: - * - * double x, y, y1l(); - * - * y = y1l( x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of the second kind, of order - * zero, of the argument. - * - * The domain is divided into the intervals [0, 4.5>, [4.5,9> 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)P9(x)/Q10(x) - * where p, q, r, s are zeros of y1(x). - * - * The third interval uses the same approximations to modulus - * and phase as j1(x), whence y1(x) = modulus * sin(phase). - * - * ACCURACY: - * - * Absolute error, when y0(x) < 1; else relative error: - * - * arithmetic domain # trials peak rms - * IEEE 0, 30 36000 2.7e-19 5.3e-20 - * - */ - -/* jnl.c - * - * Bessel function of integer order - * - * - * - * SYNOPSIS: - * - * int n; - * long double x, y, jnl(); - * - * y = jnl( n, x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order n, where n is a - * (possibly negative) integer. - * - * The ratio of jn(x) to j0(x) is computed by backward - * recurrence. First the ratio jn/jn-1 is found by a - * continued fraction expansion. Then the recurrence - * relating successive orders is applied until j0 or j1 is - * reached. - * - * If n = 0 or 1 the routine for j0 or j1 is called - * directly. - * - * - * - * ACCURACY: - * - * Absolute error: - * arithmetic domain # trials peak rms - * IEEE -30, 30 5000 3.3e-19 4.7e-20 - * - * - * Not suitable for large n or x. - * - */ - -/* ldrand.c - * - * Pseudorandom number generator - * - * - * - * SYNOPSIS: - * - * double y; - * int ldrand(); - * - * ldrand( &y ); - * - * - * - * DESCRIPTION: - * - * Yields a random number 1.0 <= y < 2.0. - * - * The three-generator congruential algorithm by Brian - * Wichmann and David Hill (BYTE magazine, March, 1987, - * pp 127-8) is used. - * - * Versions invoked by the different arithmetic compile - * time options IBMPC, and MIEEE, produce the same sequences. - * - */ - -/* log10l.c - * - * Common logarithm, long double precision - * - * - * - * SYNOPSIS: - * - * long double x, y, log10l(); - * - * y = log10l( x ); - * - * - * - * DESCRIPTION: - * - * Returns the base 10 logarithm of x. - * - * The argument is separated into its exponent and fractional - * parts. If the exponent is between -1 and +1, the logarithm - * of the fraction is approximated by - * - * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x). - * - * Otherwise, setting z = 2(x-1)/x+1), - * - * log(x) = z + z**3 P(z)/Q(z). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0.5, 2.0 30000 9.0e-20 2.6e-20 - * IEEE exp(+-10000) 30000 6.0e-20 2.3e-20 - * - * In the tests over the interval exp(+-10000), the logarithms - * of the random arguments were uniformly distributed over - * [-10000, +10000]. - * - * ERROR MESSAGES: - * - * log singularity: x = 0; returns MINLOG - * log domain: x < 0; returns MINLOG - */ - -/* log2l.c - * - * Base 2 logarithm, long double precision - * - * - * - * SYNOPSIS: - * - * long double x, y, log2l(); - * - * y = log2l( x ); - * - * - * - * DESCRIPTION: - * - * Returns the base 2 logarithm of x. - * - * The argument is separated into its exponent and fractional - * parts. If the exponent is between -1 and +1, the (natural) - * logarithm of the fraction is approximated by - * - * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x). - * - * Otherwise, setting z = 2(x-1)/x+1), - * - * log(x) = z + z**3 P(z)/Q(z). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0.5, 2.0 30000 9.8e-20 2.7e-20 - * IEEE exp(+-10000) 70000 5.4e-20 2.3e-20 - * - * In the tests over the interval exp(+-10000), the logarithms - * of the random arguments were uniformly distributed over - * [-10000, +10000]. - * - * ERROR MESSAGES: - * - * log singulari