From 1077fa4d772832f77a677ce7fb7c2d513b959e3f Mon Sep 17 00:00:00 2001 From: Eric Andersen Date: Thu, 10 May 2001 00:40:28 +0000 Subject: uClibc now has a math library. muahahahaha! -Erik --- libm/ldouble/README.txt | 3502 +++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 3502 insertions(+) create mode 100644 libm/ldouble/README.txt (limited to 'libm/ldouble/README.txt') diff --git a/libm/ldouble/README.txt b/libm/ldouble/README.txt new file mode 100644 index 000000000..30fcaad36 --- /dev/null +++ b/libm/ldouble/README.txt @@ -0,0 +1,3502 @@ +/* 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 singularity: x = 0; returns MINLOG + * log domain: x < 0; returns MINLOG + */ + +/* logl.c + * + * Natural logarithm, long double precision + * + * + * + * SYNOPSIS: + * + * long double x, y, logl(); + * + * y = logl( x ); + * + * + * + * DESCRIPTION: + * + * Returns the base e (2.718...) 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 150000 8.71e-20 2.75e-20 + * IEEE exp(+-10000) 100000 5.39e-20 2.34e-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 + */ + +/* mtherr.c + * + * Library common error handling routine + * + * + * + * SYNOPSIS: + * + * char *fctnam; + * int code; + * int mtherr(); + * + * mtherr( fctnam, code ); + * + * + * + * DESCRIPTION: + * + * This routine may be called to report one of the following + * error conditions (in the include file mconf.h). + * + * Mnemonic Value Significance + * + * DOMAIN 1 argument domain error + * SING 2 function singularity + * OVERFLOW 3 overflow range error + * UNDERFLOW 4 underflow range error + * TLOSS 5 total loss of precision + * PLOSS 6 partial loss of precision + * EDOM 33 Unix domain error code + * ERANGE 34 Unix range error code + * + * The default version of the file prints the function name, + * passed to it by the pointer fctnam, followed by the + * error condition. The display is directed to the standard + * output device. The routine then returns to the calling + * program. Users may wish to modify the program to abort by + * calling exit() under severe error conditions such as domain + * errors. + * + * Since all error conditions pass control to this function, + * the display may be easily changed, eliminated, or directed + * to an error logging device. + * + * SEE ALSO: + * + * mconf.h + * + */ + +/* nbdtrl.c + * + * Negative binomial distribution + * + * + * + * SYNOPSIS: + * + * int k, n; + * long double p, y, nbdtrl(); + * + * y = nbdtrl( k, n, p ); + * + * + * + * 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. + * + * + * + * ACCURACY: + * + * Tested at random points (k,n,p) with k and n between 1 and 10,000 + * and p between 0 and 1. + * + * arithmetic domain # trials peak rms + * Absolute error: + * IEEE 0,10000 10000 9.8e-15 2.1e-16 + * + */ + /* nbdtrcl.c + * + * Complemented negative binomial distribution + * + * + * + * SYNOPSIS: + * + * int k, n; + * long double p, y, nbdtrcl(); + * + * y = nbdtrcl( k, n, p ); + * + * + * + * 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. + * + * + * + * ACCURACY: + * + * See incbetl.c. + * + */ + /* nbdtril + * + * Functional inverse of negative binomial distribution + * + * + * + * SYNOPSIS: + * + * int k, n; + * long double p, y, nbdtril(); + * + * p = nbdtril( k, n, y ); + * + * + * + * DESCRIPTION: + * + * Finds the argument p such that nbdtr(k,n,p) is equal to y. + * + * ACCURACY: + * + * Tested at random points (a,b,y), with y between 0 and 1. + * + * a,b Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,100 + * See also incbil.c. + */ + +/* ndtril.c + * + * Inverse of Normal distribution function + * + * + * + * SYNOPSIS: + * + * long double x, y, ndtril(); + * + * x = ndtril( y ); + * + * + * + * 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. + * + * + * For small arguments 0 < y < exp(-2), the program computes + * z = sqrt( -2 log(y) ); then the approximation is + * x = z - log(z)/z - (1/z) P(1/z) / Q(1/z) . + * For larger arguments, x/sqrt(2 pi) = w + w^3 R(w^2)/S(w^2)) , + * where w = y - 0.5 . + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * Arguments uniformly distributed: + * IEEE 0, 1 5000 7.8e-19 9.9e-20 + * Arguments exponentially distributed: + * IEEE exp(-11355),-1 30000 1.7e-19 4.3e-20 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * ndtril domain x <= 0 -MAXNUML + * ndtril domain x >= 1 MAXNUML + * + */ + +/* ndtril.c + * + * Inverse of Normal distribution function + * + * + * + * SYNOPSIS: + * + * long double x, y, ndtril(); + * + * x = ndtril( y ); + * + * + * + * 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. + * + * + * For small arguments 0 < y < exp(-2), the program computes + * z = sqrt( -2 log(y) ); then the approximation is + * x = z - log(z)/z - (1/z) P(1/z) / Q(1/z) . + * For larger arguments, x/sqrt(2 pi) = w + w^3 R(w^2)/S(w^2)) , + * where w = y - 0.5 . + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * Arguments uniformly distributed: + * IEEE 0, 1 5000 7.8e-19 9.9e-20 + * Arguments exponentially distributed: + * IEEE exp(-11355),-1 30000 1.7e-19 4.3e-20 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * ndtril domain x <= 0 -MAXNUML + * ndtril domain x >= 1 MAXNUML + * + */ + +/* pdtrl.c + * + * Poisson distribution + * + * + * + * SYNOPSIS: + * + * int k; + * long double m, y, pdtrl(); + * + * y = pdtrl( k, m ); + * + * + * + * 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. + * + * + * + * ACCURACY: + * + * See igamc(). + * + */ + /* pdtrcl() + * + * Complemented poisson distribution + * + * + * + * SYNOPSIS: + * + * int k; + * long double m, y, pdtrcl(); + * + * y = pdtrcl( k, m ); + * + * + * + * 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. + * + * + * + * ACCURACY: + * + * See igam.c. + * + */ + /* pdtril() + * + * Inverse Poisson distribution + * + * + * + * SYNOPSIS: + * + * int k; + * long double m, y, pdtrl(); + * + * m = pdtril( k, y ); + * + * + * + * + * 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 ). + * + * + * + * + * ACCURACY: + * + * See igami.c. + * + * ERROR MESSAGES: + * + * message condition value returned + * pdtri domain y < 0 or y >= 1 0.0 + * k < 0 + * + */ + +/* polevll.c + * p1evll.c + * + * Evaluate polynomial + * + * + * + * SYNOPSIS: + * + * int N; + * long double x, y, coef[N+1], polevl[]; + * + * y = polevll( x, coef, N ); + * + * + * + * DESCRIPTION: + * + * Evaluates polynomial of degree N: + * + * 2 N + * y = C + C x + C x +...+ C x + * 0 1 2 N + * + * Coefficients are stored in reverse order: + * + * coef[0] = C , ..., coef[N] = C . + * N 0 + * + * The function p1evll() assumes that coef[N] = 1.0 and is + * omitted from the array. Its calling arguments are + * otherwise the same as polevll(). + * + * This module also contains the following globally declared constants: + * MAXNUML = 1.189731495357231765021263853E4932L; + * MACHEPL = 5.42101086242752217003726400434970855712890625E-20L; + * MAXLOGL = 1.1356523406294143949492E4L; + * MINLOGL = -1.1355137111933024058873E4L; + * LOGE2L = 6.9314718055994530941723E-1L; + * LOG2EL = 1.4426950408889634073599E0L; + * PIL = 3.1415926535897932384626L; + * PIO2L = 1.5707963267948966192313L; + * PIO4L = 7.8539816339744830961566E-1L; + * + * SPEED: + * + * In the interest of speed, there are no checks for out + * of bounds arithmetic. This routine is used by most of + * the functions in the library. Depending on available + * equipment features, the user may wish to rewrite the + * program in microcode or assembly language. + * + */ + +/* powil.c + * + * Real raised to integer power, long double precision + * + * + * + * SYNOPSIS: + * + * long double x, y, powil(); + * int n; + * + * y = powil( x, n ); + * + * + * + * DESCRIPTION: + * + * Returns argument x raised to the nth power. + * The routine efficiently decomposes n as a sum of powers of + * two. The desired power is a product of two-to-the-kth + * powers of x. Thus to compute the 32767 power of x requires + * 28 multiplications instead of 32767 multiplications. + * + * + * + * ACCURACY: + * + * + * Relative error: + * arithmetic x domain n domain # trials peak rms + * IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18 + * IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18 + * IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17 + * + * Returns MAXNUM on overflow, zero on underflow. + * + */ + +/* powl.c + * + * Power function, long double precision + * + * + * + * SYNOPSIS: + * + * long double x, y, z, powl(); + * + * z = powl( x, y ); + * + * + * + * DESCRIPTION: + * + * Computes x raised to the yth power. Analytically, + * + * x**y = exp( y log(x) ). + * + * Following Cody and Waite, this program uses a lookup table + * of 2**-i/32 and pseudo extended precision arithmetic to + * obtain several extra bits of accuracy in both the logarithm + * and the exponential. + * + * + * + * ACCURACY: + * + * The relative error of pow(x,y) can be estimated + * by y dl ln(2), where dl is the absolute error of + * the internally computed base 2 logarithm. At the ends + * of the approximation interval the logarithm equal 1/32 + * and its relative error is about 1 lsb = 1.1e-19. Hence + * the predicted relative error in the result is 2.3e-21 y . + * + * Relative error: + * arithmetic domain # trials peak rms + * + * IEEE +-1000 40000 2.8e-18 3.7e-19 + * .001 < x < 1000, with log(x) uniformly distributed. + * -1000 < y < 1000, y uniformly distributed. + * + * IEEE 0,8700 60000 6.5e-18 1.0e-18 + * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed. + * + * + * ERROR MESSAGES: + * + * message condition value returned + * pow overflow x**y > MAXNUM MAXNUM + * pow underflow x**y < 1/MAXNUM 0.0 + * pow domain x<0 and y noninteger 0.0 + * + */ + +/* sinhl.c + * + * Hyperbolic sine, long double precision + * + * + * + * SYNOPSIS: + * + * long double x, y, sinhl(); + * + * y = sinhl( x ); + * + * + * + * DESCRIPTION: + * + * Returns hyperbolic sine of argument in the range MINLOGL to + * MAXLOGL. + * + * The range is partitioned into two segments. If |x| <= 1, a + * rational function of the form x + x**3 P(x)/Q(x) is employed. + * Otherwise the calculation is sinh(x) = ( exp(x) - exp(-x) )/2. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -2,2 10000 1.5e-19 3.9e-20 + * IEEE +-10000 30000 1.1e-19 2.8e-20 + * + */ + +/* sinl.c + * + * Circular sine, long double precision + * + * + * + * SYNOPSIS: + * + * long double x, y, sinl(); + * + * y = sinl( x ); + * + * + * + * DESCRIPTION: + * + * Range reduction is into intervals of pi/4. The reduction + * error is nearly eliminated by contriving an extended precision + * modular arithmetic. + * + * Two polynomial approximating functions are employed. + * Between 0 and pi/4 the sine is approximated by the Cody + * and Waite polynomial form + * x + x**3 P(x**2) . + * Between pi/4 and pi/2 the cosine is represented as + * 1 - .5 x**2 + x**4 Q(x**2) . + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE +-5.5e11 200,000 1.2e-19 2.9e-20 + * + * ERROR MESSAGES: + * + * message condition value returned + * sin total loss x > 2**39 0.0 + * + * Loss of precision occurs for x > 2**39 = 5.49755813888e11. + * The routine as implemented flags a TLOSS error for + * x > 2**39 and returns 0.0. + */ + /* cosl.c + * + * Circular cosine, long double precision + * + * + * + * SYNOPSIS: + * + * long double x, y, cosl(); + * + * y = cosl( x ); + * + * + * + * DESCRIPTION: + * + * Range reduction is into intervals of pi/4. The reduction + * error is nearly eliminated by contriving an extended precision + * modular arithmetic. + * + * Two polynomial approximating functions are employed. + * Between 0 and pi/4 the cosine is approximated by + * 1 - .5 x**2 + x**4 Q(x**2) . + * Between pi/4 and pi/2 the sine is represented by the Cody + * and Waite polynomial form + * x + x**3 P(x**2) . + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE +-5.5e11 50000 1.2e-19 2.9e-20 + */ + +/* sqrtl.c + * + * Square root, long double precision + * + * + * + * SYNOPSIS: + * + * long double x, y, sqrtl(); + * + * y = sqrtl( x ); + * + * + * + * 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 three times to converge to an accurate value. + * + * Note, some arithmetic coprocessors such as the 8087 and + * 68881 produce correctly rounded square roots, which this + * routine will not. + * + * ACCURACY: + * + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,10 30000 8.1e-20 3.1e-20 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * sqrt domain x < 0 0.0 + * + */ + +/* stdtrl.c + * + * Student's t distribution + * + * + * + * SYNOPSIS: + * + * long double p, t, stdtrl(); + * int k; + * + * p = stdtrl( k, t ); + * + * + * 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 < -1.6, 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: + * + * Tested at random 1 <= k <= 100. The "domain" refers to t. + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -100,-1.6 10000 5.7e-18 9.8e-19 + * IEEE -1.6,100 10000 3.8e-18 1.0e-19 + */ + +/* stdtril.c + * + * Functional inverse of Student's t distribution + * + * + * + * SYNOPSIS: + * + * long double p, t, stdtril(); + * int k; + * + * t = stdtril( k, p ); + * + * + * DESCRIPTION: + * + * Given probability p, finds the argument t such that stdtrl(k,t) + * is equal to p. + * + * ACCURACY: + * + * Tested at random 1 <= k <= 100. The "domain" refers to p: + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,1 3500 4.2e-17 4.1e-18 + */ + +/* tanhl.c + * + * Hyperbolic tangent, long double precision + * + * + * + * SYNOPSIS: + * + * long double x, y, tanhl(); + * + * y = tanhl( x ); + * + * + * + * DESCRIPTION: + * + * Returns hyperbolic tangent of argument in the range MINLOGL to + * MAXLOGL. + * + * A rational function is used for |x| < 0.625. The form + * x + x**3 P(x)/Q(x) of Cody _& Waite is employed. + * Otherwise, + * tanh(x) = sinh(x)/cosh(x) = 1 - 2/(exp(2x) + 1). + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -2,2 30000 1.3e-19 2.4e-20 + * + */ + +/* tanl.c + * + * Circular tangent, long double precision + * + * + * + * SYNOPSIS: + * + * long double x, y, tanl(); + * + * y = tanl( x ); + * + * + * + * DESCRIPTION: + * + * Returns the circular tangent of the radian argument x. + * + * Range reduction is modulo pi/4. A rational function + * x + x**3 P(x**2)/Q(x**2) + * is employed in the basic interval [0, pi/4]. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE +-1.07e9 30000 1.9e-19 4.8e-20 + * + * ERROR MESSAGES: + * + * message condition value returned + * tan total loss x > 2^39 0.0 + * + */ + /* cotl.c + * + * Circular cotangent, long double precision + * + * + * + * SYNOPSIS: + * + * long double x, y, cotl(); + * + * y = cotl( x ); + * + * + * + * DESCRIPTION: + * + * Returns the circular cotangent of the radian argument x. + * + * Range reduction is modulo pi/4. A rational function + * x + x**3 P(x**2)/Q(x**2) + * is employed in the basic interval [0, pi/4]. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE +-1.07e9 30000 1.9e-19 5.1e-20 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * cot total loss x > 2^39 0.0 + * cot singularity x = 0 MAXNUM + * + */ + +/* unityl.c + * + * Relative error approximations for function arguments near + * unity. + * + * log1p(x) = log(1+x) + * expm1(x) = exp(x) - 1 + * cos1m(x) = cos(x) - 1 + * + */ + +/* ynl.c + * + * Bessel function of second kind of integer order + * + * + * + * SYNOPSIS: + * + * long double x, y, ynl(); + * int n; + * + * y = ynl( n, x ); + * + * + * + * DESCRIPTION: + * + * Returns Bessel function of order n, where n is a + * (possibly negative) integer. + * + * The function is evaluated by forward recurrence on + * n, starting with values computed by the routines + * y0l() and y1l(). + * + * If n = 0 or 1 the routine for y0l or y1l is called + * directly. + * + * + * + * ACCURACY: + * + * + * Absolute error, except relative error when y > 1. + * x >= 0, -30 <= n <= +30. + * arithmetic domain # trials peak rms + * IEEE -30, 30 10000 1.3e-18 1.8e-19 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * ynl singularity x = 0 MAXNUML + * ynl overflow MAXNUML + * + * Spot checked against tables for x, n between 0 and 100. + * + */ -- cgit v1.2.3