diff options
| author | Eric Andersen <andersen@codepoet.org> | 2001-05-10 00:40:28 +0000 |
|---|---|---|
| committer | Eric Andersen <andersen@codepoet.org> | 2001-05-10 00:40:28 +0000 |
| commit | 1077fa4d772832f77a677ce7fb7c2d513b959e3f (patch) | |
| tree | 579bee13fb0b58d2800206366ec2caecbb15f3fc /libm/float/README.txt | |
| parent | 22358dd7ce7bb49792204b698f01a6f69b9c8e08 (diff) | |
uClibc now has a math library. muahahahaha!
-Erik
Diffstat (limited to 'libm/float/README.txt')
| -rw-r--r-- | libm/float/README.txt | 4721 |
1 files changed, 4721 insertions, 0 deletions
diff --git a/libm/float/README.txt b/libm/float/README.txt new file mode 100644 index 000000000..30a10b083 --- /dev/null +++ b/libm/float/README.txt @@ -0,0 +1,4721 @@ +/* acoshf.c + * + * Inverse hyperbolic cosine + * + * + * + * SYNOPSIS: + * + * float x, y, acoshf(); + * + * y = acoshf( x ); + * + * + * + * DESCRIPTION: + * + * Returns inverse hyperbolic cosine of argument. + * + * If 1 <= x < 1.5, a polynomial approximation + * + * sqrt(z) * P(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 100000 1.8e-7 3.9e-8 + * IEEE 1,2000 100000 3.0e-8 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * acoshf domain |x| < 1 0.0 + * + */ + +/* airy.c + * + * Airy function + * + * + * + * SYNOPSIS: + * + * float x, ai, aip, bi, bip; + * int airyf(); + * + * airyf( x, _&ai, _&aip, _&bi, _&bip ); + * + * + * + * DESCRIPTION: + * + * Solution of the differential equation + * + * y"(x) = xy. + * + * The function returns the two independent solutions Ai, Bi + * and their first derivatives Ai'(x), Bi'(x). + * + * Evaluation is by power series summation for small x, + * by rational minimax approximations for large x. + * + * + * + * ACCURACY: + * Error criterion is absolute when function <= 1, relative + * when function > 1, except * denotes relative error criterion. + * For large negative x, the absolute error increases as x^1.5. + * For large positive x, the relative error increases as x^1.5. + * + * Arithmetic domain function # trials peak rms + * IEEE -10, 0 Ai 50000 7.0e-7 1.2e-7 + * IEEE 0, 10 Ai 50000 9.9e-6* 6.8e-7* + * IEEE -10, 0 Ai' 50000 2.4e-6 3.5e-7 + * IEEE 0, 10 Ai' 50000 8.7e-6* 6.2e-7* + * IEEE -10, 10 Bi 100000 2.2e-6 2.6e-7 + * IEEE -10, 10 Bi' 50000 2.2e-6 3.5e-7 + * + */ + +/* asinf.c + * + * Inverse circular sine + * + * + * + * SYNOPSIS: + * + * float x, y, asinf(); + * + * y = asinf( x ); + * + * + * + * DESCRIPTION: + * + * Returns radian angle between -pi/2 and +pi/2 whose sine is x. + * + * A polynomial of the form x + x**3 P(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 100000 2.5e-7 5.0e-8 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * asinf domain |x| > 1 0.0 + * + */ + /* acosf() + * + * Inverse circular cosine + * + * + * + * SYNOPSIS: + * + * float x, y, acosf(); + * + * y = acosf( 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 100000 1.4e-7 4.2e-8 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * acosf domain |x| > 1 0.0 + */ + +/* asinhf.c + * + * Inverse hyperbolic sine + * + * + * + * SYNOPSIS: + * + * float x, y, asinhf(); + * + * y = asinhf( 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 100000 2.4e-7 4.1e-8 + * + */ + +/* atanf.c + * + * Inverse circular tangent + * (arctangent) + * + * + * + * SYNOPSIS: + * + * float x, y, atanf(); + * + * y = atanf( 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 ). A polynomial approximates + * the function in this basic interval. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -10, 10 100000 1.9e-7 4.1e-8 + * + */ + /* atan2f() + * + * Quadrant correct inverse circular tangent + * + * + * + * SYNOPSIS: + * + * float x, y, z, atan2f(); + * + * z = atan2f( 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 100000 1.9e-7 4.1e-8 + * See atan.c. + * + */ + +/* atanhf.c + * + * Inverse hyperbolic tangent + * + * + * + * SYNOPSIS: + * + * float x, y, atanhf(); + * + * y = atanhf( x ); + * + * + * + * DESCRIPTION: + * + * Returns inverse hyperbolic tangent of argument in the range + * MINLOGF to MAXLOGF. + * + * If |x| < 0.5, a polynomial approximation is used. + * Otherwise, + * atanh(x) = 0.5 * log( (1+x)/(1-x) ). + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -1,1 100000 1.4e-7 3.1e-8 + * + */ + +/* bdtrf.c + * + * Binomial distribution + * + * + * + * SYNOPSIS: + * + * int k, n; + * float p, y, bdtrf(); + * + * y = bdtrf( 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: + * + * Relative error (p varies from 0 to 1): + * arithmetic domain # trials peak rms + * IEEE 0,100 2000 6.9e-5 1.1e-5 + * + * ERROR MESSAGES: + * + * message condition value returned + * bdtrf domain k < 0 0.0 + * n < k + * x < 0, x > 1 + * + */ + /* bdtrcf() + * + * Complemented binomial distribution + * + * + * + * SYNOPSIS: + * + * int k, n; + * float p, y, bdtrcf(); + * + * y = bdtrcf( 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: + * + * Relative error (p varies from 0 to 1): + * arithmetic domain # trials peak rms + * IEEE 0,100 2000 6.0e-5 1.2e-5 + * + * ERROR MESSAGES: + * + * message condition value returned + * bdtrcf domain x<0, x>1, n<k 0.0 + */ + /* bdtrif() + * + * Inverse binomial distribution + * + * + * + * SYNOPSIS: + * + * int k, n; + * float p, y, bdtrif(); + * + * p = bdtrf( k, n, y ); + * + * + * + * DESCRIPTION: + * + * Finds the event probability p such that the sum of the + * terms 0 through k of the Binomial probability density + * is equal to the given cumulative probability y. + * + * This is accomplished using the inverse beta integral + * function and the relation + * + * 1 - p = incbi( n-k, k+1, y ). + * + * + * + * + * ACCURACY: + * + * Relative error (p varies from 0 to 1): + * arithmetic domain # trials peak rms + * IEEE 0,100 2000 3.5e-5 3.3e-6 + * + * ERROR MESSAGES: + * + * message condition value returned + * bdtrif domain k < 0, n <= k 0.0 + * x < 0, x > 1 + * + */ + +/* betaf.c + * + * Beta function + * + * + * + * SYNOPSIS: + * + * float a, b, y, betaf(); + * + * y = betaf( a, b ); + * + * + * + * DESCRIPTION: + * + * - - + * | (a) | (b) + * beta( a, b ) = -----------. + * - + * | (a+b) + * + * For large arguments the logarithm of the function is + * evaluated using lgam(), then exponentiated. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,30 10000 4.0e-5 6.0e-6 + * IEEE -20,0 10000 4.9e-3 5.4e-5 + * + * ERROR MESSAGES: + * + * message condition value returned + * betaf overflow log(beta) > MAXLOG 0.0 + * a or b <0 integer 0.0 + * + */ + +/* cbrtf.c + * + * Cube root + * + * + * + * SYNOPSIS: + * + * float x, y, cbrtf(); + * + * y = cbrtf( 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 to converge to an accurate result. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,1e38 100000 7.6e-8 2.7e-8 + * + */ + +/* chbevlf.c + * + * Evaluate Chebyshev series + * + * + * + * SYNOPSIS: + * + * int N; + * float x, y, coef[N], chebevlf(); + * + * y = chbevlf( x, coef, N ); + * + * + * + * DESCRIPTION: + * + * Evaluates the series + * + * N-1 + * - ' + * y = > coef[i] T (x/2) + * - i + * i=0 + * + * of Chebyshev polynomials Ti at argument x/2. + * + * Coefficients are stored in reverse order, i.e. the zero + * order term is last in the array. Note N is the number of + * coefficients, not the order. + * + * If coefficients are for the interval a to b, x must + * have been transformed to x -> 2(2x - b - a)/(b-a) before + * entering the routine. This maps x from (a, b) to (-1, 1), + * over which the Chebyshev polynomials are defined. + * + * If the coefficients are for the inverted interval, in + * which (a, b) is mapped to (1/b, 1/a), the transformation + * required is x -> 2(2ab/x - b - a)/(b-a). If b is infinity, + * this becomes x -> 4a/x - 1. + * + * + * + * SPEED: + * + * Taking advantage of the recurrence properties of the + * Chebyshev polynomials, the routine requires one more + * addition per loop than evaluating a nested polynomial of + * the same degree. + * + */ + +/* chdtrf.c + * + * Chi-square distribution + * + * + * + * SYNOPSIS: + * + * float df, x, y, chdtrf(); + * + * y = chdtrf( 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: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,100 5000 3.2e-5 5.0e-6 + * + * ERROR MESSAGES: + * + * message condition value returned + * chdtrf domain x < 0 or v < 1 0.0 + */ + /* chdtrcf() + * + * Complemented Chi-square distribution + * + * + * + * SYNOPSIS: + * + * float v, x, y, chdtrcf(); + * + * y = chdtrcf( 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: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,100 5000 2.7e-5 3.2e-6 + * + * ERROR MESSAGES: + * + * message condition value returned + * chdtrc domain x < 0 or v < 1 0.0 + */ + /* chdtrif() + * + * Inverse of complemented Chi-square distribution + * + * + * + * SYNOPSIS: + * + * float df, x, y, chdtrif(); + * + * x = chdtrif( 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: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,100 10000 2.2e-5 8.5e-7 + * + * ERROR MESSAGES: + * + * message condition value returned + * chdtri domain y < 0 or y > 1 0.0 + * v < 1 + * + */ + +/* clogf.c + * + * Complex natural logarithm + * + * + * + * SYNOPSIS: + * + * void clogf(); + * cmplxf z, w; + * + * clogf( &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 + * IEEE -10,+10 30000 1.9e-6 6.2e-8 + * + * Larger relative error can be observed for z near 1 +i0. + * In IEEE arithmetic the peak absolute error is 3.1e-7. + * + */ + /* cexpf() + * + * Complex exponential function + * + * + * + * SYNOPSIS: + * + * void cexpf(); + * cmplxf z, w; + * + * cexpf( &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 + * IEEE -10,+10 30000 1.4e-7 4.5e-8 + * + */ + /* csinf() + * + * Complex circular sine + * + * + * + * SYNOPSIS: + * + * void csinf(); + * cmplxf z, w; + * + * csinf( &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 + * IEEE -10,+10 30000 1.9e-7 5.5e-8 + * + */ + /* ccosf() + * + * Complex circular cosine + * + * + * + * SYNOPSIS: + * + * void ccosf(); + * cmplxf z, w; + * + * ccosf( &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 + * IEEE -10,+10 30000 1.8e-7 5.5e-8 + */ + /* ctanf() + * + * Complex circular tangent + * + * + * + * SYNOPSIS: + * + * void ctanf(); + * cmplxf z, w; + * + * ctanf( &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 + * IEEE -10,+10 30000 3.3e-7 5.1e-8 + */ + /* ccotf() + * + * Complex circular cotangent + * + * + * + * SYNOPSIS: + * + * void ccotf(); + * cmplxf z, w; + * + * ccotf( &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 + * IEEE -10,+10 30000 3.6e-7 5.7e-8 + * Also tested by ctan * ccot = 1 + i0. + */ + /* casinf() + * + * Complex circular arc sine + * + * + * + * SYNOPSIS: + * + * void casinf(); + * cmplxf z, w; + * + * casinf( &z, &w ); + * + * + * + * DESCRIPTION: + * + * Inverse complex sine: + * + * 2 + * w = -i clog( iz + csqrt( 1 - z ) ). + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -10,+10 30000 1.1e-5 1.5e-6 + * Larger relative error can be observed for z near zero. + * + */ + /* cacosf() + * + * Complex circular arc cosine + * + * + * + * SYNOPSIS: + * + * void cacosf(); + * cmplxf z, w; + * + * cacosf( &z, &w ); + * + * + * + * DESCRIPTION: + * + * + * w = arccos z = PI/2 - arcsin z. + * + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -10,+10 30000 9.2e-6 1.2e-6 + * + */ + /* catan() + * + * Complex circular arc tangent + * + * + * + * SYNOPSIS: + * + * void catan(); + * cmplxf z, w; + * + * catan( &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 + * IEEE -10,+10 30000 2.3e-6 5.2e-8 + * + */ + +/* cmplxf.c + * + * Complex number arithmetic + * + * + * + * SYNOPSIS: + * + * typedef struct { + * float r; real part + * float i; imaginary part + * }cmplxf; + * + * cmplxf *a, *b, *c; + * + * caddf( a, b, c ); c = b + a + * csubf( a, b, c ); c = b - a + * cmulf( a, b, c ); c = b * a + * cdivf( a, b, c ); c = b / a + * cnegf( c ); c = -c + * cmovf( 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 + * IEEE cadd 30000 5.9e-8 2.6e-8 + * IEEE csub 30000 6.0e-8 2.6e-8 + * IEEE cmul 30000 1.1e-7 3.7e-8 + * IEEE cdiv 30000 2.1e-7 5.7e-8 + */ + +/* cabsf() + * + * Complex absolute value + * + * + * + * SYNOPSIS: + * + * float cabsf(); + * cmplxf z; + * float a; + * + * a = cabsf( &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 + * IEEE -10,+10 30000 1.2e-7 3.4e-8 + */ + /* csqrtf() + * + * Complex square root + * + * + * + * SYNOPSIS: + * + * void csqrtf(); + * cmplxf z, w; + * + * csqrtf( &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 solution + * reported 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 + * IEEE -10,+10 100000 1.8e-7 4.2e-8 + * + */ + +/* coshf.c + * + * Hyperbolic cosine + * + * + * + * SYNOPSIS: + * + * float x, y, coshf(); + * + * y = coshf( x ); + * + * + * + * DESCRIPTION: + * + * Returns hyperbolic cosine of argument in the range MINLOGF to + * MAXLOGF. + * + * cosh(x) = ( exp(x) + exp(-x) )/2. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE +-MAXLOGF 100000 1.2e-7 2.8e-8 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * coshf overflow |x| > MAXLOGF MAXNUMF + * + * + */ + +/* dawsnf.c + * + * Dawson's Integral + * + * + * + * SYNOPSIS: + * + * float x, y, dawsnf(); + * + * y = dawsnf( x ); + * + * + * + * DESCRIPTION: + * + * Approximates the integral + * + * x + * - + * 2 | | 2 + * dawsn(x) = exp( -x ) | exp( t ) dt + * | | + * - + * 0 + * + * Three different rational approximations are employed, for + * the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up. + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,10 50000 4.4e-7 6.3e-8 + * + * + */ + +/* ellief.c + * + * Incomplete elliptic integral of the second kind + * + * + * + * SYNOPSIS: + * + * float phi, m, y, ellief(); + * + * y = ellief( 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 [0, 2] and m in + * [0, 1]. + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,2 10000 4.5e-7 7.4e-8 + * + * + */ + +/* ellikf.c + * + * Incomplete elliptic integral of the first kind + * + * + * + * SYNOPSIS: + * + * float phi, m, y, ellikf(); + * + * y = ellikf( 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 phi in [0, 2] and m in + * [0, 1]. + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,2 10000 2.9e-7 5.8e-8 + * + * + */ + +/* ellpef.c + * + * Complete elliptic integral of the second kind + * + * + * + * SYNOPSIS: + * + * float m1, y, ellpef(); + * + * y = ellpef( 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 30000 1.1e-7 3.9e-8 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * ellpef domain x<0, x>1 0.0 + * + */ + +/* ellpjf.c + * + * Jacobian Elliptic Functions + * + * + * + * SYNOPSIS: + * + * float u, m, sn, cn, dn, phi; + * int ellpj(); + * + * ellpj( 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-9 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-6 2.2e-7 + * IEEE cn 10000 1.6e-6 2.2e-7 + * IEEE dn 10000 1.4e-3 1.9e-5 + * IEEE phi 10000 3.9e-7* 6.7e-8* + * + * Peak error observed in consistency check using addition + * theorem for sn(u+v) was 4e-16 (absolute). Also tested by + * the above relation to the incomplete elliptic integral. + * Accuracy deteriorates when u is large. + * + */ + +/* ellpkf.c + * + * Complete elliptic integral of the first kind + * + * + * + * SYNOPSIS: + * + * float m1, y, ellpkf(); + * + * y = ellpkf( 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 30000 1.3e-7 3.4e-8 + * + * ERROR MESSAGES: + * + * message condition value returned + * ellpkf domain x<0, x>1 0.0 + * + */ + +/* exp10f.c + * + * Base 10 exponential function + * (Common antilogarithm) + * + * + * + * SYNOPSIS: + * + * float x, y, exp10f(); + * + * y = exp10f( 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). + * A polynomial approximates 10**f. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -38,+38 100000 9.8e-8 2.8e-8 + * + * ERROR MESSAGES: + * + * message condition value returned + * exp10 underflow x < -MAXL10 0.0 + * exp10 overflow x > MAXL10 MAXNUM + * + * IEEE single arithmetic: MAXL10 = 38.230809449325611792. + * + */ + +/* exp2f.c + * + * Base 2 exponential function + * + * + * + * SYNOPSIS: + * + * float x, y, exp2f(); + * + * y = exp2f( 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 polynomial approximates 2**x in the basic range [-0.5, 0.5]. + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -127,+127 100000 1.7e-7 2.8e-8 + * + * + * See exp.c for comments on error amplification. + * + * + * ERROR MESSAGES: + * + * message condition value returned + * exp underflow x < -MAXL2 0.0 + * exp overflow x > MAXL2 MAXNUMF + * + * For IEEE arithmetic, MAXL2 = 127. + */ + +/* expf.c + * + * Exponential function + * + * + * + * SYNOPSIS: + * + * float x, y, expf(); + * + * y = expf( 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 polynomial is used to approximate exp(f) + * in the basic range [-0.5, 0.5]. + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE +- MAXLOG 100000 1.7e-7 2.8e-8 + * + * + * 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 double precision + * computer number, the result contains amplified roundoff + * error for large arguments not exactly represented. + * + * + * ERROR MESSAGES: + * + * message condition value returned + * expf underflow x < MINLOGF 0.0 + * expf overflow x > MAXLOGF MAXNUMF + * + */ + +/* expnf.c + * + * Exponential integral En + * + * + * + * SYNOPSIS: + * + * int n; + * float x, y, expnf(); + * + * y = expnf( n, x ); + * + * + * + * DESCRIPTION: + * + * Evaluates the exponential integral + * + * inf. + * - + * | | -xt + * | e + * E (x) = | ---- dt. + * n | n + * | | t + * - + * 1 + * + * + * Both n and x must be nonnegative. + * + * The routine employs either a power series, a continued + * fraction, or an asymptotic formula depending on the + * relative values of n and x. + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0, 30 10000 5.6e-7 1.2e-7 + * + */ + +/* facf.c + * + * Factorial function + * + * + * + * SYNOPSIS: + * + * float y, facf(); + * int i; + * + * y = facf( i ); + * + * + * + * DESCRIPTION: + * + * Returns factorial of i = 1 * 2 * 3 * ... * i. + * fac(0) = 1.0. + * + * Due to machine arithmetic bounds the largest value of + * i accepted is 33 in single precision arithmetic. + * Greater values, or negative ones, + * produce an error message and return MAXNUM. + * + * + * + * ACCURACY: + * + * For i < 34 the values are simply tabulated, and have + * full machine accuracy. + * + */ + +/* fdtrf.c + * + * F distribution + * + * + * + * SYNOPSIS: + * + * int df1, df2; + * float x, y, fdtrf(); + * + * y = fdtrf( 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) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ). + * + * + * The arguments a and b are greater than zero, and x + * x is nonnegative. + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,100 5000 2.2e-5 1.1e-6 + * + * ERROR MESSAGES: + * + * message condition value returned + * fdtrf domain a<0, b<0, x<0 0.0 + * + */ + /* fdtrcf() + * + * Complemented F distribution + * + * + * + * SYNOPSIS: + * + * int df1, df2; + * float x, y, fdtrcf(); + * + * y = fdtrcf( 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: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,100 5000 7.3e-5 1.2e-5 + * + * ERROR MESSAGES: + * + * message condition value returned + * fdtrcf domain a<0, b<0, x<0 0.0 + * + */ + /* fdtrif() + * + * Inverse of complemented F distribution + * + * + * + * SYNOPSIS: + * + * float df1, df2, x, y, fdtrif(); + * + * x = fdtrif( df1, df2, y ); + * + * + * + * + * 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 y. + * + * This is accomplished using the inverse beta integral + * function and the relations + * + * z = incbi( df2/2, df1/2, y ) + * 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, y ) + * x = df2 z / (df1 (1-z)). + * + * + * + * ACCURACY: + * + * arithmetic domain # trials peak rms + * Absolute error: + * IEEE 0,100 5000 4.0e-5 3.2e-6 + * Relative error: + * IEEE 0,100 5000 1.2e-3 1.8e-5 + * + * ERROR MESSAGES: + * + * message condition value returned + * fdtrif domain y <= 0 or y > 1 0.0 + * v < 1 + * + */ + +/* ceilf() + * floorf() + * frexpf() + * ldexpf() + * + * Single precision floating point numeric utilities + * + * + * + * SYNOPSIS: + * + * float x, y; + * float ceilf(), floorf(), frexpf(), ldexpf(); + * int expnt, n; + * + * y = floorf(x); + * y = ceilf(x); + * y = frexpf( x, &expnt ); + * y = ldexpf( x, n ); + * + * + * + * DESCRIPTION: + * + * All four routines return a single precision floating point + * result. + * + * sfloor() returns the largest integer less than or equal to x. + * It truncates toward minus infinity. + * + * sceil() returns the smallest integer greater than or equal + * to x. It truncates toward plus infinity. + * + * sfrexp() 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. + * + * sldexp() multiplies x by 2**n. + * + * These functions are part of the standard C run time library + * for many but not all C compilers. The ones supplied are + * written in C for either DEC or 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. + */ + +/* fresnlf.c + * + * Fresnel integral + * + * + * + * SYNOPSIS: + * + * float x, S, C; + * void fresnlf(); + * + * fresnlf( x, _&S, _&C ); + * + * + * DESCRIPTION: + * + * Evaluates the Fresnel integrals + * + * x + * - + * | | + * C(x) = | cos(pi/2 t**2) dt, + * | | + * - + * 0 + * + * x + * - + * | | + * S(x) = | sin(pi/2 t**2) dt. + * | | + * - + * 0 + * + * + * The integrals are evaluated by power series for small x. + * For x >= 1 auxiliary functions f(x) and g(x) are employed + * such that + * + * C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 ) + * S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 ) + * + * + * + * ACCURACY: + * + * Relative error. + * + * Arithmetic function domain # trials peak rms + * IEEE S(x) 0, 10 30000 1.1e-6 1.9e-7 + * IEEE C(x) 0, 10 30000 1.1e-6 2.0e-7 + */ + +/* gammaf.c + * + * Gamma function + * + * + * + * SYNOPSIS: + * + * float x, y, gammaf(); + * extern int sgngamf; + * + * y = gammaf( 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 sgngamf. + * This same variable is also filled in by the logarithmic + * gamma function lgam(). + * + * Arguments between 0 and 10 are reduced by recurrence and the + * function is approximated by a polynomial function covering + * the interval (2,3). Large arguments are handled by Stirling's + * formula. Negative arguments are made positive using + * a reflection formula. + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,-33 100,000 5.7e-7 1.0e-7 + * IEEE -33,0 100,000 6.1e-7 1.2e-7 + * + * + */ +/* lgamf() + * + * Natural logarithm of gamma function + * + * + * + * SYNOPSIS: + * + * float x, y, lgamf(); + * extern int sgngamf; + * + * y = lgamf( 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 sgngamf. + * + * For arguments greater than 6.5, the logarithm of the gamma + * function is approximated by the logarithmic version of + * Stirling's formula. Arguments between 0 and +6.5 are reduced by + * by recurrence to the interval [.75,1.25] or [1.5,2.5] of a rational + * approximation. The cosecant reflection formula is employed for + * arguments less than zero. + * + * Arguments greater than MAXLGM = 2.035093e36 return MAXNUM and an + * error message. + * + * + * + * ACCURACY: + * + * + * + * arithmetic domain # trials peak rms + * IEEE -100,+100 500,000 7.4e-7 6.8e-8 + * The error criterion was relative when the function magnitude + * was greater than one but absolute when it was less than one. + * The routine has low relative error for positive arguments. + * + * The following test used the relative error criterion. + * IEEE -2, +3 100000 4.0e-7 5.6e-8 + * + */ + +/* gdtrf.c + * + * Gamma distribution function + * + * + * + * SYNOPSIS: + * + * float a, b, x, y, gdtrf(); + * + * y = gdtrf( 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: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,100 5000 5.8e-5 3.0e-6 + * + * ERROR MESSAGES: + * + * message condition value returned + * gdtrf domain x < 0 0.0 + * + */ + /* gdtrcf.c + * + * Complemented gamma distribution function + * + * + * + * SYNOPSIS: + * + * float a, b, x, y, gdtrcf(); + * + * y = gdtrcf( 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: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,100 5000 9.1e-5 1.5e-5 + * + * ERROR MESSAGES: + * + * message condition value returned + * gdtrcf domain x < 0 0.0 + * + */ + +/* hyp2f1f.c + * + * Gauss hypergeometric function F + * 2 1 + * + * + * SYNOPSIS: + * + * float a, b, c, x, y, hyp2f1f(); + * + * y = hyp2f1f( a, b, c, x ); + * + * + * DESCRIPTION: + * + * + * hyp2f1( a, b, c, x ) = F ( a, b; c; x ) + * 2 1 + * + * inf. + * - a(a+1)...(a+k) b(b+1)...(b+k) k+1 + * = 1 + > ----------------------------- x . + * - c(c+1)...(c+k) (k+1)! + * k = 0 + * + * Cases addressed are + * Tests and escapes for negative integer a, b, or c + * Linear transformation if c - a or c - b negative integer + * Special case c = a or c = b + * Linear transformation for x near +1 + * Transformation for x < -0.5 + * Psi function expansion if x > 0.5 and c - a - b integer + * Conditionally, a recurrence on c to make c-a-b > 0 + * + * |x| > 1 is rejected. + * + * The parameters a, b, c are considered to be integer + * valued if they are within 1.0e-6 of the nearest integer. + * + * ACCURACY: + * + * Relative error (-1 < x < 1): + * arithmetic domain # trials peak rms + * IEEE 0,3 30000 5.8e-4 4.3e-6 + */ + +/* hypergf.c + * + * Confluent hypergeometric function + * + * + * + * SYNOPSIS: + * + * float a, b, x, y, hypergf(); + * + * y = hypergf( a, b, x ); + * + * + * + * DESCRIPTION: + * + * Computes the confluent hypergeometric function + * + * 1 2 + * a x a(a+1) x + * F ( a,b;x ) = 1 + ---- + --------- + ... + * 1 1 b 1! b(b+1) 2! + * + * Many higher transcendental functions are special cases of + * this power series. + * + * As is evident from the formula, b must not be a negative + * integer or zero unless a is an integer with 0 >= a > b. + * + * The routine attempts both a direct summation of the series + * and an asymptotic expansion. In each case error due to + * roundoff, cancellation, and nonconvergence is estimated. + * The result with smaller estimated error is returned. + * + * + * + * ACCURACY: + * + * Tested at random points (a, b, x), all three variables + * ranging from 0 to 30. + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,5 10000 6.6e-7 1.3e-7 + * IEEE 0,30 30000 1.1e-5 6.5e-7 + * + * Larger errors can be observed when b is near a negative + * integer or zero. Certain combinations of arguments yield + * serious cancellation error in the power series summation + * and also are not in the region of near convergence of the + * asymptotic series. An error message is printed if the + * self-estimated relative error is greater than 1.0e-3. + * + */ + +/* i0f.c + * + * Modified Bessel function of order zero + * + * + * + * SYNOPSIS: + * + * float x, y, i0(); + * + * y = i0f( x ); + * + * + * + * DESCRIPTION: + * + * Returns modified Bessel function of order zero of the + * argument. + * + * The function is defined as i0(x) = j0( ix ). + * + * The range is partitioned into the two intervals [0,8] and + * (8, infinity). Chebyshev polynomial expansions are employed + * in each interval. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,30 100000 4.0e-7 7.9e-8 + * + */ + /* i0ef.c + * + * Modified Bessel function of order zero, + * exponentially scaled + * + * + * + * SYNOPSIS: + * + * float x, y, i0ef(); + * + * y = i0ef( x ); + * + * + * + * DESCRIPTION: + * + * Returns exponentially scaled modified Bessel function + * of order zero of the argument. + * + * The function is defined as i0e(x) = exp(-|x|) j0( ix ). + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,30 100000 3.7e-7 7.0e-8 + * See i0f(). + * + */ + +/* i1f.c + * + * Modified Bessel function of order one + * + * + * + * SYNOPSIS: + * + * float x, y, i1f(); + * + * y = i1f( x ); + * + * + * + * DESCRIPTION: + * + * Returns modified Bessel function of order one of the + * argument. + * + * The function is defined as i1(x) = -i j1( ix ). + * + * The range is partitioned into the two intervals [0,8] and + * (8, infinity). Chebyshev polynomial expansions are employed + * in each interval. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0, 30 100000 1.5e-6 1.6e-7 + * + * + */ + /* i1ef.c + * + * Modified Bessel function of order one, + * exponentially scaled + * + * + * + * SYNOPSIS: + * + * float x, y, i1ef(); + * + * y = i1ef( x ); + * + * + * + * DESCRIPTION: + * + * Returns exponentially scaled modified Bessel function + * of order one of the argument. + * + * The function is defined as i1(x) = -i exp(-|x|) j1( ix ). + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0, 30 30000 1.5e-6 1.5e-7 + * See i1(). + * + */ + +/* igamf.c + * + * Incomplete gamma integral + * + * + * + * SYNOPSIS: + * + * float a, x, y, igamf(); + * + * y = igamf( 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 + * IEEE 0,30 20000 7.8e-6 5.9e-7 + * + */ + /* igamcf() + * + * Complemented incomplete gamma integral + * + * + * + * SYNOPSIS: + * + * float a, x, y, igamcf(); + * + * y = igamcf( 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 + * IEEE 0,30 30000 7.8e-6 5.9e-7 + * + */ + +/* igamif() + * + * Inverse of complemented imcomplete gamma integral + * + * + * + * SYNOPSIS: + * + * float a, x, y, igamif(); + * + * x = igamif( 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 to 100 and x from 0 to 1. + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,100 5000 1.0e-5 1.5e-6 + * + */ + +/* incbetf.c + * + * Incomplete beta integral + * + * + * SYNOPSIS: + * + * float a, b, x, y, incbetf(); + * + * y = incbetf( 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. + * If a < 1, the function calls itself recursively after a + * transformation to increase a to a+1. + * + * ACCURACY: + * + * Tested at random points (a,b,x) with a and b in the indicated + * interval and x between 0 and 1. + * + * arithmetic domain # trials peak rms + * Relative error: + * IEEE 0,30 10000 3.7e-5 5.1e-6 + * IEEE 0,100 10000 1.7e-4 2.5e-5 + * The useful domain for relative error is limited by underflow + * of the single precision exponential function. + * Absolute error: + * IEEE 0,30 100000 2.2e-5 9.6e-7 + * IEEE 0,100 10000 6.5e-5 3.7e-6 + * + * Larger errors may occur for extreme ratios of a and b. + * + * ERROR MESSAGES: + * message condition value returned + * incbetf domain x<0, x>1 0.0 + */ + +/* incbif() + * + * Inverse of imcomplete beta integral + * + * + * + * SYNOPSIS: + * + * float a, b, x, y, incbif(); + * + * x = incbif( 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 0,100 5000 2.8e-4 8.3e-6 + * + * Overflow and larger errors may occur for one of a or b near zero + * and the other large. + */ + +/* ivf.c + * + * Modified Bessel function of noninteger order + * + * + * + * SYNOPSIS: + * + * float v, x, y, ivf(); + * + * y = ivf( v, x ); + * + * + * + * DESCRIPTION: + * + * Returns modified Bessel function of order v of the + * argument. If x is negative, v must be integer valued. + * + * The function is defined as Iv(x) = Jv( ix ). It is + * here computed in terms of the confluent hypergeometric + * function, according to the formula + * + * v -x + * Iv(x) = (x/2) e hyperg( v+0.5, 2v+1, 2x ) / gamma(v+1) + * + * If v is a negative integer, then v is replaced by -v. + * + * + * ACCURACY: + * + * Tested at random points (v, x), with v between 0 and + * 30, x between 0 and 28. + * arithmetic domain # trials peak rms + * Relative error: + * IEEE 0,15 3000 4.7e-6 5.4e-7 + * Absolute error (relative when function > 1) + * IEEE 0,30 5000 8.5e-6 1.3e-6 + * + * Accuracy is diminished if v is near a negative integer. + * The useful domain for relative error is limited by overflow + * of the single precision exponential function. + * + * See also hyperg.c. + * + */ + +/* j0f.c + * + * Bessel function of order zero + * + * + * + * SYNOPSIS: + * + * float x, y, j0f(); + * + * y = j0f( x ); + * + * + * + * DESCRIPTION: + * + * Returns Bessel function of order zero of the argument. + * + * The domain is divided into the intervals [0, 2] and + * (2, infinity). In the first interval the following polynomial + * approximation is used: + * + * + * 2 2 2 + * (w - r ) (w - r ) (w - r ) P(w) + * 1 2 3 + * + * 2 + * where w = x and the three r's are zeros of the function. + * + * In the second interval, the modulus and phase are approximated + * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x) + * and Phase(x) = x + 1/x R(1/x^2) - pi/4. The function is + * + * j0(x) = Modulus(x) cos( Phase(x) ). + * + * + * + * ACCURACY: + * + * Absolute error: + * arithmetic domain # trials peak rms + * IEEE 0, 2 100000 1.3e-7 3.6e-8 + * IEEE 2, 32 100000 1.9e-7 5.4e-8 + * + */ + /* y0f.c + * + * Bessel function of the second kind, order zero + * + * + * + * SYNOPSIS: + * + * float x, y, y0f(); + * + * y = y0f( x ); + * + * + * + * DESCRIPTION: + * + * Returns Bessel function of the second kind, of order + * zero, of the argument. + * + * The domain is divided into the intervals [0, 2] and + * (2, infinity). In the first interval a rational approximation + * R(x) is employed to compute + * + * 2 2 2 + * y0(x) = (w - r ) (w - r ) (w - r ) R(x) + 2/pi ln(x) j0(x). + * 1 2 3 + * + * Thus a call to j0() is required. The three zeros are removed + * from R(x) to improve its numerical stability. + * + * In the second interval, the modulus and phase are approximated + * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x) + * and Phase(x) = x + 1/x S(1/x^2) - pi/4. Then the function is + * + * y0(x) = Modulus(x) sin( Phase(x) ). + * + * + * + * + * ACCURACY: + * + * Absolute error, when y0(x) < 1; else relative error: + * + * arithmetic domain # trials peak rms + * IEEE 0, 2 100000 2.4e-7 3.4e-8 + * IEEE 2, 32 100000 1.8e-7 5.3e-8 + * + */ + +/* j1f.c + * + * Bessel function of order one + * + * + * + * SYNOPSIS: + * + * float x, y, j1f(); + * + * y = j1f( x ); + * + * + * + * DESCRIPTION: + * + * Returns Bessel function of order one of the argument. + * + * The domain is divided into the intervals [0, 2] and + * (2, infinity). In the first interval a polynomial approximation + * 2 + * (w - r ) x P(w) + * 1 + * 2 + * is used, where w = x and r is the first zero of the function. + * + * In the second interval, the modulus and phase are approximated + * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x) + * and Phase(x) = x + 1/x R(1/x^2) - 3pi/4. The function is + * + * j0(x) = Modulus(x) cos( Phase(x) ). + * + * + * + * ACCURACY: + * + * Absolute error: + * arithmetic domain # trials peak rms + * IEEE 0, 2 100000 1.2e-7 2.5e-8 + * IEEE 2, 32 100000 2.0e-7 5.3e-8 + * + * + */ + /* y1.c + * + * Bessel function of second kind of order one + * + * + * + * SYNOPSIS: + * + * double x, y, y1(); + * + * y = y1( x ); + * + * + * + * DESCRIPTION: + * + * Returns Bessel function of the second kind of order one + * of the argument. + * + * The domain is divided into the intervals [0, 2] and + * (2, infinity). In the first interval a rational approximation + * R(x) is employed to compute + * + * 2 + * y0(x) = (w - r ) x R(x^2) + 2/pi (ln(x) j1(x) - 1/x) . + * 1 + * + * Thus a call to j1() is required. + * + * In the second interval, the modulus and phase are approximated + * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x) + * and Phase(x) = x + 1/x S(1/x^2) - 3pi/4. Then the function is + * + * y0(x) = Modulus(x) sin( Phase(x) ). + * + * + * + * + * ACCURACY: + * + * Absolute error: + * arithmetic domain # trials peak rms + * IEEE 0, 2 100000 2.2e-7 4.6e-8 + * IEEE 2, 32 100000 1.9e-7 5.3e-8 + * + * (error criterion relative when |y1| > 1). + * + */ + +/* jnf.c + * + * Bessel function of integer order + * + * + * + * SYNOPSIS: + * + * int n; + * float x, y, jnf(); + * + * y = jnf( 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 range # trials peak rms + * IEEE 0, 15 30000 3.6e-7 3.6e-8 + * + * + * Not suitable for large n or x. Use jvf() instead. + * + */ + +/* jvf.c + * + * Bessel function of noninteger order + * + * + * + * SYNOPSIS: + * + * float v, x, y, jvf(); + * + * y = jvf( v, x ); + * + * + * + * DESCRIPTION: + * + * Returns Bessel function of order v of the argument, + * where v is real. Negative x is allowed if v is an integer. + * + * Several expansions are included: the ascending power + * series, the Hankel expansion, and two transitional + * expansions for large v. If v is not too large, it + * is reduced by recurrence to a region of best accuracy. + * + * The single precision routine accepts negative v, but with + * reduced accuracy. + * + * + * + * ACCURACY: + * Results for integer v are indicated by *. + * Error criterion is absolute, except relative when |jv()| > 1. + * + * arithmetic domain # trials peak rms + * v x + * IEEE 0,125 0,125 30000 2.0e-6 2.0e-7 + * IEEE -17,0 0,125 30000 1.1e-5 4.0e-7 + * IEEE -100,0 0,125 3000 1.5e-4 7.8e-6 + */ + +/* k0f.c + * + * Modified Bessel function, third kind, order zero + * + * + * + * SYNOPSIS: + * + * float x, y, k0f(); + * + * y = k0f( x ); + * + * + * + * DESCRIPTION: + * + * Returns modified Bessel function of the third kind + * of order zero of the argument. + * + * The range is partitioned into the two intervals [0,8] and + * (8, infinity). Chebyshev polynomial expansions are employed + * in each interval. + * + * + * + * ACCURACY: + * + * Tested at 2000 random points between 0 and 8. Peak absolute + * error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15. + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0, 30 30000 7.8e-7 8.5e-8 + * + * ERROR MESSAGES: + * + * message condition value returned + * K0 domain x <= 0 MAXNUM + * + */ + /* k0ef() + * + * Modified Bessel function, third kind, order zero, + * exponentially scaled + * + * + * + * SYNOPSIS: + * + * float x, y, k0ef(); + * + * y = k0ef( x ); + * + * + * + * DESCRIPTION: + * + * Returns exponentially scaled modified Bessel function + * of the third kind of order zero of the argument. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0, 30 30000 8.1e-7 7.8e-8 + * See k0(). + * + */ + +/* k1f.c + * + * Modified Bessel function, third kind, order one + * + * + * + * SYNOPSIS: + * + * float x, y, k1f(); + * + * y = k1f( x ); + * + * + * + * DESCRIPTION: + * + * Computes the modified Bessel function of the third kind + * of order one of the argument. + * + * The range is partitioned into the two intervals [0,2] and + * (2, infinity). Chebyshev polynomial expansions are employed + * in each interval. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0, 30 30000 4.6e-7 7.6e-8 + * + * ERROR MESSAGES: + * + * message condition value returned + * k1 domain x <= 0 MAXNUM + * + */ + /* k1ef.c + * + * Modified Bessel function, third kind, order one, + * exponentially scaled + * + * + * + * SYNOPSIS: + * + * float x, y, k1ef(); + * + * y = k1ef( x ); + * + * + * + * DESCRIPTION: + * + * Returns exponentially scaled modified Bessel function + * of the third kind of order one of the argument: + * + * k1e(x) = exp(x) * k1(x). + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0, 30 30000 4.9e-7 6.7e-8 + * See k1(). + * + */ + +/* knf.c + * + * Modified Bessel function, third kind, integer order + * + * + * + * SYNOPSIS: + * + * float x, y, knf(); + * int n; + * + * y = knf( n, x ); + * + * + * + * DESCRIPTION: + * + * Returns modified Bessel function of the third kind + * of order n of the argument. + * + * The range is partitioned into the two intervals [0,9.55] and + * (9.55, infinity). An ascending power series is used in the + * low range, and an asymptotic expansion in the high range. + * + * + * + * ACCURACY: + * + * Absolute error, relative when function > 1: + * arithmetic domain # trials peak rms + * IEEE 0,30 10000 2.0e-4 3.8e-6 + * + * Error is high only near the crossover point x = 9.55 + * between the two expansions used. + */ + +/* log10f.c + * + * Common logarithm + * + * + * + * SYNOPSIS: + * + * float x, y, log10f(); + * + * y = log10f( x ); + * + * + * + * DESCRIPTION: + * + * Returns logarithm to the base 10 of x. + * + * The argument is separated into its exponent and fractional + * parts. The logarithm of the fraction is approximated by + * + * log(1+x) = x - 0.5 x**2 + x**3 P(x). + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0.5, 2.0 100000 1.3e-7 3.4e-8 + * IEEE 0, MAXNUMF 100000 1.3e-7 2.6e-8 + * + * In the tests over the interval [0, MAXNUM], the logarithms + * of the random arguments were uniformly distributed over + * [-MAXL10, MAXL10]. + * + * ERROR MESSAGES: + * + * log10f singularity: x = 0; returns -MAXL10 + * log10f domain: x < 0; returns -MAXL10 + * MAXL10 = 38.230809449325611792 + */ + +/* log2f.c + * + * Base 2 logarithm + * + * + * + * SYNOPSIS: + * + * float x, y, log2f(); + * + * y = log2f( 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 base e + * 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 exp(+-88) 100000 1.1e-7 2.4e-8 + * IEEE 0.5, 2.0 100000 1.1e-7 3.0e-8 + * + * In the tests over the interval [exp(+-88)], the logarithms + * of the random arguments were uniformly distributed. + * + * ERROR MESSAGES: + * + * log singularity: x = 0; returns MINLOGF/log(2) + * log domain: x < 0; returns MINLOGF/log(2) + */ + +/* logf.c + * + * Natural logarithm + * + * + * + * SYNOPSIS: + * + * float x, y, logf(); + * + * y = logf( 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) + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0.5, 2.0 100000 7.6e-8 2.7e-8 + * IEEE 1, MAXNUMF 100000 2.6e-8 + * + * In the tests over the interval [1, MAXNUM], the logarithms + * of the random arguments were uniformly distributed over + * [0, MAXLOGF]. + * + * ERROR MESSAGES: + * + * logf singularity: x = 0; returns MINLOG + * logf domain: x < 0; returns MINLOG + */ + +/* mtherr.c + * + * Library common error handling routine + * + * + * + * SYNOPSIS: + * + * char *fctnam; + * int code; + * void mtherr(); + * + * mtherr( fctnam, code ); + * + * + * + * DESCRIPTION: + * + * This routine may be called to report one of the following + * error conditions (in the include file math.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: + * + * math.h + * + */ + +/* nbdtrf.c + * + * Negative binomial distribution + * + * + * + * SYNOPSIS: + * + * int k, n; + * float p, y, nbdtrf(); + * + * y = nbdtrf( 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: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,100 5000 1.5e-4 1.9e-5 + * + */ + /* nbdtrcf.c + * + * Complemented negative binomial distribution + * + * + * + * SYNOPSIS: + * + * int k, n; + * float p, y, nbdtrcf(); + * + * y = nbdtrcf( 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: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,100 5000 1.4e-4 2.0e-5 + * + */ + +/* ndtrf.c + * + * Normal distribution function + * + * + * + * SYNOPSIS: + * + * float x, y, ndtrf(); + * + * y = ndtrf( x ); + * + * + * + * DESCRIPTION: + * + * Returns the area under the Gaussian probability density + * function, integrated from minus infinity to x: + * + * x + * - + * 1 | | 2 + * ndtr(x) = --------- | exp( - t /2 ) dt + * sqrt(2pi) | | + * - + * -inf. + * + * = ( 1 + erf(z) ) / 2 + * = erfc(z) / 2 + * + * where z = x/sqrt(2). Computation is via the functions + * erf and erfc. + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -13,0 50000 1.5e-5 2.6e-6 + * + * + * ERROR MESSAGES: + * + * See erfcf(). + * + */ + /* erff.c + * + * Error function + * + * + * + * SYNOPSIS: + * + * float x, y, erff(); + * + * y = erff( x ); + * + * + * + * DESCRIPTION: + * + * The integral is + * + * x + * - + * 2 | | 2 + * erf(x) = -------- | exp( - t ) dt. + * sqrt(pi) | | + * - + * 0 + * + * The magnitude of x is limited to 9.231948545 for DEC + * arithmetic; 1 or -1 is returned outside this range. + * + * For 0 <= |x| < 1, erf(x) = x * P(x**2); otherwise + * erf(x) = 1 - erfc(x). + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -9.3,9.3 50000 1.7e-7 2.8e-8 + * + */ + /* erfcf.c + * + * Complementary error function + * + * + * + * SYNOPSIS: + * + * float x, y, erfcf(); + * + * y = erfcf( x ); + * + * + * + * DESCRIPTION: + * + * + * 1 - erf(x) = + * + * inf. + * - + * 2 | | 2 + * erfc(x) = -------- | exp( - t ) dt + * sqrt(pi) | | + * - + * x + * + * + * For small x, erfc(x) = 1 - erf(x); otherwise polynomial + * approximations 1/x P(1/x**2) are computed. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -9.3,9.3 50000 3.9e-6 7.2e-7 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * erfcf underflow x**2 > MAXLOGF 0.0 + * + * + */ + +/* ndtrif.c + * + * Inverse of Normal distribution function + * + * + * + * SYNOPSIS: + * + * float x, y, ndtrif(); + * + * x = ndtrif( 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( -2.0 * log(y) ); then the approximation is + * x = z - log(z)/z - (1/z) P(1/z) / Q(1/z). + * There are two rational functions P/Q, one for 0 < y < exp(-32) + * and the other for y up to exp(-2). For larger arguments, + * w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)). + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 1e-38, 1 30000 3.6e-7 5.0e-8 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * ndtrif domain x <= 0 -MAXNUM + * ndtrif domain x >= 1 MAXNUM + * + */ + +/* pdtrf.c + * + * Poisson distribution + * + * + * + * SYNOPSIS: + * + * int k; + * float m, y, pdtrf(); + * + * y = pdtrf( 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: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,100 5000 6.9e-5 8.0e-6 + * + */ + /* pdtrcf() + * + * Complemented poisson distribution + * + * + * + * SYNOPSIS: + * + * int k; + * float m, y, pdtrcf(); + * + * y = pdtrcf( 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: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,100 5000 8.4e-5 1.2e-5 + * + */ + /* pdtrif() + * + * Inverse Poisson distribution + * + * + * + * SYNOPSIS: + * + * int k; + * float m, y, pdtrf(); + * + * m = pdtrif( 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: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,100 5000 8.7e-6 1.4e-6 + * + * ERROR MESSAGES: + * + * message condition value returned + * pdtri domain y < 0 or y >= 1 0.0 + * k < 0 + * + */ + +/* polevlf.c + * p1evlf.c + * + * Evaluate polynomial + * + * + * + * SYNOPSIS: + * + * int N; + * float x, y, coef[N+1], polevlf[]; + * + * y = polevlf( 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 p1evl() assumes that coef[N] = 1.0 and is + * omitted from the array. Its calling arguments are + * otherwise the same as polevl(). + * + * + * 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. + * + */ + +/* polynf.c + * polyrf.c + * Arithmetic operations on polynomials + * + * In the following descriptions a, b, c are polynomials of degree + * na, nb, nc respectively. The degree of a polynomial cannot + * exceed a run-time value MAXPOLF. An operation that attempts + * to use or generate a polynomial of higher degree may produce a + * result that suffers truncation at degree MAXPOL. The value of + * MAXPOL is set by calling the function + * + * polinif( maxpol ); + * + * where maxpol is the desired maximum degree. This must be + * done prior to calling any of the other functions in this module. + * Memory for internal temporary polynomial storage is allocated + * by polinif(). + * + * Each polynomial is represented by an array containing its + * coefficients, together with a separately declared integer equal + * to the degree of the polynomial. The coefficients appear in + * ascending order; that is, + * + * 2 na + * a(x) = a[0] + a[1] * x + a[2] * x + ... + a[na] * x . + * + * + * + * sum = poleva( a, na, x ); Evaluate polynomial a(t) at t = x. + * polprtf( a, na, D ); Print the coefficients of a to D digits. + * polclrf( a, na ); Set a identically equal to zero, up to a[na]. + * polmovf( a, na, b ); Set b = a. + * poladdf( a, na, b, nb, c ); c = b + a, nc = max(na,nb) + * polsubf( a, na, b, nb, c ); c = b - a, nc = max(na,nb) + * polmulf( a, na, b, nb, c ); c = b * a, nc = na+nb + * + * + * Division: + * + * i = poldivf( a, na, b, nb, c ); c = b / a, nc = MAXPOL + * + * returns i = the degree of the first nonzero coefficient of a. + * The computed quotient c must be divided by x^i. An error message + * is printed if a is identically zero. + * + * + * Change of variables: + * If a and b are polynomials, and t = a(x), then + * c(t) = b(a(x)) + * is a polynomial found by substituting a(x) for t. The + * subroutine call for this is + * + * polsbtf( a, na, b, nb, c ); + * + * + * Notes: + * poldivf() is an integer routine; polevaf() is float. + * Any of the arguments a, b, c may refer to the same array. + * + */ + +/* powf.c + * + * Power function + * + * + * + * SYNOPSIS: + * + * float x, y, z, powf(); + * + * z = powf( 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/16 and pseudo extended precision arithmetic to + * obtain an extra three bits of accuracy in both the logarithm + * and the exponential. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -10,10 100,000 1.4e-7 3.6e-8 + * 1/10 < x < 10, x uniformly distributed. + * -10 < y < 10, y uniformly distributed. + * + * + * ERROR MESSAGES: + * + * message condition value returned + * powf overflow x**y > MAXNUMF MAXNUMF + * powf underflow x**y < 1/MAXNUMF 0.0 + * powf domain x<0 and y noninteger 0.0 + * + */ + +/* powif.c + * + * Real raised to integer power + * + * + * + * SYNOPSIS: + * + * float x, y, powif(); + * int n; + * + * y = powif( 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 .04,26 -26,26 100000 1.1e-6 2.0e-7 + * IEEE 1,2 -128,128 100000 1.1e-5 1.0e-6 + * + * Returns MAXNUMF on overflow, zero on underflow. + * + */ + +/* psif.c + * + * Psi (digamma) function + * + * + * SYNOPSIS: + * + * float x, y, psif(); + * + * y = psif( x ); + * + * + * DESCRIPTION: + * + * d - + * psi(x) = -- ln | (x) + * dx + * + * is the logarithmic derivative of the gamma function. + * For integer x, + * n-1 + * - + * psi(n) = -EUL + > 1/k. + * - + * k=1 + * + * This formula is used for 0 < n <= 10. If x is negative, it + * is transformed to a positive argument by the reflection + * formula psi(1-x) = psi(x) + pi cot(pi x). + * For general positive x, the argument is made greater than 10 + * using the recurrence psi(x+1) = psi(x) + 1/x. + * Then the following asymptotic expansion is applied: + * + * inf. B + * - 2k + * psi(x) = log(x) - 1/2x - > ------- + * - 2k + * k=1 2k x + * + * where the B2k are Bernoulli numbers. + * + * ACCURACY: + * Absolute error, relative when |psi| > 1 : + * arithmetic domain # trials peak rms + * IEEE -33,0 30000 8.2e-7 1.2e-7 + * IEEE 0,33 100000 7.3e-7 7.7e-8 + * + * ERROR MESSAGES: + * message condition value returned + * psi singularity x integer <=0 MAXNUMF + */ + +/* rgammaf.c + * + * Reciprocal gamma function + * + * + * + * SYNOPSIS: + * + * float x, y, rgammaf(); + * + * y = rgammaf( x ); + * + * + * + * DESCRIPTION: + * + * Returns one divided by the gamma function of the argument. + * + * The function is approximated by a Chebyshev expansion in + * the interval [0,1]. Range reduction is by recurrence + * for arguments between -34.034 and +34.84425627277176174. + * 1/MAXNUMF is returned for positive arguments outside this + * range. + * + * The reciprocal gamma function has no singularities, + * but overflow and underflow may occur for large arguments. + * These conditions return either MAXNUMF or 1/MAXNUMF with + * appropriate sign. + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -34,+34 100000 8.9e-7 1.1e-7 + */ + +/* shichif.c + * + * Hyperbolic sine and cosine integrals + * + * + * + * SYNOPSIS: + * + * float x, Chi, Shi; + * + * shichi( x, &Chi, &Shi ); + * + * + * DESCRIPTION: + * + * Approximates the integrals + * + * x + * - + * | | cosh t - 1 + * Chi(x) = eul + ln x + | ----------- dt, + * | | t + * - + * 0 + * + * x + * - + * | | sinh t + * Shi(x) = | ------ dt + * | | t + * - + * 0 + * + * where eul = 0.57721566490153286061 is Euler's constant. + * The integrals are evaluated by power series for x < 8 + * and by Chebyshev expansions for x between 8 and 88. + * For large x, both functions approach exp(x)/2x. + * Arguments greater than 88 in magnitude return MAXNUM. + * + * + * ACCURACY: + * + * Test interval 0 to 88. + * Relative error: + * arithmetic function # trials peak rms + * IEEE Shi 20000 3.5e-7 7.0e-8 + * Absolute error, except relative when |Chi| > 1: + * IEEE Chi 20000 3.8e-7 7.6e-8 + */ + +/* sicif.c + * + * Sine and cosine integrals + * + * + * + * SYNOPSIS: + * + * float x, Ci, Si; + * + * sicif( x, &Si, &Ci ); + * + * + * DESCRIPTION: + * + * Evaluates the integrals + * + * x + * - + * | cos t - 1 + * Ci(x) = eul + ln x + | --------- dt, + * | t + * - + * 0 + * x + * - + * | sin t + * Si(x) = | ----- dt + * | t + * - + * 0 + * + * where eul = 0.57721566490153286061 is Euler's constant. + * The integrals are approximated by rational functions. + * For x > 8 auxiliary functions f(x) and g(x) are employed + * such that + * + * Ci(x) = f(x) sin(x) - g(x) cos(x) + * Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x) + * + * + * ACCURACY: + * Test interval = [0,50]. + * Absolute error, except relative when > 1: + * arithmetic function # trials peak rms + * IEEE Si 30000 2.1e-7 4.3e-8 + * IEEE Ci 30000 3.9e-7 2.2e-8 + */ + +/* sindgf.c + * + * Circular sine of angle in degrees + * + * + * + * SYNOPSIS: + * + * float x, y, sindgf(); + * + * y = sindgf( x ); + * + * + * + * DESCRIPTION: + * + * Range reduction is into intervals of 45 degrees. + * + * Two polynomial approximating functions are employed. + * Between 0 and pi/4 the sine is approximated by + * x + x**3 P(x**2). + * Between pi/4 and pi/2 the cosine is represented as + * 1 - x**2 Q(x**2). + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE +-3600 100,000 1.2e-7 3.0e-8 + * + * ERROR MESSAGES: + * + * message condition value returned + * sin total loss x > 2^24 0.0 + * + */ + +/* cosdgf.c + * + * Circular cosine of angle in degrees + * + * + * + * SYNOPSIS: + * + * float x, y, cosdgf(); + * + * y = cosdgf( x ); + * + * + * + * DESCRIPTION: + * + * Range reduction is into intervals of 45 degrees. + * + * Two polynomial approximating functions are employed. + * Between 0 and pi/4 the cosine is approximated by + * 1 - x**2 Q(x**2). + * Between pi/4 and pi/2 the sine is represented as + * x + x**3 P(x**2). + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -8192,+8192 100,000 3.0e-7 3.0e-8 + */ + +/* sinf.c + * + * Circular sine + * + * + * + * SYNOPSIS: + * + * float x, y, sinf(); + * + * y = sinf( 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 + * x + x**3 P(x**2). + * Between pi/4 and pi/2 the cosine is represented as + * 1 - x**2 Q(x**2). + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -4096,+4096 100,000 1.2e-7 3.0e-8 + * IEEE -8192,+8192 100,000 3.0e-7 3.0e-8 + * + * ERROR MESSAGES: + * + * message condition value returned + * sin total loss x > 2^24 0.0 + * + * Partial loss of accuracy begins to occur at x = 2^13 + * = 8192. Results may be meaningless for x >= 2^24 + * The routine as implemented flags a TLOSS error + * for x >= 2^24 and returns 0.0. + */ + +/* cosf.c + * + * Circular cosine + * + * + * + * SYNOPSIS: + * + * float x, y, cosf(); + * + * y = cosf( 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 - x**2 Q(x**2). + * Between pi/4 and pi/2 the sine is represented as + * x + x**3 P(x**2). + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -8192,+8192 100,000 3.0e-7 3.0e-8 + */ + +/* sinhf.c + * + * Hyperbolic sine + * + * + * + * SYNOPSIS: + * + * float x, y, sinhf(); + * + * y = sinhf( x ); + * + * + * + * DESCRIPTION: + * + * Returns hyperbolic sine of argument in the range MINLOGF to + * MAXLOGF. + * + * The range is partitioned into two segments. If |x| <= 1, a + * polynomial approximation is used. + * Otherwise the calculation is sinh(x) = ( exp(x) - exp(-x) )/2. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE +-MAXLOG 100000 1.1e-7 2.9e-8 + * + */ + +/* spencef.c + * + * Dilogarithm + * + * + * + * SYNOPSIS: + * + * float x, y, spencef(); + * + * y = spencef( x ); + * + * + * + * DESCRIPTION: + * + * Computes the integral + * + * x + * - + * | | log t + * spence(x) = - | ----- dt + * | | t - 1 + * - + * 1 + * + * for x >= 0. A rational approximation gives the integral in + * the interval (0.5, 1.5). Transformation formulas for 1/x + * and 1-x are employed outside the basic expansion range. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,4 30000 4.4e-7 6.3e-8 + * + * + */ + +/* sqrtf.c + * + * Square root + * + * + * + * SYNOPSIS: + * + * float x, y, sqrtf(); + * + * y = sqrtf( 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. + * + * + * + * ACCURACY: + * + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,1.e38 100000 8.7e-8 2.9e-8 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * sqrtf domain x < 0 0.0 + * + */ + +/* stdtrf.c + * + * Student's t distribution + * + * + * + * SYNOPSIS: + * + * float t, stdtrf(); + * short k; + * + * y = stdtrf( 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, 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: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE +/- 100 5000 2.3e-5 2.9e-6 + */ + +/* struvef.c + * + * Struve function + * + * + * + * SYNOPSIS: + * + * float v, x, y, struvef(); + * + * y = struvef( v, x ); + * + * + * + * DESCRIPTION: + * + * Computes the Struve function Hv(x) of order v, argument x. + * Negative x is rejected unless v is an integer. + * + * This module also contains the hypergeometric functions 1F2 + * and 3F0 and a routine for the Bessel function Yv(x) with + * noninteger v. + * + * + * + * ACCURACY: + * + * v varies from 0 to 10. + * Absolute error (relative error when |Hv(x)| > 1): + * arithmetic domain # trials peak rms + * IEEE -10,10 100000 9.0e-5 4.0e-6 + * + */ + +/* tandgf.c + * + * Circular tangent of angle in degrees + * + * + * + * SYNOPSIS: + * + * float x, y, tandgf(); + * + * y = tandgf( x ); + * + * + * + * DESCRIPTION: + * + * Returns the circular tangent of the radian argument x. + * + * Range reduction is into intervals of 45 degrees. + * + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE +-2^24 50000 2.4e-7 4.8e-8 + * + * ERROR MESSAGES: + * + * message condition value returned + * tanf total loss x > 2^24 0.0 + * + */ + /* cotdgf.c + * + * Circular cotangent of angle in degrees + * + * + * + * SYNOPSIS: + * + * float x, y, cotdgf(); + * + * y = cotdgf( x ); + * + * + * + * DESCRIPTION: + * + * Range reduction is into intervals of 45 degrees. + * A common routine computes either the tangent or cotangent. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE +-2^24 50000 2.4e-7 4.8e-8 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * cot total loss x > 2^24 0.0 + * cot singularity x = 0 MAXNUMF + * + */ + +/* tanf.c + * + * Circular tangent + * + * + * + * SYNOPSIS: + * + * float x, y, tanf(); + * + * y = tanf( x ); + * + * + * + * DESCRIPTION: + * + * Returns the circular tangent of the radian argument x. + * + * Range reduction is modulo pi/4. A polynomial approximation + * is employed in the basic interval [0, pi/4]. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE +-4096 100000 3.3e-7 4.5e-8 + * + * ERROR MESSAGES: + * + * message condition value returned + * tanf total loss x > 2^24 0.0 + * + */ + /* cotf.c + * + * Circular cotangent + * + * + * + * SYNOPSIS: + * + * float x, y, cotf(); + * + * y = cotf( x ); + * + * + * + * DESCRIPTION: + * + * Returns the circular cotangent of the radian argument x. + * A common routine computes either the tangent or cotangent. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE +-4096 100000 3.0e-7 4.5e-8 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * cot total loss x > 2^24 0.0 + * cot singularity x = 0 MAXNUMF + * + */ + +/* tanhf.c + * + * Hyperbolic tangent + * + * + * + * SYNOPSIS: + * + * float x, y, tanhf(); + * + * y = tanhf( x ); + * + * + * + * DESCRIPTION: + * + * Returns hyperbolic tangent of argument in the range MINLOG to + * MAXLOG. + * + * A polynomial approximation is used for |x| < 0.625. + * Otherwise, + * + * tanh(x) = sinh(x)/cosh(x) = 1 - 2/(exp(2x) + 1). + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE -2,2 100000 1.3e-7 2.6e-8 + * + */ + +/* ynf.c + * + * Bessel function of second kind of integer order + * + * + * + * SYNOPSIS: + * + * float x, y, ynf(); + * int n; + * + * y = ynf( 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 + * y0() and y1(). + * + * If n = 0 or 1 the routine for y0 or y1 is called + * directly. + * + * + * + * ACCURACY: + * + * + * Absolute error, except relative when y > 1: + * + * arithmetic domain # trials peak rms + * IEEE 0, 30 10000 2.3e-6 3.4e-7 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * yn singularity x = 0 MAXNUMF + * yn overflow MAXNUMF + * + * Spot checked against tables for x, n between 0 and 100. + * + */ + + /* zetacf.c + * + * Riemann zeta function + * + * + * + * SYNOPSIS: + * + * float x, y, zetacf(); + * + * y = zetacf( x ); + * + * + * + * DESCRIPTION: + * + * + * + * inf. + * - -x + * zetac(x) = > k , x > 1, + * - + * k=2 + * + * is related to the Riemann zeta function by + * + * Riemann zeta(x) = zetac(x) + 1. + * + * Extension of the function definition for x < 1 is implemented. + * Zero is returned for x > log2(MAXNUM). + * + * An overflow error may occur for large negative x, due to the + * gamma function in the reflection formula. + * + * ACCURACY: + * + * Tabulated values have full machine accuracy. + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 1,50 30000 5.5e-7 7.5e-8 + * + * + */ + +/* zetaf.c + * + * Riemann zeta function of two arguments + * + * + * + * SYNOPSIS: + * + * float x, q, y, zetaf(); + * + * y = zetaf( x, q ); + * + * + * + * DESCRIPTION: + * + * + * + * inf. + * - -x + * zeta(x,q) = > (k+q) + * - + * k=0 + * + * where x > 1 and q is not a negative integer or zero. + * The Euler-Maclaurin summation formula is used to obtain + * the expansion + * + * n + * - -x + * zeta(x,q) = > (k+q) + * - + * k=1 + * + * 1-x inf. B x(x+1)...(x+2j) + * (n+q) 1 - 2j + * + --------- - ------- + > -------------------- + * x-1 x - x+2j+1 + * 2(n+q) j=1 (2j)! (n+q) + * + * where the B2j are Bernoulli numbers. Note that (see zetac.c) + * zeta(x,1) = zetac(x) + 1. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * IEEE 0,25 10000 6.9e-7 1.0e-7 + * + * Large arguments may produce underflow in powf(), in which + * case the results are inaccurate. + * + * REFERENCE: + * + * Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals, + * Series, and Products, p. 1073; Academic Press, 1980. + * + */ |