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diff --git a/libm/double/README.txt b/libm/double/README.txt new file mode 100644 index 000000000..f2cb6c3dc --- /dev/null +++ b/libm/double/README.txt @@ -0,0 +1,5845 @@ +/* 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 + * + */ + +/* 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(). + */ + +/* 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 + * math.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 + * + * + */ + +/* cpmul.c + * + * Multiply two polynomials with complex coefficients + * + * + * + * SYNOPSIS: + * + * typedef struct + * { + * double r; + * double i; + * }cmplx; + * + * cmplx a[], b[], c[]; + * int da, db, dc; + * + * cpmul( a, da, b, db, c, &dc ); + * + * + * + * DESCRIPTION: + * + * The two argument polynomials are multiplied together, and + * their product is placed in c. + * + * Each polynomial is represented by its coefficients stored + * as an array of complex number structures (see the typedef). + * The degree of a is da, which must be passed to the routine + * as an argument; similarly the degree db of b is an argument. + * Array a has da + 1 elements and array b has db + 1 elements. + * Array c must have storage allocated for at least da + db + 1 + * elements. The value da + db is returned in dc; this is + * the degree of the product polynomial. + * + * Polynomial coefficients are stored in ascending order; i.e., + * a(x) = a[0]*x**0 + a[1]*x**1 + ... + a[da]*x**da. + * + * + * If desired, c may be the same as either a or b, in which + * case the input argument array is replaced by the product + * array (but only up to terms of degree da + db). + * + */ + +/* 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. + * + */ + +/* 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 10000 1.3e-12 1.8e-14 + * + * 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 + * + */ + +/* 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() + * signbit() + * isnan() + * 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. + */ + +/* 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 + * 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. + * + * 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 + */ + +/* 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) - Sn(x) ] + n -inf < x < inf + + + [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. + + Exact Smirnov statistic, for one-sided test: +double +smirnov (n, e) + int n; + double e; + + Kolmogorov's limiting distribution of two-sided test, returns + probability that sqrt(n) * max deviation > y, + or that max deviation > y/sqrt(n). + The approximation is useful for the tail of the distribution + when n is large. +double +kolmogorov (y) + double y; + + + Functional inverse of Smirnov distribution + finds e such that smirnov(n,e) = p. +double +smirnovi (n, p) + int n; + double p; + + Functional inverse of Kolmogorov statistic for two-sided test. + Finds y such that kolmogorov(y) = p. + If e = smirnovi (n,p), then kolmogi(2 * p) / sqrt(n) should + be close to e. +double +kolmogi (p) + double p; + */ + +/* Levnsn.c */ +/* Levinson-Durbin LPC + * + * | 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(). + * + */ + +/* mmmpy.c + * + * Matrix multiply + * + * + * + * SYNOPSIS: + * + * int r, c; + * double A[r*c], B[c*r], Y[r*r]; + * + * mmmpy( r, c, A, B, Y ); + * + * + * + * DESCRIPTION: + * + * Y = A B + * c-1 + * -- + * Y[i][j] = > A[i][k] B[k][j] + * -- + * k=0 + * + * Multiplies an r (rows) by c (columns) matrix A on the left + * by a c (rows) by r (columns) matrix B on the right + * to produce an r by r matrix Y. + * + * + */ + +/* mtherr.c + * + * Library common error handling routine + * + * + * + * SYNOPSIS: + * + * char *fctnam; + * int code; + * int mtherr(); + * + * mtherr( fctnam, code ); + * + * + * + * DESCRIPTION: + * + * This routine may be called to report one of the following + * error conditions (in the include file math.h). + * + * Mnemonic Value Significance + * + * DOMAIN 1 argument domain error + * SING 2 function singularity + * OVERFLOW 3 overflow range error + * UNDERFLOW 4 underflow range error + * TLOSS 5 total loss of precision + * PLOSS 6 partial loss of precision + * EDOM 33 Unix domain error code + * ERANGE 34 Unix range error code + * + * The default version of the file prints the function name, + * passed to it by the pointer fctnam, followed by the + * error condition. The display is directed to the standard + * output device. The routine then returns to the calling + * program. Users may wish to modify the program to abort by + * calling exit() under severe error conditions such as domain + * errors. + * + * Since all error conditions pass control to this function, + * the display may be easily changed, eliminated, or directed + * to an error logging device. + * + * SEE ALSO: + * + * math.h + * + */ + +/* 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. + * + * + * + */ + +/* mvmpy.c + * + * Matrix times vector + * + * + * + * SYNOPSIS: + * + * int r, c; + * double A[r*c], V[c], Y[r]; + * + * mvmpy( r, c, A, V, Y ); + * + * + * + * DESCRIPTION: + * + * c-1 + * -- + * Y[j] = > A[j][k] V[k] , j = 1, ..., r + * -- + * k=0 + * + * Multiplies the r (rows) by c (columns) matrix A on the left + * by column vector V of dimension c on the right + * to produce a (column) vector Y output of dimension r. + * + * + * + * + */ + +/* 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. + * + */ + /* nbdtrc.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. + */ + +/* nbdtrc + * + * 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: + * + * See incbet.c. + */ + /* nbdtri + * + * 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. + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * DEC -13,0 8000 2.1e-15 4.8e-16 + * IEEE -13,0 30000 3.4e-14 6.7e-15 + * + * + * ERROR MESSAGES: + * + * message condition value returned + * erfc underflow x > 37.519379347 0.0 + * + */ + /* erf.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 + * + */ + /* erfc.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. + * + * + * + * ACCURACY: + * + * Relative error: + * arithmetic domain # trials peak rms + * DEC 0, 9.2319 12000 5.1e-16 1.2e-16 + * IEEE 0,26.6417 30000 5.7e-14 1.5e-14 + * + * + * 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 + * + */ + +/* 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. + * + */ + +/* 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. + * + */ + +/* 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 */ + * 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 + * + * + */ |