/* acosh.c * * Inverse hyperbolic cosine * * * * SYNOPSIS: * * double x, y, acosh(); * * y = acosh( x ); * * * * DESCRIPTION: * * Returns inverse hyperbolic cosine of argument. * * If 1 <= x < 1.5, a rational approximation * * sqrt(z) * 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 * DEC 1,3 30000 4.2e-17 1.1e-17 * IEEE 1,3 30000 4.6e-16 8.7e-17 * * * ERROR MESSAGES: * * message condition value returned * acosh domain |x| < 1 NAN * */
/* airy.c * * Airy function * * * * SYNOPSIS: * * double x, ai, aip, bi, bip; * int airy(); * * airy( 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 10000 1.6e-15 2.7e-16 * IEEE 0, 10 Ai 10000 2.3e-14* 1.8e-15* * IEEE -10, 0 Ai' 10000 4.6e-15 7.6e-16 * IEEE 0, 10 Ai' 10000 1.8e-14* 1.5e-15* * IEEE -10, 10 Bi 30000 4.2e-15 5.3e-16 * IEEE -10, 10 Bi' 30000 4.9e-15 7.3e-16 * DEC -10, 0 Ai 5000 1.7e-16 2.8e-17 * DEC 0, 10 Ai 5000 2.1e-15* 1.7e-16* * DEC -10, 0 Ai' 5000 4.7e-16 7.8e-17 * DEC 0, 10 Ai' 12000 1.8e-15* 1.5e-16* * DEC -10, 10 Bi 10000 5.5e-16 6.8e-17 * DEC -10, 10 Bi' 7000 5.3e-16 8.7e-17 * */
/* asin.c * * Inverse circular sine * * * * SYNOPSIS: * * double x, y, asin(); * * y = asin( 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 * DEC -1, 1 40000 2.6e-17 7.1e-18 * IEEE -1, 1 10^6 1.9e-16 5.4e-17 * * * ERROR MESSAGES: * * message condition value returned * asin domain |x| > 1 NAN * */
/* acos() * * Inverse circular cosine * * * * SYNOPSIS: * * double x, y, acos(); * * y = acos( x ); * * * * DESCRIPTION: * * Returns radian angle between 0 and pi 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 * DEC -1, 1 50000 3.3e-17 8.2e-18 * IEEE -1, 1 10^6 2.2e-16 6.5e-17 * * * ERROR MESSAGES: * * message condition value returned * asin domain |x| > 1 NAN */
/* asinh.c * * Inverse hyperbolic sine * * * * SYNOPSIS: * * double x, y, asinh(); * * y = asinh( 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 * DEC -3,3 75000 4.6e-17 1.1e-17 * IEEE -1,1 30000 3.7e-16 7.8e-17 * IEEE 1,3 30000 2.5e-16 6.7e-17 * */
/* atan.c * * Inverse circular tangent * (arctangent) * * * * SYNOPSIS: * * double x, y, atan(); * * y = atan( x ); * * * * DESCRIPTION: * * Returns radian angle between -pi/2 and +pi/2 whose tangent * is x. * * Range reduction is from three intervals into the interval * from zero to 0.66. The approximant uses a rational * function of degree 4/5 of the form x + x**3 P(x)/Q(x). * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC -10, 10 50000 2.4e-17 8.3e-18 * IEEE -10, 10 10^6 1.8e-16 5.0e-17 * */
/* atan2() * * Quadrant correct inverse circular tangent * * * * SYNOPSIS: * * double x, y, z, atan2(); * * z = atan2( 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 10^6 2.5e-16 6.9e-17 * See atan.c. * */
/* atanh.c * * Inverse hyperbolic tangent * * * * SYNOPSIS: * * double x, y, atanh(); * * y = atanh( x ); * * * * DESCRIPTION: * * Returns inverse hyperbolic tangent of argument in the range * MINLOG to MAXLOG. * * 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 * DEC -1,1 50000 2.4e-17 6.4e-18 * IEEE -1,1 30000 1.9e-16 5.2e-17 * */
/* bdtr.c * * Binomial distribution * * * * SYNOPSIS: * * int k, n; * double p, y, bdtr(); * * y = bdtr( 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 (a,b,p), with p between 0 and 1. * * a,b Relative error: * arithmetic domain # trials peak rms * For p between 0.001 and 1: * IEEE 0,100 100000 4.3e-15 2.6e-16 * See also incbet.c. * * ERROR MESSAGES: * * message condition value returned * bdtr domain k < 0 0.0 * n < k * x < 0, x > 1 */
/* bdtrc() * * Complemented binomial distribution * * * * SYNOPSIS: * * int k, n; * double p, y, bdtrc(); * * y = bdtrc( 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: * * Tested at random points (a,b,p). * * a,b Relative error: * arithmetic domain # trials peak rms * For p between 0.001 and 1: * IEEE 0,100 100000 6.7e-15 8.2e-16 * For p between 0 and .001: * IEEE 0,100 100000 1.5e-13 2.7e-15 * * ERROR MESSAGES: * * message condition value returned * bdtrc domain x<0, x>1, n<k 0.0 */
/* bdtri() * * Inverse binomial distribution * * * * SYNOPSIS: * * int k, n; * double p, y, bdtri(); * * p = bdtr( 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: * * Tested at random points (a,b,p). * * a,b Relative error: * arithmetic domain # trials peak rms * For p between 0.001 and 1: * IEEE 0,100 100000 2.3e-14 6.4e-16 * IEEE 0,10000 100000 6.6e-12 1.2e-13 * For p between 10^-6 and 0.001: * IEEE 0,100 100000 2.0e-12 1.3e-14 * IEEE 0,10000 100000 1.5e-12 3.2e-14 * See also incbi.c. * * ERROR MESSAGES: * * message condition value returned * bdtri domain k < 0, n <= k 0.0 * x < 0, x > 1 */
/* beta.c * * Beta function * * * * SYNOPSIS: * * double a, b, y, beta(); * * y = beta( 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 * DEC 0,30 1700 7.7e-15 1.5e-15 * IEEE 0,30 30000 8.1e-14 1.1e-14 * * ERROR MESSAGES: * * message condition value returned * beta overflow log(beta) > MAXLOG 0.0 * a or b <0 integer 0.0 * */
/* btdtr.c * * Beta distribution * * * * SYNOPSIS: * * double a, b, x, y, btdtr(); * * y = btdtr( 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 * * * This function is identical to the incomplete beta * integral function incbet(a, b, x). * * The complemented function is * * 1 - P(1-x) = incbet( b, a, x ); * * * ACCURACY: * * See incbet.c. * */
/* cbrt.c * * Cube root * * * * SYNOPSIS: * * double x, y, cbrt(); * * y = cbrt( 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 * DEC -10,10 200000 1.8e-17 6.2e-18 * IEEE 0,1e308 30000 1.5e-16 5.0e-17 * */
/* chbevl.c * * Evaluate Chebyshev series * * * * SYNOPSIS: * * int N; * double x, y, coef[N], chebevl(); * * y = chbevl( 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. * */
/* chdtr.c * * Chi-square distribution * * * * SYNOPSIS: * * double df, x, y, chdtr(); * * y = chdtr( 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 */
/* chdtrc() * * Complemented Chi-square distribution * * * * SYNOPSIS: * * double v, x, y, chdtrc(); * * y = chdtrc( 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 */
/* chdtri() * * Inverse of complemented Chi-square distribution * * * * SYNOPSIS: * * double df, x, y, chdtri(); * * x = chdtri( 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 * */
/* cheby.c * * Program to calculate coefficients of the Chebyshev polynomial * expansion of a given input function. The algorithm computes * the discrete Fourier cosine transform of the function evaluated * at unevenly spaced points. Library routine chbevl.c uses the * coefficients to calculate an approximate value of the original * function. */
/* clog.c * * Complex natural logarithm * * * * SYNOPSIS: * * void clog(); * cmplx z, w; * * clog( &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. */
/* cexp() * * Complex exponential function * * * * SYNOPSIS: * * void cexp(); * cmplx z, w; * * cexp( &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 * */
/* csin() * * Complex circular sine * * * * SYNOPSIS: * * void csin(); * cmplx z, w; * * csin( &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. * */
/* ccos() * * Complex circular cosine * * * * SYNOPSIS: * * void ccos(); * cmplx z, w; * * ccos( &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 */
/* ctan() * * Complex circular tangent * * * * SYNOPSIS: * * void ctan(); * cmplx z, w; * * ctan( &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. */
/* ccot() * * Complex circular cotangent * * * * SYNOPSIS: * * void ccot(); * cmplx z, w; * * ccot( &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. */
/* casin() * * Complex circular arc sine * * * * SYNOPSIS: * * void casin(); * cmplx z, w; * * casin( &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. */
/* cacos() * * Complex circular arc cosine * * * * SYNOPSIS: * * void cacos(); * cmplx z, w; * * cacos( &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 */
/* catan() * * Complex circular arc tangent * * * * SYNOPSIS: * * void catan(); * cmplx 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 * 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(). */
/* csinh * * Complex hyperbolic sine * * * * SYNOPSIS: * * void csinh(); * cmplx z, w; * * csinh( &z, &w ); * * * DESCRIPTION: * * csinh z = (cexp(z) - cexp(-z))/2 * = sinh x * cos y + i cosh x * sin y . * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -10,+10 30000 3.1e-16 8.2e-17 * */
/* casinh * * Complex inverse hyperbolic sine * * * * SYNOPSIS: * * void casinh(); * cmplx z, w; * * casinh (&z, &w); * * * * DESCRIPTION: * * casinh z = -i casin iz . * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -10,+10 30000 1.8e-14 2.6e-15 * */
/* ccosh * * Complex hyperbolic cosine * * * * SYNOPSIS: * * void ccosh(); * cmplx z, w; * * ccosh (&z, &w); * * * * DESCRIPTION: * * ccosh(z) = cosh x cos y + i sinh x sin y . * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -10,+10 30000 2.9e-16 8.1e-17 * */
/* cacosh * * Complex inverse hyperbolic cosine * * * * SYNOPSIS: * * void cacosh(); * cmplx z, w; * * cacosh (&z, &w); * * * * DESCRIPTION: * * acosh z = i acos z . * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -10,+10 30000 1.6e-14 2.1e-15 * */
/* ctanh * * Complex hyperbolic tangent * * * * SYNOPSIS: * * void ctanh(); * cmplx z, w; * * ctanh (&z, &w); * * * * DESCRIPTION: * * tanh z = (sinh 2x + i sin 2y) / (cosh 2x + cos 2y) . * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -10,+10 30000 1.7e-14 2.4e-16 * */
/* catanh * * Complex inverse hyperbolic tangent * * * * SYNOPSIS: * * void catanh(); * cmplx z, w; * * catanh (&z, &w); * * * * DESCRIPTION: * * Inverse tanh, equal to -i catan (iz); * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -10,+10 30000 2.3e-16 6.2e-17 * */
/* cpow * * Complex power function * * * * SYNOPSIS: * * void cpow(); * cmplx a, z, w; * * cpow (&a, &z, &w); * * * * DESCRIPTION: * * Raises complex A to the complex Zth power. * Definition is per AMS55 # 4.2.8, * analytically equivalent to cpow(a,z) = cexp(z clog(a)). * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -10,+10 30000 9.4e-15 1.5e-15 * */
/* cmplx.c
*
* Complex number arithmetic
*
*
*
* SYNOPSIS:
*
* typedef struct {
* double r; real part
* double i; imaginary part
* }cmplx;
*
* cmplx *a, *b, *c;
*
* cadd( a, b, c ); c = b + a
* csub( a, b, c ); c = b - a
* cmul( a, b, c ); c = b * a
* cdiv( a, b, c ); c = b / a
* cneg( c ); c = -c
* cmov( 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
*/
/* cabs() * * Complex absolute value * * * * SYNOPSIS: * * double cabs(); * cmplx z; * 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 */
/* csqrt() * * Complex square root * * * * SYNOPSIS: * * void csqrt(); * cmplx z, w; * * csqrt( &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. */
/* const.c * * Globally declared constants * * * * SYNOPSIS: * * extern double nameofconstant; * * * * * DESCRIPTION: * * This file contains a number of mathematical constants and * also some needed size parameters of the computer arithmetic. * The values are supplied as arrays of hexadecimal integers * for IEEE arithmetic; arrays of octal constants for DEC * arithmetic; and in a normal decimal scientific notation for * other machines. The particular notation used is determined * by a symbol (DEC, IBMPC, or UNK) defined in the include file * mconf.h. * * The default size parameters are as follows. * * For DEC and UNK modes: * MACHEP = 1.38777878078144567553E-17 2**-56 * MAXLOG = 8.8029691931113054295988E1 log(2**127) * MINLOG = -8.872283911167299960540E1 log(2**-128) * MAXNUM = 1.701411834604692317316873e38 2**127 * * For IEEE arithmetic (IBMPC): * MACHEP = 1.11022302462515654042E-16 2**-53 * MAXLOG = 7.09782712893383996843E2 log(2**1024) * MINLOG = -7.08396418532264106224E2 log(2**-1022) * MAXNUM = 1.7976931348623158E308 2**1024 * * The global symbols for mathematical constants are * PI = 3.14159265358979323846 pi * PIO2 = 1.57079632679489661923 pi/2 * PIO4 = 7.85398163397448309616E-1 pi/4 * SQRT2 = 1.41421356237309504880 sqrt(2) * SQRTH = 7.07106781186547524401E-1 sqrt(2)/2 * LOG2E = 1.4426950408889634073599 1/log(2) * SQ2OPI = 7.9788456080286535587989E-1 sqrt( 2/pi ) * LOGE2 = 6.93147180559945309417E-1 log(2) * LOGSQ2 = 3.46573590279972654709E-1 log(2)/2 * THPIO4 = 2.35619449019234492885 3*pi/4 * TWOOPI = 6.36619772367581343075535E-1 2/pi * * These lists are subject to change. */
/* cosh.c * * Hyperbolic cosine * * * * SYNOPSIS: * * double x, y, cosh(); * * y = cosh( x ); * * * * DESCRIPTION: * * Returns hyperbolic cosine of argument in the range MINLOG to * MAXLOG. * * cosh(x) = ( exp(x) + exp(-x) )/2. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC +- 88 50000 4.0e-17 7.7e-18 * IEEE +-MAXLOG 30000 2.6e-16 5.7e-17 * * * ERROR MESSAGES: * * message condition value returned * cosh overflow |x| > MAXLOG MAXNUM * * */
/* dawsn.c * * Dawson's Integral * * * * SYNOPSIS: * * double x, y, dawsn(); * * y = dawsn( 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 10000 6.9e-16 1.0e-16 * DEC 0,10 6000 7.4e-17 1.4e-17 * * */
/* drand.c * * Pseudorandom number generator * * * * SYNOPSIS: * * double y, drand(); * * drand( &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. The period, given by them, is * 6953607871644. * * Versions invoked by the different arithmetic compile * time options DEC, IBMPC, and MIEEE, produce * approximately the same sequences, differing only in the * least significant bits of the numbers. The UNK option * implements the algorithm as recommended in the BYTE * article. It may be used on all computers. However, * the low order bits of a double precision number may * not be adequately random, and may vary due to arithmetic * implementation details on different computers. * * The other compile options generate an additional random * integer that overwrites the low order bits of the double * precision number. This reduces the period by a factor of * two but tends to overcome the problems mentioned. * */
/* ei.c * * Exponential integral * * * SYNOPSIS: * * double x, y, ei(); * * y = ei( x ); * * * * DESCRIPTION: * * x * - t * | | e * Ei(x) = -|- --- dt . * | | t * - * -inf * * Not defined for x <= 0. * See also expn.c. * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE 0,100 50000 8.6e-16 1.3e-16 * */
/* eigens.c * * Eigenvalues and eigenvectors of a real symmetric matrix * * * * SYNOPSIS: * * int n; * double A[n*(n+1)/2], EV[n*n], E[n]; * void eigens( A, EV, E, n ); * * * * DESCRIPTION: * * The algorithm is due to J. vonNeumann. * * A[] is a symmetric matrix stored in lower triangular form. * That is, A[ row, column ] = A[ (row*row+row)/2 + column ] * or equivalently with row and column interchanged. The * indices row and column run from 0 through n-1. * * EV[] is the output matrix of eigenvectors stored columnwise. * That is, the elements of each eigenvector appear in sequential * memory order. The jth element of the ith eigenvector is * EV[ n*i+j ] = EV[i][j]. * * E[] is the output matrix of eigenvalues. The ith element * of E corresponds to the ith eigenvector (the ith row of EV). * * On output, the matrix A will have been diagonalized and its * orginal contents are destroyed. * * ACCURACY: * * The error is controlled by an internal parameter called RANGE * which is set to 1e-10. After diagonalization, the * off-diagonal elements of A will have been reduced by * this factor. * * ERROR MESSAGES: * * None. * */
/* ellie.c * * Incomplete elliptic integral of the second kind * * * * SYNOPSIS: * * double phi, m, y, ellie(); * * y = ellie( 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 * DEC 0,2 2000 1.9e-16 3.4e-17 * IEEE -10,10 150000 3.3e-15 1.4e-16 * * */
/* ellik.c * * Incomplete elliptic integral of the first kind * * * * SYNOPSIS: * * double phi, m, y, ellik(); * * y = ellik( 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 200000 7.4e-16 1.0e-16 * * */
/* ellpe.c * * Complete elliptic integral of the second kind * * * * SYNOPSIS: * * double m1, y, ellpe(); * * y = ellpe( 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 * DEC 0, 1 13000 3.1e-17 9.4e-18 * IEEE 0, 1 10000 2.1e-16 7.3e-17 * * * ERROR MESSAGES: * * message condition value returned * ellpe domain x<0, x>1 0.0 * */
/* ellpj.c * * Jacobian Elliptic Functions * * * * SYNOPSIS: * * double 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 * DEC sn 1800 4.5e-16 8.7e-17 * IEEE phi 10000 9.2e-16* 1.4e-16* * IEEE sn 50000 4.1e-15 4.6e-16 * IEEE cn 40000 3.6e-15 4.4e-16 * IEEE dn 100000 3.9e-15 1.7e-16 * * Larger errors occur for m near 1. * 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. * */
/* ellpk.c * * Complete elliptic integral of the first kind * * * * SYNOPSIS: * * double m1, y, ellpk(); * * y = ellpk( 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 * DEC 0,1 16000 3.5e-17 1.1e-17 * IEEE 0,1 30000 2.5e-16 6.8e-17 * * ERROR MESSAGES: * * message condition value returned * ellpk domain x<0, x>1 0.0 * */
/* euclid.c
*
* Rational arithmetic routines
*
*
*
* SYNOPSIS:
*
*
* typedef struct
* {
* double n; numerator
* double d; denominator
* }fract;
*
* radd( a, b, c ) c = b + a
* rsub( a, b, c ) c = b - a
* rmul( a, b, c ) c = b * a
* rdiv( a, b, c ) c = b / a
* euclid( &n, &d ) Reduce n/d to lowest terms,
* return greatest common divisor.
*
* Arguments of the routines are pointers to the structures.
* The double precision numbers are assumed, without checking,
* to be integer valued. Overflow conditions are reported.
*/
/* exp.c * * Exponential function * * * * SYNOPSIS: * * double x, y, exp(); * * y = exp( 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 1 + 2x P(x**2)/( Q(x**2) - P(x**2) ) * of degree 2/3 is used to approximate exp(f) in the basic * interval [-0.5, 0.5]. * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC +- 88 50000 2.8e-17 7.0e-18 * IEEE +- 708 40000 2.0e-16 5.6e-17 * * * 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 * exp underflow x < MINLOG 0.0 * exp overflow x > MAXLOG INFINITY * */
/* exp10.c * * Base 10 exponential function * (Common antilogarithm) * * * * SYNOPSIS: * * double x, y, exp10(); * * y = exp10( 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 -307,+307 30000 2.2e-16 5.5e-17 * Test result from an earlier version (2.1): * DEC -38,+38 70000 3.1e-17 7.0e-18 * * ERROR MESSAGES: * * message condition value returned * exp10 underflow x < -MAXL10 0.0 * exp10 overflow x > MAXL10 MAXNUM * * DEC arithmetic: MAXL10 = 38.230809449325611792. * IEEE arithmetic: MAXL10 = 308.2547155599167. * */
/* exp2.c * * Base 2 exponential function * * * * SYNOPSIS: * * double x, y, exp2(); * * y = exp2( 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 -1022,+1024 30000 1.8e-16 5.4e-17 * * * See exp.c for comments on error amplification. * * * ERROR MESSAGES: * * message condition value returned * exp underflow x < -MAXL2 0.0 * exp overflow x > MAXL2 MAXNUM * * For DEC arithmetic, MAXL2 = 127. * For IEEE arithmetic, MAXL2 = 1024. */
/* expn.c * * Exponential integral En * * * * SYNOPSIS: * * int n; * double x, y, expn(); * * y = expn( 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 * DEC 0, 30 5000 2.0e-16 4.6e-17 * IEEE 0, 30 10000 1.7e-15 3.6e-16 * */
/* expx2.c * * Exponential of squared argument * * * * SYNOPSIS: * * double x, y, expx2(); * int sign; * * y = expx2( x, sign ); * * * * DESCRIPTION: * * Computes y = exp(x*x) while suppressing error amplification * that would ordinarily arise from the inexactness of the * exponential argument x*x. * * If sign < 0, the result is inverted; i.e., y = exp(-x*x) . * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -26.6, 26.6 10^7 3.9e-16 8.9e-17 * */
/* fabs.c * * Absolute value * * * * SYNOPSIS: * * double x, y; * * y = fabs( x ); * * * * DESCRIPTION: * * Returns the absolute value of the argument. * */
/* fac.c * * Factorial function * * * * SYNOPSIS: * * double y, fac(); * int i; * * y = fac( 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 DEC arithmetic or 170 in IEEE * 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. If i > 55, fac(i) = gamma(i+1); * see gamma.c. * * Relative error: * arithmetic domain peak * IEEE 0, 170 1.4e-15 * DEC 0, 33 1.4e-17 * */
/* fdtr.c * * F distribution * * * * SYNOPSIS: * * int df1, df2; * double x, y, fdtr(); * * y = fdtr( 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 is * nonnegative. * * ACCURACY: * * Tested at random points (a,b,x). * * x a,b Relative error: * arithmetic domain domain # trials peak rms * IEEE 0,1 0,100 100000 9.8e-15 1.7e-15 * IEEE 1,5 0,100 100000 6.5e-15 3.5e-16 * IEEE 0,1 1,10000 100000 2.2e-11 3.3e-12 * IEEE 1,5 1,10000 100000 1.1e-11 1.7e-13 * See also incbet.c. * * * ERROR MESSAGES: * * message condition value returned * fdtr domain a<0, b<0, x<0 0.0 * */
/* fdtrc() * * Complemented F distribution * * * * SYNOPSIS: * * int df1, df2; * double x, y, fdtrc(); * * y = fdtrc( 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 * * * The incomplete beta integral is used, according to the * formula * * P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ). * * * 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 100000 3.7e-14 5.9e-16 * IEEE 1,5 1,100 100000 8.0e-15 1.6e-15 * IEEE 0,1 1,10000 100000 1.8e-11 3.5e-13 * IEEE 1,5 1,10000 100000 2.0e-11 3.0e-12 * See also incbet.c. * * ERROR MESSAGES: * * message condition value returned * fdtrc domain a<0, b<0, x<0 0.0 * */
/* fdtri() * * Inverse of complemented F distribution * * * * SYNOPSIS: * * int df1, df2; * double x, p, fdtri(); * * x = fdtri( 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: * * 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 100000 8.3e-15 4.7e-16 * IEEE 1,10000 100000 2.1e-11 1.4e-13 * For p between 10^-6 and 10^-3: * IEEE 1,100 50000 1.3e-12 8.4e-15 * IEEE 1,10000 50000 3.0e-12 4.8e-14 * See also fdtrc.c. * * ERROR MESSAGES: * * message condition value returned * fdtri domain p <= 0 or p > 1 0.0 * v < 1 * */
/* fftr.c * * FFT of Real Valued Sequence * * * * SYNOPSIS: * * double x[], sine[]; * int m; * * fftr( x, m, sine ); * * * * DESCRIPTION: * * Computes the (complex valued) discrete Fourier transform of * the real valued sequence x[]. The input sequence x[] contains * n = 2**m samples. The program fills array sine[k] with * n/4 + 1 values of sin( 2 PI k / n ). * * Data format for complex valued output is real part followed * by imaginary part. The output is developed in the input * array x[]. * * The algorithm takes advantage of the fact that the FFT of an * n point real sequence can be obtained from an n/2 point * complex FFT. * * A radix 2 FFT algorithm is used. * * Execution time on an LSI-11/23 with floating point chip * is 1.0 sec for n = 256. * * * * REFERENCE: * * E. Oran Brigham, The Fast Fourier Transform; * Prentice-Hall, Inc., 1974 * */
/* ceil() * floor() * frexp() * ldexp() * * Floating point numeric utilities * * * * SYNOPSIS: * * double ceil(), floor(), frexp(), ldexp(); * double x, y; * int expnt, n; * * y = floor(x); * y = ceil(x); * y = frexp( x, &expnt ); * y = ldexp( x, n ); * * * * DESCRIPTION: * * All four routines return a double precision floating point * result. * * floor() returns the largest integer less than or equal to x. * It truncates toward minus infinity. * * ceil() returns the smallest integer greater than or equal * to x. It truncates toward plus infinity. * * frexp() 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. * * ldexp() 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. */
/* fresnl.c * * Fresnel integral * * * * SYNOPSIS: * * double x, S, C; * void fresnl(); * * fresnl( x, &S, &C ); * * * DESCRIPTION: * * Evaluates the Fresnel integrals * * x * - * | | * C(x) = | cos(pi/2 t**2) dt, * | | * - * 0 * * x * - * | | * S(x) = | sin(pi/2 t**2) dt. * | | * - * 0 * * * The integrals are evaluated by a power series for x < 1. * For 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 10000 2.0e-15 3.2e-16 * IEEE C(x) 0, 10 10000 1.8e-15 3.3e-16 * DEC S(x) 0, 10 6000 2.2e-16 3.9e-17 * DEC C(x) 0, 10 5000 2.3e-16 3.9e-17 */
/* gamma.c * * Gamma function * * * * SYNOPSIS: * * double x, y, gamma(); * extern int sgngam; * * y = gamma( 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| <= 34 are reduced by recurrence and the function * approximated by a rational function of degree 6/7 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 * DEC -34, 34 10000 1.3e-16 2.5e-17 * IEEE -170,-33 20000 2.3e-15 3.3e-16 * IEEE -33, 33 20000 9.4e-16 2.2e-16 * IEEE 33, 171.6 20000 2.3e-15 3.2e-16 * * Error for arguments outside the test range will be larger * owing to error amplification by the exponential function. * */
/* lgam() * * Natural logarithm of gamma function * * * * SYNOPSIS: * * double x, y, lgam(); * extern int sgngam; * * y = lgam( 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 13, 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 MAXLGM return MAXNUM and an error * message. MAXLGM = 2.035093e36 for DEC * arithmetic or 2.556348e305 for IEEE arithmetic. * * * * ACCURACY: * * * arithmetic domain # trials peak rms * DEC 0, 3 7000 5.2e-17 1.3e-17 * DEC 2.718, 2.035e36 5000 3.9e-17 9.9e-18 * IEEE 0, 3 28000 5.4e-16 1.1e-16 * IEEE 2.718, 2.556e305 40000 3.5e-16 8.3e-17 * The error criterion was relative when the function magnitude * was greater than one but absolute when it was less than one. * * The following test used the relative error criterion, though * at certain points the relative error could be much higher than * indicated. * IEEE -200, -4 10000 4.8e-16 1.3e-16 * */
/* gdtr.c * * Gamma distribution function * * * * SYNOPSIS: * * double a, b, x, y, gdtr(); * * y = gdtr( 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 * gdtr domain x < 0 0.0 * */
/* gdtrc.c * * Complemented gamma distribution function * * * * SYNOPSIS: * * double a, b, x, y, gdtrc(); * * y = gdtrc( 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 * gdtrc 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 */
/* hyp2f1.c * * Gauss hypergeometric function F * 2 1 * * * SYNOPSIS: * * double a, b, c, x, y, hyp2f1(); * * y = hyp2f1( 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-14 of the nearest integer * (1.0e-13 for IEEE arithmetic). * * ACCURACY: * * * Relative error (-1 < x < 1): * arithmetic domain # trials peak rms * IEEE -1,7 230000 1.2e-11 5.2e-14 * * Several special cases also tested with a, b, c in * the range -7 to 7. * * ERROR MESSAGES: * * A "partial loss of precision" message is printed if * the internally estimated relative error exceeds 1^-12. * A "singularity" message is printed on overflow or * in cases not addressed (such as x < -1). */
/* hyperg.c * * Confluent hypergeometric function * * * * SYNOPSIS: * * double a, b, x, y, hyperg(); * * y = hyperg( 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 * DEC 0,30 2000 1.2e-15 1.3e-16 qtst1: 21800 max = 1.4200E-14 rms = 1.0841E-15 ave = -5.3640E-17 ltstd: 25500 max = 1.2759e-14 rms = 3.7155e-16 ave = 1.5384e-18 * IEEE 0,30 30000 1.8e-14 1.1e-15 * * Larger errors can be observed when b is near a negative * integer or zero. Certain combinations of arguments yield * serious cancellation error in the power series summation * and also are not in the region of near convergence of the * asymptotic series. An error message is printed if the * self-estimated relative error is greater than 1.0e-12. * */
/* i0.c * * Modified Bessel function of order zero * * * * SYNOPSIS: * * double x, y, i0(); * * y = i0( 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 * DEC 0,30 6000 8.2e-17 1.9e-17 * IEEE 0,30 30000 5.8e-16 1.4e-16 * */
/* i0e.c * * Modified Bessel function of order zero, * exponentially scaled * * * * SYNOPSIS: * * double x, y, i0e(); * * y = i0e( 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 30000 5.4e-16 1.2e-16 * See i0(). * */
/* i1.c * * Modified Bessel function of order one * * * * SYNOPSIS: * * double x, y, i1(); * * y = i1( 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 * DEC 0, 30 3400 1.2e-16 2.3e-17 * IEEE 0, 30 30000 1.9e-15 2.1e-16 * * */
/* i1e.c * * Modified Bessel function of order one, * exponentially scaled * * * * SYNOPSIS: * * double x, y, i1e(); * * y = i1e( 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 2.0e-15 2.0e-16 * See i1(). * */
/* igam.c * * Incomplete gamma integral * * * * SYNOPSIS: * * double a, x, y, igam(); * * y = igam( 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 200000 3.6e-14 2.9e-15 * IEEE 0,100 300000 9.9e-14 1.5e-14 */
/* igamc() * * Complemented incomplete gamma integral * * * * SYNOPSIS: * * double a, x, y, igamc(); * * y = igamc( 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: * * Tested at random a, x. * a x Relative error: * arithmetic domain domain # trials peak rms * IEEE 0.5,100 0,100 200000 1.9e-14 1.7e-15 * IEEE 0.01,0.5 0,100 200000 1.4e-13 1.6e-15 */
/* igami() * * Inverse of complemented imcomplete gamma integral * * * * SYNOPSIS: * * double a, x, p, igami(); * * x = igami( a, p ); * * DESCRIPTION: * * Given p, the function finds x such that * * igamc( a, x ) = p. * * It is valid in the right-hand tail of the distribution, p < 0.5. * Starting with the approximate value * * 3 * x = a t * * where * * t = 1 - d - ndtri(p) sqrt(d) * * and * * d = 1/9a, * * the routine performs up to 10 Newton iterations to find the * root of igamc(a,x) - p = 0. * * ACCURACY: * * Tested at random a, p in the intervals indicated. * * a p Relative error: * arithmetic domain domain # trials peak rms * IEEE 0.5,100 0,0.5 100000 1.0e-14 1.7e-15 * IEEE 0.01,0.5 0,0.5 100000 9.0e-14 3.4e-15 * IEEE 0.5,10000 0,0.5 20000 2.3e-13 3.8e-14 */
/* incbet.c * * Incomplete beta integral * * * SYNOPSIS: * * double a, b, x, y, incbet(); * * y = incbet( 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 uniformly distributed random points (a,b,x) with a and b * in "domain" and x between 0 and 1. * Relative error * arithmetic domain # trials peak rms * IEEE 0,5 10000 6.9e-15 4.5e-16 * IEEE 0,85 250000 2.2e-13 1.7e-14 * IEEE 0,1000 30000 5.3e-12 6.3e-13 * IEEE 0,10000 250000 9.3e-11 7.1e-12 * IEEE 0,100000 10000 8.7e-10 4.8e-11 * Outputs smaller than the IEEE gradual underflow threshold * were excluded from these statistics. * * ERROR MESSAGES: * message condition value returned * incbet domain x<0, x>1 0.0 * incbet underflow 0.0 */
/* incbi() * * Inverse of imcomplete beta integral * * * * SYNOPSIS: * * double a, b, x, y, incbi(); * * x = incbi( a, b, y ); * * * * DESCRIPTION: * * Given y, the function finds x such that * * incbet( a, b, x ) = y . * * The routine performs interval halving or 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 50000 5.8e-12 1.3e-13 * IEEE 0,1 .25,100 100000 1.8e-13 3.9e-15 * IEEE 0,1 0,5 50000 1.1e-12 5.5e-15 * VAX 0,1 .5,100 25000 3.5e-14 1.1e-15 * With a and b constrained to half-integer or integer values: * IEEE 0,1 .5,10000 50000 5.8e-12 1.1e-13 * IEEE 0,1 .5,100 100000 1.7e-14 7.9e-16 * With a = .5, b constrained to half-integer or integer values: * IEEE 0,1 .5,10000 10000 8.3e-11 1.0e-11 */
/* isnan() * signbit() * isfinite() * * Floating point numeric utilities * * * * SYNOPSIS: * * double ceil(), floor(), frexp(), ldexp(); * int signbit(), isnan(), isfinite(); * double x, y; * int expnt, n; * * y = floor(x); * y = ceil(x); * y = frexp( x, &expnt ); * y = ldexp( x, n ); * n = signbit(x); * n = isnan(x); * n = isfinite(x); * * * * DESCRIPTION: * * All four routines return a double precision floating point * result. * * floor() returns the largest integer less than or equal to x. * It truncates toward minus infinity. * * ceil() returns the smallest integer greater than or equal * to x. It truncates toward plus infinity. * * frexp() 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. * * ldexp() multiplies x by 2**n. * * signbit(x) returns 1 if the sign bit of x is 1, else 0. * * 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. */
/* iv.c * * Modified Bessel function of noninteger order * * * * SYNOPSIS: * * double v, x, y, iv(); * * y = iv( 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. * Relative error: * arithmetic domain # trials peak rms * DEC 0,30 2000 3.1e-15 5.4e-16 * IEEE 0,30 10000 1.7e-14 2.7e-15 * * Accuracy is diminished if v is near a negative integer. * * See also hyperg.c. * */
/* j0.c * * Bessel function of order zero * * * * SYNOPSIS: * * double x, y, j0(); * * y = j0( x ); * * * * DESCRIPTION: * * Returns Bessel function of order zero of the argument. * * The domain is divided into the intervals [0, 5] and * (5, infinity). In the first interval the following rational * approximation is used: * * * 2 2 * (w - r ) (w - r ) P (w) / Q (w) * 1 2 3 8 * * 2 * where w = x and the two r's are zeros of the function. * * In the second interval, the Hankel asymptotic expansion * is employed with two rational functions of degree 6/6 * and 7/7. * * * * ACCURACY: * * Absolute error: * arithmetic domain # trials peak rms * DEC 0, 30 10000 4.4e-17 6.3e-18 * IEEE 0, 30 60000 4.2e-16 1.1e-16 * */
/* y0.c * * Bessel function of the second kind, order zero * * * * SYNOPSIS: * * double x, y, y0(); * * y = y0( x ); * * * * DESCRIPTION: * * Returns Bessel function of the second kind, of order * zero, of the argument. * * The domain is divided into the intervals [0, 5] and * (5, infinity). In the first interval a rational approximation * R(x) is employed to compute * y0(x) = R(x) + 2 * log(x) * j0(x) / PI. * Thus a call to j0() is required. * * In the second interval, the Hankel asymptotic expansion * is employed with two rational functions of degree 6/6 * and 7/7. * * * * ACCURACY: * * Absolute error, when y0(x) < 1; else relative error: * * arithmetic domain # trials peak rms * DEC 0, 30 9400 7.0e-17 7.9e-18 * IEEE 0, 30 30000 1.3e-15 1.6e-16 * */
/* j1.c * * Bessel function of order one * * * * SYNOPSIS: * * double x, y, j1(); * * y = j1( x ); * * * * DESCRIPTION: * * Returns Bessel function of order one of the argument. * * The domain is divided into the intervals [0, 8] and * (8, infinity). In the first interval a 24 term Chebyshev * expansion is used. In the second, the asymptotic * trigonometric representation is employed using two * rational functions of degree 5/5. * * * * ACCURACY: * * Absolute error: * arithmetic domain # trials peak rms * DEC 0, 30 10000 4.0e-17 1.1e-17 * IEEE 0, 30 30000 2.6e-16 1.1e-16 * * */
/* 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, 8] and * (8, infinity). In the first interval a 25 term Chebyshev * expansion is used, and a call to j1() is required. * In the second, the asymptotic trigonometric representation * is employed using two rational functions of degree 5/5. * * * * ACCURACY: * * Absolute error: * arithmetic domain # trials peak rms * DEC 0, 30 10000 8.6e-17 1.3e-17 * IEEE 0, 30 30000 1.0e-15 1.3e-16 * * (error criterion relative when |y1| > 1). * */
/* jn.c * * Bessel function of integer order * * * * SYNOPSIS: * * int n; * double x, y, jn(); * * y = jn( 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 * DEC 0, 30 5500 6.9e-17 9.3e-18 * IEEE 0, 30 5000 4.4e-16 7.9e-17 * * * Not suitable for large n or x. Use jv() instead. * */
/* jv.c * * Bessel function of noninteger order * * * * SYNOPSIS: * * double v, x, y, jv(); * * y = jv( 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 transitional expansions give 12D accuracy for v > 500. * * * * ACCURACY: * Results for integer v are indicated by *, where x and v * both vary from -125 to +125. Otherwise, * x ranges from 0 to 125, v ranges as indicated by "domain." * Error criterion is absolute, except relative when |jv()| > 1. * * arithmetic v domain x domain # trials peak rms * IEEE 0,125 0,125 100000 4.6e-15 2.2e-16 * IEEE -125,0 0,125 40000 5.4e-11 3.7e-13 * IEEE 0,500 0,500 20000 4.4e-15 4.0e-16 * Integer v: * IEEE -125,125 -125,125 50000 3.5e-15* 1.9e-16* * */
/* k0.c * * Modified Bessel function, third kind, order zero * * * * SYNOPSIS: * * double x, y, k0(); * * y = k0( 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 * DEC 0, 30 3100 1.3e-16 2.1e-17 * IEEE 0, 30 30000 1.2e-15 1.6e-16 * * ERROR MESSAGES: * * message condition value returned * K0 domain x <= 0 MAXNUM * */
/* k0e() * * Modified Bessel function, third kind, order zero, * exponentially scaled * * * * SYNOPSIS: * * double x, y, k0e(); * * y = k0e( 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 1.4e-15 1.4e-16 * See k0(). * */
/* k1.c * * Modified Bessel function, third kind, order one * * * * SYNOPSIS: * * double x, y, k1(); * * y = k1( 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 * DEC 0, 30 3300 8.9e-17 2.2e-17 * IEEE 0, 30 30000 1.2e-15 1.6e-16 * * ERROR MESSAGES: * * message condition value returned * k1 domain x <= 0 MAXNUM * */
/* k1e.c * * Modified Bessel function, third kind, order one, * exponentially scaled * * * * SYNOPSIS: * * double x, y, k1e(); * * y = k1e( 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 7.8e-16 1.2e-16 * See k1(). * */
/* kn.c * * Modified Bessel function, third kind, integer order * * * * SYNOPSIS: * * double x, y, kn(); * int n; * * y = kn( 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: * * Relative error: * arithmetic domain # trials peak rms * DEC 0,30 3000 1.3e-9 5.8e-11 * IEEE 0,30 90000 1.8e-8 3.0e-10 * * Error is high only near the crossover point x = 9.55 * between the two expansions used. */
/* Re Kolmogorov statistics, here is Birnbaum and Tingey's formula for the
distribution of D+, the maximum of all positive deviations between a
theoretical distribution function P(x) and an empirical one Sn(x)
from n samples.
+
D = sup [P(x) - S (x)]
n -inf < x < inf n
[n(1-e)]
+ - v-1 n-v
Pr{D > e} = > C e (e + v/n) (1 - e - v/n)
n - n v
v=0
[n(1-e)] is the largest integer not exceeding n(1-e).
nCv is the number of combinations of n things taken v at a time. */
/* lmdif.c * * The purpose of lmdif is to minimize the sum of the squares of * M nonlinear functions in N variables by a modification of * the Levenberg-Marquardt algorithm. The user must provide a * subroutine that calculates the functions. The Jacobian is * then calculated numerically by a forward-difference approximation. * * Refer to the source code for information on the use of the routine. * * This is a C language translation of the Fortran version of * the corresponding routine from Argonne National Laboratories * MINPACK subroutine suite. * */
/* Levnsn.c */ /* Levinson-Durbin LPC * linear predictive coding * * | R0 R1 R2 ... RN-1 | | A1 | | -R1 | * | R1 R0 R1 ... RN-2 | | A2 | | -R2 | * | R2 R1 R0 ... RN-3 | | A3 | = | -R3 | * | ... | | ...| | ... | * | RN-1 RN-2... R0 | | AN | | -RN | * * Ref: John Makhoul, "Linear Prediction, A Tutorial Review" * Proc. IEEE Vol. 63, PP 561-580 April, 1975. * * R is the input autocorrelation function. R0 is the zero lag * term. A is the output array of predictor coefficients. Note * that a filter impulse response has a coefficient of 1.0 preceding * A1. E is an array of mean square error for each prediction order * 1 to N. REFL is an output array of the reflection coefficients. */
/* log.c * * Natural logarithm * * * * SYNOPSIS: * * double x, y, log(); * * y = log( 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 1.44e-16 5.06e-17 * IEEE +-MAXNUM 30000 1.20e-16 4.78e-17 * DEC 0, 10 170000 1.8e-17 6.3e-18 * * In the tests over the interval [+-MAXNUM], the logarithms * of the random arguments were uniformly distributed over * [0, MAXLOG]. * * ERROR MESSAGES: * * log singularity: x = 0; returns -INFINITY * log domain: x < 0; returns NAN */
/* log10.c * * Common logarithm * * * * SYNOPSIS: * * double x, y, log10(); * * y = log10( 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)/Q(x). * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE 0.5, 2.0 30000 1.5e-16 5.0e-17 * IEEE 0, MAXNUM 30000 1.4e-16 4.8e-17 * DEC 1, MAXNUM 50000 2.5e-17 6.0e-18 * * In the tests over the interval [1, MAXNUM], the logarithms * of the random arguments were uniformly distributed over * [0, MAXLOG]. * * ERROR MESSAGES: * * log10 singularity: x = 0; returns -INFINITY * log10 domain: x < 0; returns NAN */
/* log2.c * * Base 2 logarithm * * * * SYNOPSIS: * * double x, y, log2(); * * y = log2( 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 0.5, 2.0 30000 2.0e-16 5.5e-17 * IEEE exp(+-700) 40000 1.3e-16 4.6e-17 * * In the tests over the interval [exp(+-700)], the logarithms * of the random arguments were uniformly distributed. * * ERROR MESSAGES: * * log2 singularity: x = 0; returns -INFINITY * log2 domain: x < 0; returns NAN */
/* lrand.c * * Pseudorandom number generator * * * * SYNOPSIS: * * long y, drand(); * * drand( &y ); * * * * DESCRIPTION: * * Yields a long integer random number. * * The three-generator congruential algorithm by Brian * Wichmann and David Hill (BYTE magazine, March, 1987, * pp 127-8) is used. The period, given by them, is * 6953607871644. * * */
/* lsqrt.c * * Integer square root * * * * SYNOPSIS: * * long x, y; * long lsqrt(); * * y = lsqrt( x ); * * * * DESCRIPTION: * * Returns a long integer square root of the long integer * argument. The computation is by binary long division. * * The largest possible result is lsqrt(2,147,483,647) * = 46341. * * If x < 0, the square root of |x| is returned, and an * error message is printed. * * * ACCURACY: * * An extra, roundoff, bit is computed; hence the result * is the nearest integer to the actual square root. * NOTE: only DEC arithmetic is currently supported. * */
/* minv.c * * Matrix inversion * * * * SYNOPSIS: * * int n, errcod; * double A[n*n], X[n*n]; * double B[n]; * int IPS[n]; * int minv(); * * errcod = minv( A, X, n, B, IPS ); * * * * DESCRIPTION: * * Finds the inverse of the n by n matrix A. The result goes * to X. B and IPS are scratch pad arrays of length n. * The contents of matrix A are destroyed. * * The routine returns nonzero on error; error messages are printed * by subroutine simq(). * */
/* mtransp.c * * Matrix transpose * * * * SYNOPSIS: * * int n; * double A[n*n], T[n*n]; * * mtransp( n, A, T ); * * * * DESCRIPTION: * * * T[r][c] = A[c][r] * * * Transposes the n by n square matrix A and puts the result in T. * The output, T, may occupy the same storage as A. * * * */
/* nbdtr.c * * Negative binomial distribution * * * * SYNOPSIS: * * int k, n; * double p, y, nbdtr(); * * y = nbdtr( 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 (a,b,p), with p between 0 and 1. * * a,b Relative error: * arithmetic domain # trials peak rms * IEEE 0,100 100000 1.7e-13 8.8e-15 * See also incbet.c. * */
/* nbdtr.c * * Complemented negative binomial distribution * * * * SYNOPSIS: * * int k, n; * double p, y, nbdtrc(); * * y = nbdtrc( 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: * * Tested at random points (a,b,p), with p between 0 and 1. * * a,b Relative error: * arithmetic domain # trials peak rms * IEEE 0,100 100000 1.7e-13 8.8e-15 * See also incbet.c. */
/* nbdtr.c * * Functional inverse of negative binomial distribution * * * * SYNOPSIS: * * int k, n; * double p, y, nbdtri(); * * p = nbdtri( 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 100000 1.5e-14 8.5e-16 * See also incbi.c. */
/* ndtr.c * * Normal distribution function * * * * SYNOPSIS: * * double x, y, ndtr(); * * y = ndtr( 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 with care to avoid error amplification in computing exp(-x^2). * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE -13,0 30000 1.3e-15 2.2e-16 * * * ERROR MESSAGES: * * message condition value returned * erfc underflow x > 37.519379347 0.0 * */
/* ndtr.c * * Error function * * * * SYNOPSIS: * * double x, y, erf(); * * y = erf( 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 * P4(x**2)/Q5(x**2); otherwise * erf(x) = 1 - erfc(x). * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC 0,1 14000 4.7e-17 1.5e-17 * IEEE 0,1 30000 3.7e-16 1.0e-16 * */
/* ndtr.c * * Complementary error function * * * * SYNOPSIS: * * double x, y, erfc(); * * y = erfc( 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 rational * approximations are computed. * * A special function expx2.c is used to suppress error amplification * in computing exp(-x^2). * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * IEEE 0,26.6417 30000 1.3e-15 2.2e-16 * * * ERROR MESSAGES: * * message condition value returned * erfc underflow x > 9.231948545 (DEC) 0.0 * * */
/* ndtri.c * * Inverse of Normal distribution function * * * * SYNOPSIS: * * double x, y, ndtri(); * * x = ndtri( 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 * DEC 0.125, 1 5500 9.5e-17 2.1e-17 * DEC 6e-39, 0.135 3500 5.7e-17 1.3e-17 * IEEE 0.125, 1 20000 7.2e-16 1.3e-16 * IEEE 3e-308, 0.135 50000 4.6e-16 9.8e-17 * * * ERROR MESSAGES: * * message condition value returned * ndtri domain x <= 0 -MAXNUM * ndtri domain x >= 1 MAXNUM * */
/* pdtr.c * * Poisson distribution * * * * SYNOPSIS: * * int k; * double m, y, pdtr(); * * y = pdtr( 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(). * */
/* pdtrc() * * Complemented poisson distribution * * * * SYNOPSIS: * * int k; * double m, y, pdtrc(); * * y = pdtrc( 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. * */
/* pdtri() * * Inverse Poisson distribution * * * * SYNOPSIS: * * int k; * double m, y, pdtr(); * * m = pdtri( 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 * */
/* planck.c * * Integral of Planck's black body radiation formula * * * * SYNOPSIS: * * double lambda, T, y, plancki(); * * y = plancki( lambda, T ); * * * * DESCRIPTION: * * Evaluates the definite integral, from wavelength 0 to lambda, * of Planck's radiation formula * -5 * c1 lambda * E = ------------------ * c2/(lambda T) * e - 1 * * Physical constants c1 and c2 (see below) are built in * to the function program. They are scaled to provide a result * in watts per square meter. Argument T represents temperature in degrees * Kelvin; lambda is wavelength in meters. * * The integral is expressed in closed form, in terms of polylogarithms * (see polylog.c). * * The total area under the curve is * (-1/8) (42 zeta(4) - 12 pi^2 zeta(2) + pi^4 ) c1 (T/c2)^4 * = (pi^4 / 15) c1 (T/c2)^4 * = sigma T^4 * * * CONSTANTS: * * First radiation constant c1 = 2 pi h c^2 = 3.741 771 53 (17) e-16 W m2 * Second radiation constant c2 = h c / k = 0.014 387 770 (13) m K * Stefan-Boltzmann constant sigma = 5.670 373 (21) e-8 W m^-2 K^-4 * Wien wavelength displacement law constant wien = 2.8977721 (26) e-3 m K * These are NIST values as of 2010. * * * ACCURACY: * * The left tail of the function experiences some relative error * amplification in computing the dominant term exp(-c2/(lambda T)). * For the right-hand tail see planckc, below. * * Relative error. * The domain refers to lambda T / c2. * arithmetic domain # trials peak rms * IEEE 0.1, 10 50000 7.1e-15 5.4e-16 * */
/* polevl.c * p1evl.c * * Evaluate polynomial * * * * SYNOPSIS: * * int N; * double x, y, coef[N+1], polevl[]; * * y = polevl( 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. * */
/* polmisc.c * Square root, sine, cosine, and arctangent of polynomial. * See polyn.c for data structures and discussion. */
/* polrt.c
*
* Find roots of a polynomial
*
*
*
* SYNOPSIS:
*
* typedef struct
* {
* double r;
* double i;
* }cmplx;
*
* double xcof[], cof[];
* int m;
* cmplx root[];
*
* polrt( xcof, cof, m, root )
*
*
*
* DESCRIPTION:
*
* Iterative determination of the roots of a polynomial of
* degree m whose coefficient vector is xcof[]. The
* coefficients are arranged in ascending order; i.e., the
* coefficient of x**m is xcof[m].
*
* The array cof[] is working storage the same size as xcof[].
* root[] is the output array containing the complex roots.
*
*
* ACCURACY:
*
* Termination depends on evaluation of the polynomial at
* the trial values of the roots. The values of multiple roots
* or of roots that are nearly equal may have poor relative
* accuracy after the first root in the neighborhood has been
* found.
*
*/
/* polylog.c * * Polylogarithms * * * * SYNOPSIS: * * double x, y, polylog(); * int n; * * y = polylog( n, x ); * * * The polylogarithm of order n is defined by the series * * * inf k * - x * Li (x) = > --- . * n - n * k=1 k * * * For x = 1, * * inf * - 1 * Li (1) = > --- = Riemann zeta function (n) . * n - n * k=1 k * * * When n = 2, the function is the dilogarithm, related to Spence's integral: * * x 1-x * - - * | | -ln(1-t) | | ln t * Li (x) = | -------- dt = | ------ dt = spence(1-x) . * 2 | | t | | 1 - t * - - * 0 1 * * * See also the program cpolylog.c for the complex polylogarithm, * whose definition is extended to x > 1. * * References: * * Lewin, L., _Polylogarithms and Associated Functions_, * North Holland, 1981. * * Lewin, L., ed., _Structural Properties of Polylogarithms_, * American Mathematical Society, 1991. * * * ACCURACY: * * Relative error: * arithmetic domain n # trials peak rms * IEEE 0, 1 2 50000 6.2e-16 8.0e-17 * IEEE 0, 1 3 100000 2.5e-16 6.6e-17 * IEEE 0, 1 4 30000 1.7e-16 4.9e-17 * IEEE 0, 1 5 30000 5.1e-16 7.8e-17 * */
/* polyn.c * polyr.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 MAXPOL. An operation that attempts * to use or generate a polynomial of higher degree may produce a * result that suffers truncation at degree MAXPOL. The value of * MAXPOL is set by calling the function * * polini( maxpol ); * * where maxpol is the desired maximum degree. This must be * done prior to calling any of the other functions in this module. * Memory for internal temporary polynomial storage is allocated * by polini(). * * Each polynomial is represented by an array containing its * coefficients, together with a separately declared integer equal * to the degree of the polynomial. The coefficients appear in * ascending order; that is, * * 2 na * a(x) = a[0] + a[1] * x + a[2] * x + ... + a[na] * x . * * * * sum = poleva( a, na, x ); Evaluate polynomial a(t) at t = x. * polprt( a, na, D ); Print the coefficients of a to D digits. * polclr( a, na ); Set a identically equal to zero, up to a[na]. * polmov( a, na, b ); Set b = a. * poladd( a, na, b, nb, c ); c = b + a, nc = max(na,nb) * polsub( a, na, b, nb, c ); c = b - a, nc = max(na,nb) * polmul( a, na, b, nb, c ); c = b * a, nc = na+nb * * * Division: * * i = poldiv( a, na, b, nb, c ); c = b / a, nc = MAXPOL * * returns i = the degree of the first nonzero coefficient of a. * The computed quotient c must be divided by x^i. An error message * is printed if a is identically zero. * * * Change of variables: * If a and b are polynomials, and t = a(x), then * c(t) = b(a(x)) * is a polynomial found by substituting a(x) for t. The * subroutine call for this is * * polsbt( a, na, b, nb, c ); * * * Notes: * poldiv() is an integer routine; poleva() is double. * Any of the arguments a, b, c may refer to the same array. * */
/* Arithmetic operations on polynomials with rational coefficients * * In the following descriptions a, b, c are polynomials of degree * na, nb, nc respectively. The degree of a polynomial cannot * exceed a run-time value MAXPOL. An operation that attempts * to use or generate a polynomial of higher degree may produce a * result that suffers truncation at degree MAXPOL. The value of * MAXPOL is set by calling the function * * polini( maxpol ); * * where maxpol is the desired maximum degree. This must be * done prior to calling any of the other functions in this module. * Memory for internal temporary polynomial storage is allocated * by polini(). * * Each polynomial is represented by an array containing its * coefficients, together with a separately declared integer equal * to the degree of the polynomial. The coefficients appear in * ascending order; that is, * * 2 na * a(x) = a[0] + a[1] * x + a[2] * x + ... + a[na] * x . * * * * `a', `b', `c' are arrays of fracts. * poleva( a, na, &x, &sum ); Evaluate polynomial a(t) at t = x. * polprt( a, na, D ); Print the coefficients of a to D digits. * polclr( a, na ); Set a identically equal to zero, up to a[na]. * polmov( a, na, b ); Set b = a. * poladd( a, na, b, nb, c ); c = b + a, nc = max(na,nb) * polsub( a, na, b, nb, c ); c = b - a, nc = max(na,nb) * polmul( a, na, b, nb, c ); c = b * a, nc = na+nb * * * Division: * * i = poldiv( a, na, b, nb, c ); c = b / a, nc = MAXPOL * * returns i = the degree of the first nonzero coefficient of a. * The computed quotient c must be divided by x^i. An error message * is printed if a is identically zero. * * * Change of variables: * If a and b are polynomials, and t = a(x), then * c(t) = b(a(x)) * is a polynomial found by substituting a(x) for t. The * subroutine call for this is * * polsbt( a, na, b, nb, c ); * * * Notes: * poldiv() is an integer routine; poleva() is double. * Any of the arguments a, b, c may refer to the same array. * */
/* pow.c * * Power function * * * * SYNOPSIS: * * double x, y, z, pow(); * * z = pow( 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 -26,26 30000 4.2e-16 7.7e-17 * DEC -26,26 60000 4.8e-17 9.1e-18 * 1/26 < x < 26, with log(x) uniformly distributed. * -26 < y < 26, y uniformly distributed. * IEEE 0,8700 30000 1.5e-14 2.1e-15 * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed. * * * ERROR MESSAGES: * * message condition value returned * pow overflow x**y > MAXNUM INFINITY * pow underflow x**y < 1/MAXNUM 0.0 * pow domain x<0 and y noninteger 0.0 * */
/* powi.c * * Real raised to integer power * * * * SYNOPSIS: * * double x, y, powi(); * int n; * * y = powi( 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 * DEC .04,26 -26,26 100000 2.7e-16 4.3e-17 * IEEE .04,26 -26,26 50000 2.0e-15 3.8e-16 * IEEE 1,2 -1022,1023 50000 8.6e-14 1.6e-14 * * Returns MAXNUM on overflow, zero on underflow. * */
/* psi.c * * Psi (digamma) function * * * SYNOPSIS: * * double x, y, psi(); * * y = psi( 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: * Relative error (except absolute when |psi| < 1): * arithmetic domain # trials peak rms * DEC 0,30 2500 1.7e-16 2.0e-17 * IEEE 0,30 30000 1.3e-15 1.4e-16 * IEEE -30,0 40000 1.5e-15 2.2e-16 * * ERROR MESSAGES: * message condition value returned * psi singularity x integer <=0 MAXNUM */
/* revers.c * * Reversion of power series * * * * SYNOPSIS: * * extern int MAXPOL; * int n; * double x[n+1], y[n+1]; * * polini(n); * revers( y, x, n ); * * Note, polini() initializes the polynomial arithmetic subroutines; * see polyn.c. * * * DESCRIPTION: * * If * * inf * - i * y(x) = > a x * - i * i=1 * * then * * inf * - j * x(y) = > A y , * - j * j=1 * * where * 1 * A = --- * 1 a * 1 * * etc. The coefficients of x(y) are found by expanding * * inf inf * - - i * x(y) = > A > a x * - j - i * j=1 i=1 * * and setting each coefficient of x , higher than the first, * to zero. * * * * RESTRICTIONS: * * y[0] must be zero, and y[1] must be nonzero. * */
/* rgamma.c * * Reciprocal gamma function * * * * SYNOPSIS: * * double x, y, rgamma(); * * y = rgamma( 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/MAXNUM is returned for positive arguments outside this * range. For arguments less than -34.034 the cosecant * reflection formula is applied; lograrithms are employed * to avoid unnecessary overflow. * * The reciprocal gamma function has no singularities, * but overflow and underflow may occur for large arguments. * These conditions return either MAXNUM or 1/MAXNUM with * appropriate sign. * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC -30,+30 4000 1.2e-16 1.8e-17 * IEEE -30,+30 30000 1.1e-15 2.0e-16 * For arguments less than -34.034 the peak error is on the * order of 5e-15 (DEC), excepting overflow or underflow. */
/* round.c * * Round double to nearest or even integer valued double * * * * SYNOPSIS: * * double x, y, round(); * * y = round(x); * * * * DESCRIPTION: * * Returns the nearest integer to x as a double precision * floating point result. If x ends in 0.5 exactly, the * nearest even integer is chosen. * * * * ACCURACY: * * If x is greater than 1/(2*MACHEP), its closest machine * representation is already an integer, so rounding does * not change it. */
/* shichi.c * * Hyperbolic sine and cosine integrals * * * * SYNOPSIS: * * double x, Chi, Shi, shichi(); * * 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 * DEC Shi 3000 9.1e-17 * IEEE Shi 30000 6.9e-16 1.6e-16 * Absolute error, except relative when |Chi| > 1: * DEC Chi 2500 9.3e-17 * IEEE Chi 30000 8.4e-16 1.4e-16 */
/* sici.c * * Sine and cosine integrals * * * * SYNOPSIS: * * double x, Ci, Si, sici(); * * sici( 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 4.4e-16 7.3e-17 * IEEE Ci 30000 6.9e-16 5.1e-17 * DEC Si 5000 4.4e-17 9.0e-18 * DEC Ci 5300 7.9e-17 5.2e-18 */
/* simpsn.c */ /* simpsn.c * Numerical integration of function tabulated * at equally spaced arguments */
/* simq.c * * Solution of simultaneous linear equations AX = B * by Gaussian elimination with partial pivoting * * * * SYNOPSIS: * * double A[n*n], B[n], X[n]; * int n, flag; * int IPS[]; * int simq(); * * ercode = simq( A, B, X, n, flag, IPS ); * * * * DESCRIPTION: * * B, X, IPS are vectors of length n. * A is an n x n matrix (i.e., a vector of length n*n), * stored row-wise: that is, A(i,j) = A[ij], * where ij = i*n + j, which is the transpose of the normal * column-wise storage. * * The contents of matrix A are destroyed. * * Set flag=0 to solve. * Set flag=-1 to do a new back substitution for different B vector * using the same A matrix previously reduced when flag=0. * * The routine returns nonzero on error; messages are printed. * * * ACCURACY: * * Depends on the conditioning (range of eigenvalues) of matrix A. * * * REFERENCE: * * Computer Solution of Linear Algebraic Systems, * by George E. Forsythe and Cleve B. Moler; Prentice-Hall, 1967. * */
/* sin.c * * Circular sine * * * * SYNOPSIS: * * double x, y, sin(); * * y = sin( 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 * DEC 0, 10 150000 3.0e-17 7.8e-18 * IEEE -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17 * * ERROR MESSAGES: * * message condition value returned * sin total loss x > 1.073741824e9 0.0 * * Partial loss of accuracy begins to occur at x = 2**30 * = 1.074e9. The loss is not gradual, but jumps suddenly to * about 1 part in 10e7. Results may be meaningless for * x > 2**49 = 5.6e14. The routine as implemented flags a * TLOSS error for x > 2**30 and returns 0.0. */
/* cos.c * * Circular cosine * * * * SYNOPSIS: * * double x, y, cos(); * * y = cos( 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 -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17 * DEC 0,+1.07e9 17000 3.0e-17 7.2e-18 */
/* sincos.c * * Circular sine and cosine of argument in degrees * Table lookup and interpolation algorithm * * * * SYNOPSIS: * * double x, sine, cosine, flg, sincos(); * * sincos( x, &sine, &cosine, flg ); * * * * DESCRIPTION: * * Returns both the sine and the cosine of the argument x. * Several different compile time options and minimax * approximations are supplied to permit tailoring the * tradeoff between computation speed and accuracy. * * Since range reduction is time consuming, the reduction * of x modulo 360 degrees is also made optional. * * sin(i) is internally tabulated for 0 <= i <= 90 degrees. * Approximation polynomials, ranging from linear interpolation * to cubics in (x-i)**2, compute the sine and cosine * of the residual x-i which is between -0.5 and +0.5 degree. * In the case of the high accuracy options, the residual * and the tabulated values are combined using the trigonometry * formulas for sin(A+B) and cos(A+B). * * Compile time options are supplied for 5, 11, or 17 decimal * relative accuracy (ACC5, ACC11, ACC17 respectively). * A subroutine flag argument "flg" chooses betwen this * accuracy and table lookup only (peak absolute error * = 0.0087). * * If the argument flg = 1, then the tabulated value is * returned for the nearest whole number of degrees. The * approximation polynomials are not computed. At * x = 0.5 deg, the absolute error is then sin(0.5) = 0.0087. * * An intermediate speed and precision can be obtained using * the compile time option LINTERP and flg = 1. This yields * a linear interpolation using a slope estimated from the sine * or cosine at the nearest integer argument. The peak absolute * error with this option is 3.8e-5. Relative error at small * angles is about 1e-5. * * If flg = 0, then the approximation polynomials are computed * and applied. * * * * SPEED: * * Relative speed comparisons follow for 6MHz IBM AT clone * and Microsoft C version 4.0. These figures include * software overhead of do loop and function calls. * Since system hardware and software vary widely, the * numbers should be taken as representative only. * * flg=0 flg=0 flg=1 flg=1 * ACC11 ACC5 LINTERP Lookup only * In-line 8087 (/FPi) * sin(), cos() 1.0 1.0 1.0 1.0 * * In-line 8087 (/FPi) * sincos() 1.1 1.4 1.9 3.0 * * Software (/FPa) * sin(), cos() 0.19 0.19 0.19 0.19 * * Software (/FPa) * sincos() 0.39 0.50 0.73 1.7 * * * * ACCURACY: * * The accurate approximations are designed with a relative error * criterion. The absolute error is greatest at x = 0.5 degree. * It decreases from a local maximum at i+0.5 degrees to full * machine precision at each integer i degrees. With the * ACC5 option, the relative error of 6.3e-6 is equivalent to * an absolute angular error of 0.01 arc second in the argument * at x = i+0.5 degrees. For small angles < 0.5 deg, the ACC5 * accuracy is 6.3e-6 (.00063%) of reading; i.e., the absolute * error decreases in proportion to the argument. This is true * for both the sine and cosine approximations, since the latter * is for the function 1 - cos(x). * * If absolute error is of most concern, use the compile time * option ABSERR to obtain an absolute error of 2.7e-8 for ACC5 * precision. This is about half the absolute error of the * relative precision option. In this case the relative error * for small angles will increase to 9.5e-6 -- a reasonable * tradeoff. */
/* sindg.c * * Circular sine of angle in degrees * * * * SYNOPSIS: * * double x, y, sindg(); * * y = sindg( 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 P(x**2). * * * * ACCURACY: * * Relative error: * arithmetic domain # trials peak rms * DEC +-1000 3100 3.3e-17 9.0e-18 * IEEE +-1000 30000 2.3e-16 5.6e-17 * * ERROR MESSAGES: * * message condition value returned * sindg total loss x > 8.0e14 (DEC) 0.0 * x > 1.0e14 (IEEE) * */
/* cosdg.c * * Circular cosine of angle in degrees * * * * SYNOPSIS: * * double x, y, cosdg(); * * y = cosdg( 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 P(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 * DEC +-1000 3400 3.5e-17 9.1e-18 * IEEE +-1000 30000 2.1e-16 5.7e-17 * See also sin(). * */
/* sinh.c * * Hyperbolic sine * * * * SYNOPSIS: * * double x, y, sinh(); * * y = sinh( x ); * * * * DESCRIPTION: * * Returns hyperbolic sine of argument in the range MINLOG to * MAXLOG. * * 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 * DEC +- 88 50000 4.0e-17 7.7e-18 * IEEE +-MAXLOG 30000 2.6e-16 5.7e-17 * */
/* spence.c * * Dilogarithm * * * * SYNOPSIS: * * double x, y, spence(); * * y = spence( 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 3.9e-15 5.4e-16 * DEC 0,4 3000 2.5e-16 4.5e-17 * * */
/* sqrt.c * * Square root * * * * SYNOPSIS: * * double x, y, sqrt(); * * y = sqrt( 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 * DEC 0, 10 60000 2.1e-17 7.9e-18 * IEEE 0,1.7e308 30000 1.7e-16 6.3e-17 * * * ERROR MESSAGES: * * message condition value returned * sqrt domain x < 0 0.0 * */
/* stdtr.c * * Student's t distribution * * * * SYNOPSIS: * * double t, stdtr(); * short k; * * y = stdtr( 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 < -2, this is the method of computation. For higher t, * a direct method is derived from integration by parts. * Since the function is symmetric about t=0, the area under the * right tail of the density is found by calling the function * with -t instead of t. * * ACCURACY: * * Tested at random 1 <= k <= 25. The "domain" refers to t. * Relative error: * arithmetic domain # trials peak rms * IEEE -100,-2 50000 5.9e-15 1.4e-15 * IEEE -2,100 500000 2.7e-15 4.9e-17 */
/* stdtri.c * * Functional inverse of Student's t distribution * * * * SYNOPSIS: * * double p, t, stdtri(); * int k; * * t = stdtri( k, p ); * * * DESCRIPTION: * * Given probability p, finds the argument t such that stdtr(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 .001,.999 25000 5.7e-15 8.0e-16 * IEEE 10^-6,.001 25000 2.0e-12 2.9e-14 */
/* struve.c * * Struve function * * * * SYNOPSIS: * * double v, x, y, struve(); * * y = struve( 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: * * Not accurately characterized, but spot checked against tables. * */
/* tan.c * * Circular tangent * * * * SYNOPSIS: * * double x, y, tan(); * * y = tan( 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 * DEC +-1.07e9 44000 4.1e-17 1.0e-17 * IEEE +-1.07e9 30000 2.9e-16 8.1e-17 * * ERROR MESSAGES: * * message condition value returned * tan total loss x > 1.073741824e9 0.0 * */
/* cot.c * * Circular cotangent * * * * SYNOPSIS: * * double x, y, cot(); * * y = cot( 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 2.9e-16 8.2e-17 * * * ERROR MESSAGES: * * message condition value returned * cot total loss x > 1.073741824e9 0.0 * cot singularity x = 0 INFINITY * */
/* tandg.c * * Circular tangent of argument in degrees * * * * SYNOPSIS: * * double x, y, tandg(); * * y = tandg( x ); * * * * DESCRIPTION: * * Returns the circular tangent of the argument x in degrees. * * 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 * DEC 0,10 8000 3.4e-17 1.2e-17 * IEEE 0,10 30000 3.2e-16 8.4e-17 * * ERROR MESSAGES: * * message condition value returned * tandg total loss x > 8.0e14 (DEC) 0.0 * x > 1.0e14 (IEEE) * tandg singularity x = 180 k + 90 MAXNUM */
/* cotdg.c * * Circular cotangent of argument in degrees * * * * SYNOPSIS: * * double x, y, cotdg(); * * y = cotdg( x ); * * * * DESCRIPTION: * * Returns the circular cotangent of the argument x in degrees. * * 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]. * * * ERROR MESSAGES: * * message condition value returned * cotdg total loss x > 8.0e14 (DEC) 0.0 * x > 1.0e14 (IEEE) * cotdg singularity x = 180 k MAXNUM */
/* tanh.c * * Hyperbolic tangent * * * * SYNOPSIS: * * double x, y, tanh(); * * y = tanh( x ); * * * * DESCRIPTION: * * Returns hyperbolic tangent of argument in the range MINLOG to * MAXLOG. * * 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 * DEC -2,2 50000 3.3e-17 6.4e-18 * IEEE -2,2 30000 2.5e-16 5.8e-17 * */
/* unity.c * * Relative error approximations for function arguments near * unity. * * log1p(x) = log(1+x) * expm1(x) = exp(x) - 1 * cosm1(x) = cos(x) - 1 * */
/* yn.c * * Bessel function of second kind of integer order * * * * SYNOPSIS: * * double x, y, yn(); * int n; * * y = yn( 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 * DEC 0, 30 2200 2.9e-16 5.3e-17 * IEEE 0, 30 30000 3.4e-15 4.3e-16 * * * ERROR MESSAGES: * * message condition value returned * yn singularity x = 0 MAXNUM * yn overflow MAXNUM * * Spot checked against tables for x, n between 0 and 100. * */
/* zeta.c * * Riemann zeta function of two arguments * * * * SYNOPSIS: * * double x, q, y, zeta(); * * y = zeta( 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: * * * * REFERENCE: * * Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals, * Series, and Products, p. 1073; Academic Press, 1980. * */
/* zetac.c * * Riemann zeta function * * * * SYNOPSIS: * * double x, y, zetac(); * * y = zetac( 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 10000 9.8e-16 1.3e-16 * DEC 1,50 2000 1.1e-16 1.9e-17 * * */
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