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-/* acoshf.c
- *
- * Inverse hyperbolic cosine
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, acoshf();
- *
- * y = acoshf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns inverse hyperbolic cosine of argument.
- *
- * If 1 <= x < 1.5, a polynomial approximation
- *
- * sqrt(z) * P(z)
- *
- * where z = x-1, is used. Otherwise,
- *
- * acosh(x) = log( x + sqrt( (x-1)(x+1) ).
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 1,3 100000 1.8e-7 3.9e-8
- * IEEE 1,2000 100000 3.0e-8
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * acoshf domain |x| < 1 0.0
- *
- */
-
-/* airy.c
- *
- * Airy function
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, ai, aip, bi, bip;
- * int airyf();
- *
- * airyf( x, _&ai, _&aip, _&bi, _&bip );
- *
- *
- *
- * DESCRIPTION:
- *
- * Solution of the differential equation
- *
- * y"(x) = xy.
- *
- * The function returns the two independent solutions Ai, Bi
- * and their first derivatives Ai'(x), Bi'(x).
- *
- * Evaluation is by power series summation for small x,
- * by rational minimax approximations for large x.
- *
- *
- *
- * ACCURACY:
- * Error criterion is absolute when function <= 1, relative
- * when function > 1, except * denotes relative error criterion.
- * For large negative x, the absolute error increases as x^1.5.
- * For large positive x, the relative error increases as x^1.5.
- *
- * Arithmetic domain function # trials peak rms
- * IEEE -10, 0 Ai 50000 7.0e-7 1.2e-7
- * IEEE 0, 10 Ai 50000 9.9e-6* 6.8e-7*
- * IEEE -10, 0 Ai' 50000 2.4e-6 3.5e-7
- * IEEE 0, 10 Ai' 50000 8.7e-6* 6.2e-7*
- * IEEE -10, 10 Bi 100000 2.2e-6 2.6e-7
- * IEEE -10, 10 Bi' 50000 2.2e-6 3.5e-7
- *
- */
-
-/* asinf.c
- *
- * Inverse circular sine
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, asinf();
- *
- * y = asinf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns radian angle between -pi/2 and +pi/2 whose sine is x.
- *
- * A polynomial of the form x + x**3 P(x**2)
- * is used for |x| in the interval [0, 0.5]. If |x| > 0.5 it is
- * transformed by the identity
- *
- * asin(x) = pi/2 - 2 asin( sqrt( (1-x)/2 ) ).
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -1, 1 100000 2.5e-7 5.0e-8
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * asinf domain |x| > 1 0.0
- *
- */
- /* acosf()
- *
- * Inverse circular cosine
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, acosf();
- *
- * y = acosf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns radian angle between -pi/2 and +pi/2 whose cosine
- * is x.
- *
- * Analytically, acos(x) = pi/2 - asin(x). However if |x| is
- * near 1, there is cancellation error in subtracting asin(x)
- * from pi/2. Hence if x < -0.5,
- *
- * acos(x) = pi - 2.0 * asin( sqrt((1+x)/2) );
- *
- * or if x > +0.5,
- *
- * acos(x) = 2.0 * asin( sqrt((1-x)/2) ).
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -1, 1 100000 1.4e-7 4.2e-8
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * acosf domain |x| > 1 0.0
- */
-
-/* asinhf.c
- *
- * Inverse hyperbolic sine
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, asinhf();
- *
- * y = asinhf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns inverse hyperbolic sine of argument.
- *
- * If |x| < 0.5, the function is approximated by a rational
- * form x + x**3 P(x)/Q(x). Otherwise,
- *
- * asinh(x) = log( x + sqrt(1 + x*x) ).
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -3,3 100000 2.4e-7 4.1e-8
- *
- */
-
-/* atanf.c
- *
- * Inverse circular tangent
- * (arctangent)
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, atanf();
- *
- * y = atanf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns radian angle between -pi/2 and +pi/2 whose tangent
- * is x.
- *
- * Range reduction is from four intervals into the interval
- * from zero to tan( pi/8 ). A polynomial approximates
- * the function in this basic interval.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -10, 10 100000 1.9e-7 4.1e-8
- *
- */
- /* atan2f()
- *
- * Quadrant correct inverse circular tangent
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, z, atan2f();
- *
- * z = atan2f( y, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns radian angle whose tangent is y/x.
- * Define compile time symbol ANSIC = 1 for ANSI standard,
- * range -PI < z <= +PI, args (y,x); else ANSIC = 0 for range
- * 0 to 2PI, args (x,y).
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -10, 10 100000 1.9e-7 4.1e-8
- * See atan.c.
- *
- */
-
-/* atanhf.c
- *
- * Inverse hyperbolic tangent
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, atanhf();
- *
- * y = atanhf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns inverse hyperbolic tangent of argument in the range
- * MINLOGF to MAXLOGF.
- *
- * If |x| < 0.5, a polynomial approximation is used.
- * Otherwise,
- * atanh(x) = 0.5 * log( (1+x)/(1-x) ).
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -1,1 100000 1.4e-7 3.1e-8
- *
- */
-
-/* bdtrf.c
- *
- * Binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * float p, y, bdtrf();
- *
- * y = bdtrf( k, n, p );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the sum of the terms 0 through k of the Binomial
- * probability density:
- *
- * k
- * -- ( n ) j n-j
- * > ( ) p (1-p)
- * -- ( j )
- * j=0
- *
- * The terms are not summed directly; instead the incomplete
- * beta integral is employed, according to the formula
- *
- * y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ).
- *
- * The arguments must be positive, with p ranging from 0 to 1.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error (p varies from 0 to 1):
- * arithmetic domain # trials peak rms
- * IEEE 0,100 2000 6.9e-5 1.1e-5
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * bdtrf domain k < 0 0.0
- * n < k
- * x < 0, x > 1
- *
- */
- /* bdtrcf()
- *
- * Complemented binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * float p, y, bdtrcf();
- *
- * y = bdtrcf( k, n, p );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the sum of the terms k+1 through n of the Binomial
- * probability density:
- *
- * n
- * -- ( n ) j n-j
- * > ( ) p (1-p)
- * -- ( j )
- * j=k+1
- *
- * The terms are not summed directly; instead the incomplete
- * beta integral is employed, according to the formula
- *
- * y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ).
- *
- * The arguments must be positive, with p ranging from 0 to 1.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error (p varies from 0 to 1):
- * arithmetic domain # trials peak rms
- * IEEE 0,100 2000 6.0e-5 1.2e-5
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * bdtrcf domain x<0, x>1, n<k 0.0
- */
- /* bdtrif()
- *
- * Inverse binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * float p, y, bdtrif();
- *
- * p = bdtrf( k, n, y );
- *
- *
- *
- * DESCRIPTION:
- *
- * Finds the event probability p such that the sum of the
- * terms 0 through k of the Binomial probability density
- * is equal to the given cumulative probability y.
- *
- * This is accomplished using the inverse beta integral
- * function and the relation
- *
- * 1 - p = incbi( n-k, k+1, y ).
- *
- *
- *
- *
- * ACCURACY:
- *
- * Relative error (p varies from 0 to 1):
- * arithmetic domain # trials peak rms
- * IEEE 0,100 2000 3.5e-5 3.3e-6
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * bdtrif domain k < 0, n <= k 0.0
- * x < 0, x > 1
- *
- */
-
-/* betaf.c
- *
- * Beta function
- *
- *
- *
- * SYNOPSIS:
- *
- * float a, b, y, betaf();
- *
- * y = betaf( a, b );
- *
- *
- *
- * DESCRIPTION:
- *
- * - -
- * | (a) | (b)
- * beta( a, b ) = -----------.
- * -
- * | (a+b)
- *
- * For large arguments the logarithm of the function is
- * evaluated using lgam(), then exponentiated.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,30 10000 4.0e-5 6.0e-6
- * IEEE -20,0 10000 4.9e-3 5.4e-5
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * betaf overflow log(beta) > MAXLOG 0.0
- * a or b <0 integer 0.0
- *
- */
-
-/* cbrtf.c
- *
- * Cube root
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, cbrtf();
- *
- * y = cbrtf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the cube root of the argument, which may be negative.
- *
- * Range reduction involves determining the power of 2 of
- * the argument. A polynomial of degree 2 applied to the
- * mantissa, and multiplication by the cube root of 1, 2, or 4
- * approximates the root to within about 0.1%. Then Newton's
- * iteration is used to converge to an accurate result.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,1e38 100000 7.6e-8 2.7e-8
- *
- */
-
-/* chbevlf.c
- *
- * Evaluate Chebyshev series
- *
- *
- *
- * SYNOPSIS:
- *
- * int N;
- * float x, y, coef[N], chebevlf();
- *
- * y = chbevlf( x, coef, N );
- *
- *
- *
- * DESCRIPTION:
- *
- * Evaluates the series
- *
- * N-1
- * - '
- * y = > coef[i] T (x/2)
- * - i
- * i=0
- *
- * of Chebyshev polynomials Ti at argument x/2.
- *
- * Coefficients are stored in reverse order, i.e. the zero
- * order term is last in the array. Note N is the number of
- * coefficients, not the order.
- *
- * If coefficients are for the interval a to b, x must
- * have been transformed to x -> 2(2x - b - a)/(b-a) before
- * entering the routine. This maps x from (a, b) to (-1, 1),
- * over which the Chebyshev polynomials are defined.
- *
- * If the coefficients are for the inverted interval, in
- * which (a, b) is mapped to (1/b, 1/a), the transformation
- * required is x -> 2(2ab/x - b - a)/(b-a). If b is infinity,
- * this becomes x -> 4a/x - 1.
- *
- *
- *
- * SPEED:
- *
- * Taking advantage of the recurrence properties of the
- * Chebyshev polynomials, the routine requires one more
- * addition per loop than evaluating a nested polynomial of
- * the same degree.
- *
- */
-
-/* chdtrf.c
- *
- * Chi-square distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * float df, x, y, chdtrf();
- *
- * y = chdtrf( df, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the area under the left hand tail (from 0 to x)
- * of the Chi square probability density function with
- * v degrees of freedom.
- *
- *
- * inf.
- * -
- * 1 | | v/2-1 -t/2
- * P( x | v ) = ----------- | t e dt
- * v/2 - | |
- * 2 | (v/2) -
- * x
- *
- * where x is the Chi-square variable.
- *
- * The incomplete gamma integral is used, according to the
- * formula
- *
- * y = chdtr( v, x ) = igam( v/2.0, x/2.0 ).
- *
- *
- * The arguments must both be positive.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,100 5000 3.2e-5 5.0e-6
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * chdtrf domain x < 0 or v < 1 0.0
- */
- /* chdtrcf()
- *
- * Complemented Chi-square distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * float v, x, y, chdtrcf();
- *
- * y = chdtrcf( v, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the area under the right hand tail (from x to
- * infinity) of the Chi square probability density function
- * with v degrees of freedom:
- *
- *
- * inf.
- * -
- * 1 | | v/2-1 -t/2
- * P( x | v ) = ----------- | t e dt
- * v/2 - | |
- * 2 | (v/2) -
- * x
- *
- * where x is the Chi-square variable.
- *
- * The incomplete gamma integral is used, according to the
- * formula
- *
- * y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ).
- *
- *
- * The arguments must both be positive.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,100 5000 2.7e-5 3.2e-6
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * chdtrc domain x < 0 or v < 1 0.0
- */
- /* chdtrif()
- *
- * Inverse of complemented Chi-square distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * float df, x, y, chdtrif();
- *
- * x = chdtrif( df, y );
- *
- *
- *
- *
- * DESCRIPTION:
- *
- * Finds the Chi-square argument x such that the integral
- * from x to infinity of the Chi-square density is equal
- * to the given cumulative probability y.
- *
- * This is accomplished using the inverse gamma integral
- * function and the relation
- *
- * x/2 = igami( df/2, y );
- *
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,100 10000 2.2e-5 8.5e-7
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * chdtri domain y < 0 or y > 1 0.0
- * v < 1
- *
- */
-
-/* clogf.c
- *
- * Complex natural logarithm
- *
- *
- *
- * SYNOPSIS:
- *
- * void clogf();
- * cmplxf z, w;
- *
- * clogf( &z, &w );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns complex logarithm to the base e (2.718...) of
- * the complex argument x.
- *
- * If z = x + iy, r = sqrt( x**2 + y**2 ),
- * then
- * w = log(r) + i arctan(y/x).
- *
- * The arctangent ranges from -PI to +PI.
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -10,+10 30000 1.9e-6 6.2e-8
- *
- * Larger relative error can be observed for z near 1 +i0.
- * In IEEE arithmetic the peak absolute error is 3.1e-7.
- *
- */
- /* cexpf()
- *
- * Complex exponential function
- *
- *
- *
- * SYNOPSIS:
- *
- * void cexpf();
- * cmplxf z, w;
- *
- * cexpf( &z, &w );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the exponential of the complex argument z
- * into the complex result w.
- *
- * If
- * z = x + iy,
- * r = exp(x),
- *
- * then
- *
- * w = r cos y + i r sin y.
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -10,+10 30000 1.4e-7 4.5e-8
- *
- */
- /* csinf()
- *
- * Complex circular sine
- *
- *
- *
- * SYNOPSIS:
- *
- * void csinf();
- * cmplxf z, w;
- *
- * csinf( &z, &w );
- *
- *
- *
- * DESCRIPTION:
- *
- * If
- * z = x + iy,
- *
- * then
- *
- * w = sin x cosh y + i cos x sinh y.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -10,+10 30000 1.9e-7 5.5e-8
- *
- */
- /* ccosf()
- *
- * Complex circular cosine
- *
- *
- *
- * SYNOPSIS:
- *
- * void ccosf();
- * cmplxf z, w;
- *
- * ccosf( &z, &w );
- *
- *
- *
- * DESCRIPTION:
- *
- * If
- * z = x + iy,
- *
- * then
- *
- * w = cos x cosh y - i sin x sinh y.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -10,+10 30000 1.8e-7 5.5e-8
- */
- /* ctanf()
- *
- * Complex circular tangent
- *
- *
- *
- * SYNOPSIS:
- *
- * void ctanf();
- * cmplxf z, w;
- *
- * ctanf( &z, &w );
- *
- *
- *
- * DESCRIPTION:
- *
- * If
- * z = x + iy,
- *
- * then
- *
- * sin 2x + i sinh 2y
- * w = --------------------.
- * cos 2x + cosh 2y
- *
- * On the real axis the denominator is zero at odd multiples
- * of PI/2. The denominator is evaluated by its Taylor
- * series near these points.
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -10,+10 30000 3.3e-7 5.1e-8
- */
- /* ccotf()
- *
- * Complex circular cotangent
- *
- *
- *
- * SYNOPSIS:
- *
- * void ccotf();
- * cmplxf z, w;
- *
- * ccotf( &z, &w );
- *
- *
- *
- * DESCRIPTION:
- *
- * If
- * z = x + iy,
- *
- * then
- *
- * sin 2x - i sinh 2y
- * w = --------------------.
- * cosh 2y - cos 2x
- *
- * On the real axis, the denominator has zeros at even
- * multiples of PI/2. Near these points it is evaluated
- * by a Taylor series.
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -10,+10 30000 3.6e-7 5.7e-8
- * Also tested by ctan * ccot = 1 + i0.
- */
- /* casinf()
- *
- * Complex circular arc sine
- *
- *
- *
- * SYNOPSIS:
- *
- * void casinf();
- * cmplxf z, w;
- *
- * casinf( &z, &w );
- *
- *
- *
- * DESCRIPTION:
- *
- * Inverse complex sine:
- *
- * 2
- * w = -i clog( iz + csqrt( 1 - z ) ).
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -10,+10 30000 1.1e-5 1.5e-6
- * Larger relative error can be observed for z near zero.
- *
- */
- /* cacosf()
- *
- * Complex circular arc cosine
- *
- *
- *
- * SYNOPSIS:
- *
- * void cacosf();
- * cmplxf z, w;
- *
- * cacosf( &z, &w );
- *
- *
- *
- * DESCRIPTION:
- *
- *
- * w = arccos z = PI/2 - arcsin z.
- *
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -10,+10 30000 9.2e-6 1.2e-6
- *
- */
- /* catan()
- *
- * Complex circular arc tangent
- *
- *
- *
- * SYNOPSIS:
- *
- * void catan();
- * cmplxf z, w;
- *
- * catan( &z, &w );
- *
- *
- *
- * DESCRIPTION:
- *
- * If
- * z = x + iy,
- *
- * then
- * 1 ( 2x )
- * Re w = - arctan(-----------) + k PI
- * 2 ( 2 2)
- * (1 - x - y )
- *
- * ( 2 2)
- * 1 (x + (y+1) )
- * Im w = - log(------------)
- * 4 ( 2 2)
- * (x + (y-1) )
- *
- * Where k is an arbitrary integer.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -10,+10 30000 2.3e-6 5.2e-8
- *
- */
-
-/* cmplxf.c
- *
- * Complex number arithmetic
- *
- *
- *
- * SYNOPSIS:
- *
- * typedef struct {
- * float r; real part
- * float i; imaginary part
- * }cmplxf;
- *
- * cmplxf *a, *b, *c;
- *
- * caddf( a, b, c ); c = b + a
- * csubf( a, b, c ); c = b - a
- * cmulf( a, b, c ); c = b * a
- * cdivf( a, b, c ); c = b / a
- * cnegf( c ); c = -c
- * cmovf( b, c ); c = b
- *
- *
- *
- * DESCRIPTION:
- *
- * Addition:
- * c.r = b.r + a.r
- * c.i = b.i + a.i
- *
- * Subtraction:
- * c.r = b.r - a.r
- * c.i = b.i - a.i
- *
- * Multiplication:
- * c.r = b.r * a.r - b.i * a.i
- * c.i = b.r * a.i + b.i * a.r
- *
- * Division:
- * d = a.r * a.r + a.i * a.i
- * c.r = (b.r * a.r + b.i * a.i)/d
- * c.i = (b.i * a.r - b.r * a.i)/d
- * ACCURACY:
- *
- * In DEC arithmetic, the test (1/z) * z = 1 had peak relative
- * error 3.1e-17, rms 1.2e-17. The test (y/z) * (z/y) = 1 had
- * peak relative error 8.3e-17, rms 2.1e-17.
- *
- * Tests in the rectangle {-10,+10}:
- * Relative error:
- * arithmetic function # trials peak rms
- * IEEE cadd 30000 5.9e-8 2.6e-8
- * IEEE csub 30000 6.0e-8 2.6e-8
- * IEEE cmul 30000 1.1e-7 3.7e-8
- * IEEE cdiv 30000 2.1e-7 5.7e-8
- */
-
-/* cabsf()
- *
- * Complex absolute value
- *
- *
- *
- * SYNOPSIS:
- *
- * float cabsf();
- * cmplxf z;
- * float a;
- *
- * a = cabsf( &z );
- *
- *
- *
- * DESCRIPTION:
- *
- *
- * If z = x + iy
- *
- * then
- *
- * a = sqrt( x**2 + y**2 ).
- *
- * Overflow and underflow are avoided by testing the magnitudes
- * of x and y before squaring. If either is outside half of
- * the floating point full scale range, both are rescaled.
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -10,+10 30000 1.2e-7 3.4e-8
- */
- /* csqrtf()
- *
- * Complex square root
- *
- *
- *
- * SYNOPSIS:
- *
- * void csqrtf();
- * cmplxf z, w;
- *
- * csqrtf( &z, &w );
- *
- *
- *
- * DESCRIPTION:
- *
- *
- * If z = x + iy, r = |z|, then
- *
- * 1/2
- * Im w = [ (r - x)/2 ] ,
- *
- * Re w = y / 2 Im w.
- *
- *
- * Note that -w is also a square root of z. The solution
- * reported is always in the upper half plane.
- *
- * Because of the potential for cancellation error in r - x,
- * the result is sharpened by doing a Heron iteration
- * (see sqrt.c) in complex arithmetic.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -10,+10 100000 1.8e-7 4.2e-8
- *
- */
-
-/* coshf.c
- *
- * Hyperbolic cosine
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, coshf();
- *
- * y = coshf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns hyperbolic cosine of argument in the range MINLOGF to
- * MAXLOGF.
- *
- * cosh(x) = ( exp(x) + exp(-x) )/2.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE +-MAXLOGF 100000 1.2e-7 2.8e-8
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * coshf overflow |x| > MAXLOGF MAXNUMF
- *
- *
- */
-
-/* dawsnf.c
- *
- * Dawson's Integral
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, dawsnf();
- *
- * y = dawsnf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Approximates the integral
- *
- * x
- * -
- * 2 | | 2
- * dawsn(x) = exp( -x ) | exp( t ) dt
- * | |
- * -
- * 0
- *
- * Three different rational approximations are employed, for
- * the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up.
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,10 50000 4.4e-7 6.3e-8
- *
- *
- */
-
-/* ellief.c
- *
- * Incomplete elliptic integral of the second kind
- *
- *
- *
- * SYNOPSIS:
- *
- * float phi, m, y, ellief();
- *
- * y = ellief( phi, m );
- *
- *
- *
- * DESCRIPTION:
- *
- * Approximates the integral
- *
- *
- * phi
- * -
- * | |
- * | 2
- * E(phi\m) = | sqrt( 1 - m sin t ) dt
- * |
- * | |
- * -
- * 0
- *
- * of amplitude phi and modulus m, using the arithmetic -
- * geometric mean algorithm.
- *
- *
- *
- * ACCURACY:
- *
- * Tested at random arguments with phi in [0, 2] and m in
- * [0, 1].
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,2 10000 4.5e-7 7.4e-8
- *
- *
- */
-
-/* ellikf.c
- *
- * Incomplete elliptic integral of the first kind
- *
- *
- *
- * SYNOPSIS:
- *
- * float phi, m, y, ellikf();
- *
- * y = ellikf( phi, m );
- *
- *
- *
- * DESCRIPTION:
- *
- * Approximates the integral
- *
- *
- *
- * phi
- * -
- * | |
- * | dt
- * F(phi\m) = | ------------------
- * | 2
- * | | sqrt( 1 - m sin t )
- * -
- * 0
- *
- * of amplitude phi and modulus m, using the arithmetic -
- * geometric mean algorithm.
- *
- *
- *
- *
- * ACCURACY:
- *
- * Tested at random points with phi in [0, 2] and m in
- * [0, 1].
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,2 10000 2.9e-7 5.8e-8
- *
- *
- */
-
-/* ellpef.c
- *
- * Complete elliptic integral of the second kind
- *
- *
- *
- * SYNOPSIS:
- *
- * float m1, y, ellpef();
- *
- * y = ellpef( m1 );
- *
- *
- *
- * DESCRIPTION:
- *
- * Approximates the integral
- *
- *
- * pi/2
- * -
- * | | 2
- * E(m) = | sqrt( 1 - m sin t ) dt
- * | |
- * -
- * 0
- *
- * Where m = 1 - m1, using the approximation
- *
- * P(x) - x log x Q(x).
- *
- * Though there are no singularities, the argument m1 is used
- * rather than m for compatibility with ellpk().
- *
- * E(1) = 1; E(0) = pi/2.
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0, 1 30000 1.1e-7 3.9e-8
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * ellpef domain x<0, x>1 0.0
- *
- */
-
-/* ellpjf.c
- *
- * Jacobian Elliptic Functions
- *
- *
- *
- * SYNOPSIS:
- *
- * float u, m, sn, cn, dn, phi;
- * int ellpj();
- *
- * ellpj( u, m, _&sn, _&cn, _&dn, _&phi );
- *
- *
- *
- * DESCRIPTION:
- *
- *
- * Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m),
- * and dn(u|m) of parameter m between 0 and 1, and real
- * argument u.
- *
- * These functions are periodic, with quarter-period on the
- * real axis equal to the complete elliptic integral
- * ellpk(1.0-m).
- *
- * Relation to incomplete elliptic integral:
- * If u = ellik(phi,m), then sn(u|m) = sin(phi),
- * and cn(u|m) = cos(phi). Phi is called the amplitude of u.
- *
- * Computation is by means of the arithmetic-geometric mean
- * algorithm, except when m is within 1e-9 of 0 or 1. In the
- * latter case with m close to 1, the approximation applies
- * only for phi < pi/2.
- *
- * ACCURACY:
- *
- * Tested at random points with u between 0 and 10, m between
- * 0 and 1.
- *
- * Absolute error (* = relative error):
- * arithmetic function # trials peak rms
- * IEEE sn 10000 1.7e-6 2.2e-7
- * IEEE cn 10000 1.6e-6 2.2e-7
- * IEEE dn 10000 1.4e-3 1.9e-5
- * IEEE phi 10000 3.9e-7* 6.7e-8*
- *
- * Peak error observed in consistency check using addition
- * theorem for sn(u+v) was 4e-16 (absolute). Also tested by
- * the above relation to the incomplete elliptic integral.
- * Accuracy deteriorates when u is large.
- *
- */
-
-/* ellpkf.c
- *
- * Complete elliptic integral of the first kind
- *
- *
- *
- * SYNOPSIS:
- *
- * float m1, y, ellpkf();
- *
- * y = ellpkf( m1 );
- *
- *
- *
- * DESCRIPTION:
- *
- * Approximates the integral
- *
- *
- *
- * pi/2
- * -
- * | |
- * | dt
- * K(m) = | ------------------
- * | 2
- * | | sqrt( 1 - m sin t )
- * -
- * 0
- *
- * where m = 1 - m1, using the approximation
- *
- * P(x) - log x Q(x).
- *
- * The argument m1 is used rather than m so that the logarithmic
- * singularity at m = 1 will be shifted to the origin; this
- * preserves maximum accuracy.
- *
- * K(0) = pi/2.
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,1 30000 1.3e-7 3.4e-8
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * ellpkf domain x<0, x>1 0.0
- *
- */
-
-/* exp10f.c
- *
- * Base 10 exponential function
- * (Common antilogarithm)
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, exp10f();
- *
- * y = exp10f( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns 10 raised to the x power.
- *
- * Range reduction is accomplished by expressing the argument
- * as 10**x = 2**n 10**f, with |f| < 0.5 log10(2).
- * A polynomial approximates 10**f.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -38,+38 100000 9.8e-8 2.8e-8
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * exp10 underflow x < -MAXL10 0.0
- * exp10 overflow x > MAXL10 MAXNUM
- *
- * IEEE single arithmetic: MAXL10 = 38.230809449325611792.
- *
- */
-
-/* exp2f.c
- *
- * Base 2 exponential function
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, exp2f();
- *
- * y = exp2f( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns 2 raised to the x power.
- *
- * Range reduction is accomplished by separating the argument
- * into an integer k and fraction f such that
- * x k f
- * 2 = 2 2.
- *
- * A polynomial approximates 2**x in the basic range [-0.5, 0.5].
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -127,+127 100000 1.7e-7 2.8e-8
- *
- *
- * See exp.c for comments on error amplification.
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * exp underflow x < -MAXL2 0.0
- * exp overflow x > MAXL2 MAXNUMF
- *
- * For IEEE arithmetic, MAXL2 = 127.
- */
-
-/* expf.c
- *
- * Exponential function
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, expf();
- *
- * y = expf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns e (2.71828...) raised to the x power.
- *
- * Range reduction is accomplished by separating the argument
- * into an integer k and fraction f such that
- *
- * x k f
- * e = 2 e.
- *
- * A polynomial is used to approximate exp(f)
- * in the basic range [-0.5, 0.5].
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE +- MAXLOG 100000 1.7e-7 2.8e-8
- *
- *
- * Error amplification in the exponential function can be
- * a serious matter. The error propagation involves
- * exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ),
- * which shows that a 1 lsb error in representing X produces
- * a relative error of X times 1 lsb in the function.
- * While the routine gives an accurate result for arguments
- * that are exactly represented by a double precision
- * computer number, the result contains amplified roundoff
- * error for large arguments not exactly represented.
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * expf underflow x < MINLOGF 0.0
- * expf overflow x > MAXLOGF MAXNUMF
- *
- */
-
-/* expnf.c
- *
- * Exponential integral En
- *
- *
- *
- * SYNOPSIS:
- *
- * int n;
- * float x, y, expnf();
- *
- * y = expnf( n, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Evaluates the exponential integral
- *
- * inf.
- * -
- * | | -xt
- * | e
- * E (x) = | ---- dt.
- * n | n
- * | | t
- * -
- * 1
- *
- *
- * Both n and x must be nonnegative.
- *
- * The routine employs either a power series, a continued
- * fraction, or an asymptotic formula depending on the
- * relative values of n and x.
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0, 30 10000 5.6e-7 1.2e-7
- *
- */
-
-/* facf.c
- *
- * Factorial function
- *
- *
- *
- * SYNOPSIS:
- *
- * float y, facf();
- * int i;
- *
- * y = facf( i );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns factorial of i = 1 * 2 * 3 * ... * i.
- * fac(0) = 1.0.
- *
- * Due to machine arithmetic bounds the largest value of
- * i accepted is 33 in single precision arithmetic.
- * Greater values, or negative ones,
- * produce an error message and return MAXNUM.
- *
- *
- *
- * ACCURACY:
- *
- * For i < 34 the values are simply tabulated, and have
- * full machine accuracy.
- *
- */
-
-/* fdtrf.c
- *
- * F distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int df1, df2;
- * float x, y, fdtrf();
- *
- * y = fdtrf( df1, df2, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the area from zero to x under the F density
- * function (also known as Snedcor's density or the
- * variance ratio density). This is the density
- * of x = (u1/df1)/(u2/df2), where u1 and u2 are random
- * variables having Chi square distributions with df1
- * and df2 degrees of freedom, respectively.
- *
- * The incomplete beta integral is used, according to the
- * formula
- *
- * P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ).
- *
- *
- * The arguments a and b are greater than zero, and x
- * x is nonnegative.
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,100 5000 2.2e-5 1.1e-6
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * fdtrf domain a<0, b<0, x<0 0.0
- *
- */
- /* fdtrcf()
- *
- * Complemented F distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int df1, df2;
- * float x, y, fdtrcf();
- *
- * y = fdtrcf( df1, df2, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the area from x to infinity under the F density
- * function (also known as Snedcor's density or the
- * variance ratio density).
- *
- *
- * inf.
- * -
- * 1 | | a-1 b-1
- * 1-P(x) = ------ | t (1-t) dt
- * B(a,b) | |
- * -
- * x
- *
- * (See fdtr.c.)
- *
- * The incomplete beta integral is used, according to the
- * formula
- *
- * P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ).
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,100 5000 7.3e-5 1.2e-5
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * fdtrcf domain a<0, b<0, x<0 0.0
- *
- */
- /* fdtrif()
- *
- * Inverse of complemented F distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * float df1, df2, x, y, fdtrif();
- *
- * x = fdtrif( df1, df2, y );
- *
- *
- *
- *
- * DESCRIPTION:
- *
- * Finds the F density argument x such that the integral
- * from x to infinity of the F density is equal to the
- * given probability y.
- *
- * This is accomplished using the inverse beta integral
- * function and the relations
- *
- * z = incbi( df2/2, df1/2, y )
- * x = df2 (1-z) / (df1 z).
- *
- * Note: the following relations hold for the inverse of
- * the uncomplemented F distribution:
- *
- * z = incbi( df1/2, df2/2, y )
- * x = df2 z / (df1 (1-z)).
- *
- *
- *
- * ACCURACY:
- *
- * arithmetic domain # trials peak rms
- * Absolute error:
- * IEEE 0,100 5000 4.0e-5 3.2e-6
- * Relative error:
- * IEEE 0,100 5000 1.2e-3 1.8e-5
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * fdtrif domain y <= 0 or y > 1 0.0
- * v < 1
- *
- */
-
-/* ceilf()
- * floorf()
- * frexpf()
- * ldexpf()
- *
- * Single precision floating point numeric utilities
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y;
- * float ceilf(), floorf(), frexpf(), ldexpf();
- * int expnt, n;
- *
- * y = floorf(x);
- * y = ceilf(x);
- * y = frexpf( x, &expnt );
- * y = ldexpf( x, n );
- *
- *
- *
- * DESCRIPTION:
- *
- * All four routines return a single precision floating point
- * result.
- *
- * sfloor() returns the largest integer less than or equal to x.
- * It truncates toward minus infinity.
- *
- * sceil() returns the smallest integer greater than or equal
- * to x. It truncates toward plus infinity.
- *
- * sfrexp() extracts the exponent from x. It returns an integer
- * power of two to expnt and the significand between 0.5 and 1
- * to y. Thus x = y * 2**expn.
- *
- * sldexp() multiplies x by 2**n.
- *
- * These functions are part of the standard C run time library
- * for many but not all C compilers. The ones supplied are
- * written in C for either DEC or IEEE arithmetic. They should
- * be used only if your compiler library does not already have
- * them.
- *
- * The IEEE versions assume that denormal numbers are implemented
- * in the arithmetic. Some modifications will be required if
- * the arithmetic has abrupt rather than gradual underflow.
- */
-
-/* fresnlf.c
- *
- * Fresnel integral
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, S, C;
- * void fresnlf();
- *
- * fresnlf( x, _&S, _&C );
- *
- *
- * DESCRIPTION:
- *
- * Evaluates the Fresnel integrals
- *
- * x
- * -
- * | |
- * C(x) = | cos(pi/2 t**2) dt,
- * | |
- * -
- * 0
- *
- * x
- * -
- * | |
- * S(x) = | sin(pi/2 t**2) dt.
- * | |
- * -
- * 0
- *
- *
- * The integrals are evaluated by power series for small x.
- * For x >= 1 auxiliary functions f(x) and g(x) are employed
- * such that
- *
- * C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 )
- * S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 )
- *
- *
- *
- * ACCURACY:
- *
- * Relative error.
- *
- * Arithmetic function domain # trials peak rms
- * IEEE S(x) 0, 10 30000 1.1e-6 1.9e-7
- * IEEE C(x) 0, 10 30000 1.1e-6 2.0e-7
- */
-
-/* gammaf.c
- *
- * Gamma function
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, gammaf();
- * extern int sgngamf;
- *
- * y = gammaf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns gamma function of the argument. The result is
- * correctly signed, and the sign (+1 or -1) is also
- * returned in a global (extern) variable named sgngamf.
- * This same variable is also filled in by the logarithmic
- * gamma function lgam().
- *
- * Arguments between 0 and 10 are reduced by recurrence and the
- * function is approximated by a polynomial function covering
- * the interval (2,3). Large arguments are handled by Stirling's
- * formula. Negative arguments are made positive using
- * a reflection formula.
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,-33 100,000 5.7e-7 1.0e-7
- * IEEE -33,0 100,000 6.1e-7 1.2e-7
- *
- *
- */
-/* lgamf()
- *
- * Natural logarithm of gamma function
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, lgamf();
- * extern int sgngamf;
- *
- * y = lgamf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the base e (2.718...) logarithm of the absolute
- * value of the gamma function of the argument.
- * The sign (+1 or -1) of the gamma function is returned in a
- * global (extern) variable named sgngamf.
- *
- * For arguments greater than 6.5, the logarithm of the gamma
- * function is approximated by the logarithmic version of
- * Stirling's formula. Arguments between 0 and +6.5 are reduced by
- * by recurrence to the interval [.75,1.25] or [1.5,2.5] of a rational
- * approximation. The cosecant reflection formula is employed for
- * arguments less than zero.
- *
- * Arguments greater than MAXLGM = 2.035093e36 return MAXNUM and an
- * error message.
- *
- *
- *
- * ACCURACY:
- *
- *
- *
- * arithmetic domain # trials peak rms
- * IEEE -100,+100 500,000 7.4e-7 6.8e-8
- * The error criterion was relative when the function magnitude
- * was greater than one but absolute when it was less than one.
- * The routine has low relative error for positive arguments.
- *
- * The following test used the relative error criterion.
- * IEEE -2, +3 100000 4.0e-7 5.6e-8
- *
- */
-
-/* gdtrf.c
- *
- * Gamma distribution function
- *
- *
- *
- * SYNOPSIS:
- *
- * float a, b, x, y, gdtrf();
- *
- * y = gdtrf( a, b, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the integral from zero to x of the gamma probability
- * density function:
- *
- *
- * x
- * b -
- * a | | b-1 -at
- * y = ----- | t e dt
- * - | |
- * | (b) -
- * 0
- *
- * The incomplete gamma integral is used, according to the
- * relation
- *
- * y = igam( b, ax ).
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,100 5000 5.8e-5 3.0e-6
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * gdtrf domain x < 0 0.0
- *
- */
- /* gdtrcf.c
- *
- * Complemented gamma distribution function
- *
- *
- *
- * SYNOPSIS:
- *
- * float a, b, x, y, gdtrcf();
- *
- * y = gdtrcf( a, b, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the integral from x to infinity of the gamma
- * probability density function:
- *
- *
- * inf.
- * b -
- * a | | b-1 -at
- * y = ----- | t e dt
- * - | |
- * | (b) -
- * x
- *
- * The incomplete gamma integral is used, according to the
- * relation
- *
- * y = igamc( b, ax ).
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,100 5000 9.1e-5 1.5e-5
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * gdtrcf domain x < 0 0.0
- *
- */
-
-/* hyp2f1f.c
- *
- * Gauss hypergeometric function F
- * 2 1
- *
- *
- * SYNOPSIS:
- *
- * float a, b, c, x, y, hyp2f1f();
- *
- * y = hyp2f1f( a, b, c, x );
- *
- *
- * DESCRIPTION:
- *
- *
- * hyp2f1( a, b, c, x ) = F ( a, b; c; x )
- * 2 1
- *
- * inf.
- * - a(a+1)...(a+k) b(b+1)...(b+k) k+1
- * = 1 + > ----------------------------- x .
- * - c(c+1)...(c+k) (k+1)!
- * k = 0
- *
- * Cases addressed are
- * Tests and escapes for negative integer a, b, or c
- * Linear transformation if c - a or c - b negative integer
- * Special case c = a or c = b
- * Linear transformation for x near +1
- * Transformation for x < -0.5
- * Psi function expansion if x > 0.5 and c - a - b integer
- * Conditionally, a recurrence on c to make c-a-b > 0
- *
- * |x| > 1 is rejected.
- *
- * The parameters a, b, c are considered to be integer
- * valued if they are within 1.0e-6 of the nearest integer.
- *
- * ACCURACY:
- *
- * Relative error (-1 < x < 1):
- * arithmetic domain # trials peak rms
- * IEEE 0,3 30000 5.8e-4 4.3e-6
- */
-
-/* hypergf.c
- *
- * Confluent hypergeometric function
- *
- *
- *
- * SYNOPSIS:
- *
- * float a, b, x, y, hypergf();
- *
- * y = hypergf( a, b, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Computes the confluent hypergeometric function
- *
- * 1 2
- * a x a(a+1) x
- * F ( a,b;x ) = 1 + ---- + --------- + ...
- * 1 1 b 1! b(b+1) 2!
- *
- * Many higher transcendental functions are special cases of
- * this power series.
- *
- * As is evident from the formula, b must not be a negative
- * integer or zero unless a is an integer with 0 >= a > b.
- *
- * The routine attempts both a direct summation of the series
- * and an asymptotic expansion. In each case error due to
- * roundoff, cancellation, and nonconvergence is estimated.
- * The result with smaller estimated error is returned.
- *
- *
- *
- * ACCURACY:
- *
- * Tested at random points (a, b, x), all three variables
- * ranging from 0 to 30.
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,5 10000 6.6e-7 1.3e-7
- * IEEE 0,30 30000 1.1e-5 6.5e-7
- *
- * Larger errors can be observed when b is near a negative
- * integer or zero. Certain combinations of arguments yield
- * serious cancellation error in the power series summation
- * and also are not in the region of near convergence of the
- * asymptotic series. An error message is printed if the
- * self-estimated relative error is greater than 1.0e-3.
- *
- */
-
-/* i0f.c
- *
- * Modified Bessel function of order zero
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, i0();
- *
- * y = i0f( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns modified Bessel function of order zero of the
- * argument.
- *
- * The function is defined as i0(x) = j0( ix ).
- *
- * The range is partitioned into the two intervals [0,8] and
- * (8, infinity). Chebyshev polynomial expansions are employed
- * in each interval.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,30 100000 4.0e-7 7.9e-8
- *
- */
- /* i0ef.c
- *
- * Modified Bessel function of order zero,
- * exponentially scaled
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, i0ef();
- *
- * y = i0ef( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns exponentially scaled modified Bessel function
- * of order zero of the argument.
- *
- * The function is defined as i0e(x) = exp(-|x|) j0( ix ).
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,30 100000 3.7e-7 7.0e-8
- * See i0f().
- *
- */
-
-/* i1f.c
- *
- * Modified Bessel function of order one
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, i1f();
- *
- * y = i1f( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns modified Bessel function of order one of the
- * argument.
- *
- * The function is defined as i1(x) = -i j1( ix ).
- *
- * The range is partitioned into the two intervals [0,8] and
- * (8, infinity). Chebyshev polynomial expansions are employed
- * in each interval.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0, 30 100000 1.5e-6 1.6e-7
- *
- *
- */
- /* i1ef.c
- *
- * Modified Bessel function of order one,
- * exponentially scaled
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, i1ef();
- *
- * y = i1ef( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns exponentially scaled modified Bessel function
- * of order one of the argument.
- *
- * The function is defined as i1(x) = -i exp(-|x|) j1( ix ).
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0, 30 30000 1.5e-6 1.5e-7
- * See i1().
- *
- */
-
-/* igamf.c
- *
- * Incomplete gamma integral
- *
- *
- *
- * SYNOPSIS:
- *
- * float a, x, y, igamf();
- *
- * y = igamf( a, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * The function is defined by
- *
- * x
- * -
- * 1 | | -t a-1
- * igam(a,x) = ----- | e t dt.
- * - | |
- * | (a) -
- * 0
- *
- *
- * In this implementation both arguments must be positive.
- * The integral is evaluated by either a power series or
- * continued fraction expansion, depending on the relative
- * values of a and x.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,30 20000 7.8e-6 5.9e-7
- *
- */
- /* igamcf()
- *
- * Complemented incomplete gamma integral
- *
- *
- *
- * SYNOPSIS:
- *
- * float a, x, y, igamcf();
- *
- * y = igamcf( a, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * The function is defined by
- *
- *
- * igamc(a,x) = 1 - igam(a,x)
- *
- * inf.
- * -
- * 1 | | -t a-1
- * = ----- | e t dt.
- * - | |
- * | (a) -
- * x
- *
- *
- * In this implementation both arguments must be positive.
- * The integral is evaluated by either a power series or
- * continued fraction expansion, depending on the relative
- * values of a and x.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,30 30000 7.8e-6 5.9e-7
- *
- */
-
-/* igamif()
- *
- * Inverse of complemented imcomplete gamma integral
- *
- *
- *
- * SYNOPSIS:
- *
- * float a, x, y, igamif();
- *
- * x = igamif( a, y );
- *
- *
- *
- * DESCRIPTION:
- *
- * Given y, the function finds x such that
- *
- * igamc( a, x ) = y.
- *
- * Starting with the approximate value
- *
- * 3
- * x = a t
- *
- * where
- *
- * t = 1 - d - ndtri(y) sqrt(d)
- *
- * and
- *
- * d = 1/9a,
- *
- * the routine performs up to 10 Newton iterations to find the
- * root of igamc(a,x) - y = 0.
- *
- *
- * ACCURACY:
- *
- * Tested for a ranging from 0 to 100 and x from 0 to 1.
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,100 5000 1.0e-5 1.5e-6
- *
- */
-
-/* incbetf.c
- *
- * Incomplete beta integral
- *
- *
- * SYNOPSIS:
- *
- * float a, b, x, y, incbetf();
- *
- * y = incbetf( a, b, x );
- *
- *
- * DESCRIPTION:
- *
- * Returns incomplete beta integral of the arguments, evaluated
- * from zero to x. The function is defined as
- *
- * x
- * - -
- * | (a+b) | | a-1 b-1
- * ----------- | t (1-t) dt.
- * - - | |
- * | (a) | (b) -
- * 0
- *
- * The domain of definition is 0 <= x <= 1. In this
- * implementation a and b are restricted to positive values.
- * The integral from x to 1 may be obtained by the symmetry
- * relation
- *
- * 1 - incbet( a, b, x ) = incbet( b, a, 1-x ).
- *
- * The integral is evaluated by a continued fraction expansion.
- * If a < 1, the function calls itself recursively after a
- * transformation to increase a to a+1.
- *
- * ACCURACY:
- *
- * Tested at random points (a,b,x) with a and b in the indicated
- * interval and x between 0 and 1.
- *
- * arithmetic domain # trials peak rms
- * Relative error:
- * IEEE 0,30 10000 3.7e-5 5.1e-6
- * IEEE 0,100 10000 1.7e-4 2.5e-5
- * The useful domain for relative error is limited by underflow
- * of the single precision exponential function.
- * Absolute error:
- * IEEE 0,30 100000 2.2e-5 9.6e-7
- * IEEE 0,100 10000 6.5e-5 3.7e-6
- *
- * Larger errors may occur for extreme ratios of a and b.
- *
- * ERROR MESSAGES:
- * message condition value returned
- * incbetf domain x<0, x>1 0.0
- */
-
-/* incbif()
- *
- * Inverse of imcomplete beta integral
- *
- *
- *
- * SYNOPSIS:
- *
- * float a, b, x, y, incbif();
- *
- * x = incbif( a, b, y );
- *
- *
- *
- * DESCRIPTION:
- *
- * Given y, the function finds x such that
- *
- * incbet( a, b, x ) = y.
- *
- * the routine performs up to 10 Newton iterations to find the
- * root of incbet(a,b,x) - y = 0.
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * x a,b
- * arithmetic domain domain # trials peak rms
- * IEEE 0,1 0,100 5000 2.8e-4 8.3e-6
- *
- * Overflow and larger errors may occur for one of a or b near zero
- * and the other large.
- */
-
-/* ivf.c
- *
- * Modified Bessel function of noninteger order
- *
- *
- *
- * SYNOPSIS:
- *
- * float v, x, y, ivf();
- *
- * y = ivf( v, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns modified Bessel function of order v of the
- * argument. If x is negative, v must be integer valued.
- *
- * The function is defined as Iv(x) = Jv( ix ). It is
- * here computed in terms of the confluent hypergeometric
- * function, according to the formula
- *
- * v -x
- * Iv(x) = (x/2) e hyperg( v+0.5, 2v+1, 2x ) / gamma(v+1)
- *
- * If v is a negative integer, then v is replaced by -v.
- *
- *
- * ACCURACY:
- *
- * Tested at random points (v, x), with v between 0 and
- * 30, x between 0 and 28.
- * arithmetic domain # trials peak rms
- * Relative error:
- * IEEE 0,15 3000 4.7e-6 5.4e-7
- * Absolute error (relative when function > 1)
- * IEEE 0,30 5000 8.5e-6 1.3e-6
- *
- * Accuracy is diminished if v is near a negative integer.
- * The useful domain for relative error is limited by overflow
- * of the single precision exponential function.
- *
- * See also hyperg.c.
- *
- */
-
-/* j0f.c
- *
- * Bessel function of order zero
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, j0f();
- *
- * y = j0f( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of order zero of the argument.
- *
- * The domain is divided into the intervals [0, 2] and
- * (2, infinity). In the first interval the following polynomial
- * approximation is used:
- *
- *
- * 2 2 2
- * (w - r ) (w - r ) (w - r ) P(w)
- * 1 2 3
- *
- * 2
- * where w = x and the three r's are zeros of the function.
- *
- * In the second interval, the modulus and phase are approximated
- * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x)
- * and Phase(x) = x + 1/x R(1/x^2) - pi/4. The function is
- *
- * j0(x) = Modulus(x) cos( Phase(x) ).
- *
- *
- *
- * ACCURACY:
- *
- * Absolute error:
- * arithmetic domain # trials peak rms
- * IEEE 0, 2 100000 1.3e-7 3.6e-8
- * IEEE 2, 32 100000 1.9e-7 5.4e-8
- *
- */
- /* y0f.c
- *
- * Bessel function of the second kind, order zero
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, y0f();
- *
- * y = y0f( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of the second kind, of order
- * zero, of the argument.
- *
- * The domain is divided into the intervals [0, 2] and
- * (2, infinity). In the first interval a rational approximation
- * R(x) is employed to compute
- *
- * 2 2 2
- * y0(x) = (w - r ) (w - r ) (w - r ) R(x) + 2/pi ln(x) j0(x).
- * 1 2 3
- *
- * Thus a call to j0() is required. The three zeros are removed
- * from R(x) to improve its numerical stability.
- *
- * In the second interval, the modulus and phase are approximated
- * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x)
- * and Phase(x) = x + 1/x S(1/x^2) - pi/4. Then the function is
- *
- * y0(x) = Modulus(x) sin( Phase(x) ).
- *
- *
- *
- *
- * ACCURACY:
- *
- * Absolute error, when y0(x) < 1; else relative error:
- *
- * arithmetic domain # trials peak rms
- * IEEE 0, 2 100000 2.4e-7 3.4e-8
- * IEEE 2, 32 100000 1.8e-7 5.3e-8
- *
- */
-
-/* j1f.c
- *
- * Bessel function of order one
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, j1f();
- *
- * y = j1f( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of order one of the argument.
- *
- * The domain is divided into the intervals [0, 2] and
- * (2, infinity). In the first interval a polynomial approximation
- * 2
- * (w - r ) x P(w)
- * 1
- * 2
- * is used, where w = x and r is the first zero of the function.
- *
- * In the second interval, the modulus and phase are approximated
- * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x)
- * and Phase(x) = x + 1/x R(1/x^2) - 3pi/4. The function is
- *
- * j0(x) = Modulus(x) cos( Phase(x) ).
- *
- *
- *
- * ACCURACY:
- *
- * Absolute error:
- * arithmetic domain # trials peak rms
- * IEEE 0, 2 100000 1.2e-7 2.5e-8
- * IEEE 2, 32 100000 2.0e-7 5.3e-8
- *
- *
- */
- /* y1.c
- *
- * Bessel function of second kind of order one
- *
- *
- *
- * SYNOPSIS:
- *
- * double x, y, y1();
- *
- * y = y1( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of the second kind of order one
- * of the argument.
- *
- * The domain is divided into the intervals [0, 2] and
- * (2, infinity). In the first interval a rational approximation
- * R(x) is employed to compute
- *
- * 2
- * y0(x) = (w - r ) x R(x^2) + 2/pi (ln(x) j1(x) - 1/x) .
- * 1
- *
- * Thus a call to j1() is required.
- *
- * In the second interval, the modulus and phase are approximated
- * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x)
- * and Phase(x) = x + 1/x S(1/x^2) - 3pi/4. Then the function is
- *
- * y0(x) = Modulus(x) sin( Phase(x) ).
- *
- *
- *
- *
- * ACCURACY:
- *
- * Absolute error:
- * arithmetic domain # trials peak rms
- * IEEE 0, 2 100000 2.2e-7 4.6e-8
- * IEEE 2, 32 100000 1.9e-7 5.3e-8
- *
- * (error criterion relative when |y1| > 1).
- *
- */
-
-/* jnf.c
- *
- * Bessel function of integer order
- *
- *
- *
- * SYNOPSIS:
- *
- * int n;
- * float x, y, jnf();
- *
- * y = jnf( n, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of order n, where n is a
- * (possibly negative) integer.
- *
- * The ratio of jn(x) to j0(x) is computed by backward
- * recurrence. First the ratio jn/jn-1 is found by a
- * continued fraction expansion. Then the recurrence
- * relating successive orders is applied until j0 or j1 is
- * reached.
- *
- * If n = 0 or 1 the routine for j0 or j1 is called
- * directly.
- *
- *
- *
- * ACCURACY:
- *
- * Absolute error:
- * arithmetic range # trials peak rms
- * IEEE 0, 15 30000 3.6e-7 3.6e-8
- *
- *
- * Not suitable for large n or x. Use jvf() instead.
- *
- */
-
-/* jvf.c
- *
- * Bessel function of noninteger order
- *
- *
- *
- * SYNOPSIS:
- *
- * float v, x, y, jvf();
- *
- * y = jvf( v, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of order v of the argument,
- * where v is real. Negative x is allowed if v is an integer.
- *
- * Several expansions are included: the ascending power
- * series, the Hankel expansion, and two transitional
- * expansions for large v. If v is not too large, it
- * is reduced by recurrence to a region of best accuracy.
- *
- * The single precision routine accepts negative v, but with
- * reduced accuracy.
- *
- *
- *
- * ACCURACY:
- * Results for integer v are indicated by *.
- * Error criterion is absolute, except relative when |jv()| > 1.
- *
- * arithmetic domain # trials peak rms
- * v x
- * IEEE 0,125 0,125 30000 2.0e-6 2.0e-7
- * IEEE -17,0 0,125 30000 1.1e-5 4.0e-7
- * IEEE -100,0 0,125 3000 1.5e-4 7.8e-6
- */
-
-/* k0f.c
- *
- * Modified Bessel function, third kind, order zero
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, k0f();
- *
- * y = k0f( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns modified Bessel function of the third kind
- * of order zero of the argument.
- *
- * The range is partitioned into the two intervals [0,8] and
- * (8, infinity). Chebyshev polynomial expansions are employed
- * in each interval.
- *
- *
- *
- * ACCURACY:
- *
- * Tested at 2000 random points between 0 and 8. Peak absolute
- * error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15.
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0, 30 30000 7.8e-7 8.5e-8
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * K0 domain x <= 0 MAXNUM
- *
- */
- /* k0ef()
- *
- * Modified Bessel function, third kind, order zero,
- * exponentially scaled
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, k0ef();
- *
- * y = k0ef( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns exponentially scaled modified Bessel function
- * of the third kind of order zero of the argument.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0, 30 30000 8.1e-7 7.8e-8
- * See k0().
- *
- */
-
-/* k1f.c
- *
- * Modified Bessel function, third kind, order one
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, k1f();
- *
- * y = k1f( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Computes the modified Bessel function of the third kind
- * of order one of the argument.
- *
- * The range is partitioned into the two intervals [0,2] and
- * (2, infinity). Chebyshev polynomial expansions are employed
- * in each interval.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0, 30 30000 4.6e-7 7.6e-8
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * k1 domain x <= 0 MAXNUM
- *
- */
- /* k1ef.c
- *
- * Modified Bessel function, third kind, order one,
- * exponentially scaled
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, k1ef();
- *
- * y = k1ef( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns exponentially scaled modified Bessel function
- * of the third kind of order one of the argument:
- *
- * k1e(x) = exp(x) * k1(x).
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0, 30 30000 4.9e-7 6.7e-8
- * See k1().
- *
- */
-
-/* knf.c
- *
- * Modified Bessel function, third kind, integer order
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, knf();
- * int n;
- *
- * y = knf( n, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns modified Bessel function of the third kind
- * of order n of the argument.
- *
- * The range is partitioned into the two intervals [0,9.55] and
- * (9.55, infinity). An ascending power series is used in the
- * low range, and an asymptotic expansion in the high range.
- *
- *
- *
- * ACCURACY:
- *
- * Absolute error, relative when function > 1:
- * arithmetic domain # trials peak rms
- * IEEE 0,30 10000 2.0e-4 3.8e-6
- *
- * Error is high only near the crossover point x = 9.55
- * between the two expansions used.
- */
-
-/* log10f.c
- *
- * Common logarithm
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, log10f();
- *
- * y = log10f( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns logarithm to the base 10 of x.
- *
- * The argument is separated into its exponent and fractional
- * parts. The logarithm of the fraction is approximated by
- *
- * log(1+x) = x - 0.5 x**2 + x**3 P(x).
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0.5, 2.0 100000 1.3e-7 3.4e-8
- * IEEE 0, MAXNUMF 100000 1.3e-7 2.6e-8
- *
- * In the tests over the interval [0, MAXNUM], the logarithms
- * of the random arguments were uniformly distributed over
- * [-MAXL10, MAXL10].
- *
- * ERROR MESSAGES:
- *
- * log10f singularity: x = 0; returns -MAXL10
- * log10f domain: x < 0; returns -MAXL10
- * MAXL10 = 38.230809449325611792
- */
-
-/* log2f.c
- *
- * Base 2 logarithm
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, log2f();
- *
- * y = log2f( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the base 2 logarithm of x.
- *
- * The argument is separated into its exponent and fractional
- * parts. If the exponent is between -1 and +1, the base e
- * logarithm of the fraction is approximated by
- *
- * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
- *
- * Otherwise, setting z = 2(x-1)/x+1),
- *
- * log(x) = z + z**3 P(z)/Q(z).
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE exp(+-88) 100000 1.1e-7 2.4e-8
- * IEEE 0.5, 2.0 100000 1.1e-7 3.0e-8
- *
- * In the tests over the interval [exp(+-88)], the logarithms
- * of the random arguments were uniformly distributed.
- *
- * ERROR MESSAGES:
- *
- * log singularity: x = 0; returns MINLOGF/log(2)
- * log domain: x < 0; returns MINLOGF/log(2)
- */
-
-/* logf.c
- *
- * Natural logarithm
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, logf();
- *
- * y = logf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the base e (2.718...) logarithm of x.
- *
- * The argument is separated into its exponent and fractional
- * parts. If the exponent is between -1 and +1, the logarithm
- * of the fraction is approximated by
- *
- * log(1+x) = x - 0.5 x**2 + x**3 P(x)
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0.5, 2.0 100000 7.6e-8 2.7e-8
- * IEEE 1, MAXNUMF 100000 2.6e-8
- *
- * In the tests over the interval [1, MAXNUM], the logarithms
- * of the random arguments were uniformly distributed over
- * [0, MAXLOGF].
- *
- * ERROR MESSAGES:
- *
- * logf singularity: x = 0; returns MINLOG
- * logf domain: x < 0; returns MINLOG
- */
-
-/* mtherr.c
- *
- * Library common error handling routine
- *
- *
- *
- * SYNOPSIS:
- *
- * char *fctnam;
- * int code;
- * void mtherr();
- *
- * mtherr( fctnam, code );
- *
- *
- *
- * DESCRIPTION:
- *
- * This routine may be called to report one of the following
- * error conditions (in the include file math.h).
- *
- * Mnemonic Value Significance
- *
- * DOMAIN 1 argument domain error
- * SING 2 function singularity
- * OVERFLOW 3 overflow range error
- * UNDERFLOW 4 underflow range error
- * TLOSS 5 total loss of precision
- * PLOSS 6 partial loss of precision
- * EDOM 33 Unix domain error code
- * ERANGE 34 Unix range error code
- *
- * The default version of the file prints the function name,
- * passed to it by the pointer fctnam, followed by the
- * error condition. The display is directed to the standard
- * output device. The routine then returns to the calling
- * program. Users may wish to modify the program to abort by
- * calling exit() under severe error conditions such as domain
- * errors.
- *
- * Since all error conditions pass control to this function,
- * the display may be easily changed, eliminated, or directed
- * to an error logging device.
- *
- * SEE ALSO:
- *
- * math.h
- *
- */
-
-/* nbdtrf.c
- *
- * Negative binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * float p, y, nbdtrf();
- *
- * y = nbdtrf( k, n, p );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the sum of the terms 0 through k of the negative
- * binomial distribution:
- *
- * k
- * -- ( n+j-1 ) n j
- * > ( ) p (1-p)
- * -- ( j )
- * j=0
- *
- * In a sequence of Bernoulli trials, this is the probability
- * that k or fewer failures precede the nth success.
- *
- * The terms are not computed individually; instead the incomplete
- * beta integral is employed, according to the formula
- *
- * y = nbdtr( k, n, p ) = incbet( n, k+1, p ).
- *
- * The arguments must be positive, with p ranging from 0 to 1.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,100 5000 1.5e-4 1.9e-5
- *
- */
- /* nbdtrcf.c
- *
- * Complemented negative binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * float p, y, nbdtrcf();
- *
- * y = nbdtrcf( k, n, p );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the sum of the terms k+1 to infinity of the negative
- * binomial distribution:
- *
- * inf
- * -- ( n+j-1 ) n j
- * > ( ) p (1-p)
- * -- ( j )
- * j=k+1
- *
- * The terms are not computed individually; instead the incomplete
- * beta integral is employed, according to the formula
- *
- * y = nbdtrc( k, n, p ) = incbet( k+1, n, 1-p ).
- *
- * The arguments must be positive, with p ranging from 0 to 1.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,100 5000 1.4e-4 2.0e-5
- *
- */
-
-/* ndtrf.c
- *
- * Normal distribution function
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, ndtrf();
- *
- * y = ndtrf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the area under the Gaussian probability density
- * function, integrated from minus infinity to x:
- *
- * x
- * -
- * 1 | | 2
- * ndtr(x) = --------- | exp( - t /2 ) dt
- * sqrt(2pi) | |
- * -
- * -inf.
- *
- * = ( 1 + erf(z) ) / 2
- * = erfc(z) / 2
- *
- * where z = x/sqrt(2). Computation is via the functions
- * erf and erfc.
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -13,0 50000 1.5e-5 2.6e-6
- *
- *
- * ERROR MESSAGES:
- *
- * See erfcf().
- *
- */
- /* erff.c
- *
- * Error function
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, erff();
- *
- * y = erff( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * The integral is
- *
- * x
- * -
- * 2 | | 2
- * erf(x) = -------- | exp( - t ) dt.
- * sqrt(pi) | |
- * -
- * 0
- *
- * The magnitude of x is limited to 9.231948545 for DEC
- * arithmetic; 1 or -1 is returned outside this range.
- *
- * For 0 <= |x| < 1, erf(x) = x * P(x**2); otherwise
- * erf(x) = 1 - erfc(x).
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -9.3,9.3 50000 1.7e-7 2.8e-8
- *
- */
- /* erfcf.c
- *
- * Complementary error function
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, erfcf();
- *
- * y = erfcf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- *
- * 1 - erf(x) =
- *
- * inf.
- * -
- * 2 | | 2
- * erfc(x) = -------- | exp( - t ) dt
- * sqrt(pi) | |
- * -
- * x
- *
- *
- * For small x, erfc(x) = 1 - erf(x); otherwise polynomial
- * approximations 1/x P(1/x**2) are computed.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -9.3,9.3 50000 3.9e-6 7.2e-7
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * erfcf underflow x**2 > MAXLOGF 0.0
- *
- *
- */
-
-/* ndtrif.c
- *
- * Inverse of Normal distribution function
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, ndtrif();
- *
- * x = ndtrif( y );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the argument, x, for which the area under the
- * Gaussian probability density function (integrated from
- * minus infinity to x) is equal to y.
- *
- *
- * For small arguments 0 < y < exp(-2), the program computes
- * z = sqrt( -2.0 * log(y) ); then the approximation is
- * x = z - log(z)/z - (1/z) P(1/z) / Q(1/z).
- * There are two rational functions P/Q, one for 0 < y < exp(-32)
- * and the other for y up to exp(-2). For larger arguments,
- * w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 1e-38, 1 30000 3.6e-7 5.0e-8
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * ndtrif domain x <= 0 -MAXNUM
- * ndtrif domain x >= 1 MAXNUM
- *
- */
-
-/* pdtrf.c
- *
- * Poisson distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k;
- * float m, y, pdtrf();
- *
- * y = pdtrf( k, m );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the sum of the first k terms of the Poisson
- * distribution:
- *
- * k j
- * -- -m m
- * > e --
- * -- j!
- * j=0
- *
- * The terms are not summed directly; instead the incomplete
- * gamma integral is employed, according to the relation
- *
- * y = pdtr( k, m ) = igamc( k+1, m ).
- *
- * The arguments must both be positive.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,100 5000 6.9e-5 8.0e-6
- *
- */
- /* pdtrcf()
- *
- * Complemented poisson distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k;
- * float m, y, pdtrcf();
- *
- * y = pdtrcf( k, m );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the sum of the terms k+1 to infinity of the Poisson
- * distribution:
- *
- * inf. j
- * -- -m m
- * > e --
- * -- j!
- * j=k+1
- *
- * The terms are not summed directly; instead the incomplete
- * gamma integral is employed, according to the formula
- *
- * y = pdtrc( k, m ) = igam( k+1, m ).
- *
- * The arguments must both be positive.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,100 5000 8.4e-5 1.2e-5
- *
- */
- /* pdtrif()
- *
- * Inverse Poisson distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k;
- * float m, y, pdtrf();
- *
- * m = pdtrif( k, y );
- *
- *
- *
- *
- * DESCRIPTION:
- *
- * Finds the Poisson variable x such that the integral
- * from 0 to x of the Poisson density is equal to the
- * given probability y.
- *
- * This is accomplished using the inverse gamma integral
- * function and the relation
- *
- * m = igami( k+1, y ).
- *
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,100 5000 8.7e-6 1.4e-6
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * pdtri domain y < 0 or y >= 1 0.0
- * k < 0
- *
- */
-
-/* polevlf.c
- * p1evlf.c
- *
- * Evaluate polynomial
- *
- *
- *
- * SYNOPSIS:
- *
- * int N;
- * float x, y, coef[N+1], polevlf[];
- *
- * y = polevlf( x, coef, N );
- *
- *
- *
- * DESCRIPTION:
- *
- * Evaluates polynomial of degree N:
- *
- * 2 N
- * y = C + C x + C x +...+ C x
- * 0 1 2 N
- *
- * Coefficients are stored in reverse order:
- *
- * coef[0] = C , ..., coef[N] = C .
- * N 0
- *
- * The function p1evl() assumes that coef[N] = 1.0 and is
- * omitted from the array. Its calling arguments are
- * otherwise the same as polevl().
- *
- *
- * SPEED:
- *
- * In the interest of speed, there are no checks for out
- * of bounds arithmetic. This routine is used by most of
- * the functions in the library. Depending on available
- * equipment features, the user may wish to rewrite the
- * program in microcode or assembly language.
- *
- */
-
-/* polynf.c
- * polyrf.c
- * Arithmetic operations on polynomials
- *
- * In the following descriptions a, b, c are polynomials of degree
- * na, nb, nc respectively. The degree of a polynomial cannot
- * exceed a run-time value MAXPOLF. An operation that attempts
- * to use or generate a polynomial of higher degree may produce a
- * result that suffers truncation at degree MAXPOL. The value of
- * MAXPOL is set by calling the function
- *
- * polinif( maxpol );
- *
- * where maxpol is the desired maximum degree. This must be
- * done prior to calling any of the other functions in this module.
- * Memory for internal temporary polynomial storage is allocated
- * by polinif().
- *
- * Each polynomial is represented by an array containing its
- * coefficients, together with a separately declared integer equal
- * to the degree of the polynomial. The coefficients appear in
- * ascending order; that is,
- *
- * 2 na
- * a(x) = a[0] + a[1] * x + a[2] * x + ... + a[na] * x .
- *
- *
- *
- * sum = poleva( a, na, x ); Evaluate polynomial a(t) at t = x.
- * polprtf( a, na, D ); Print the coefficients of a to D digits.
- * polclrf( a, na ); Set a identically equal to zero, up to a[na].
- * polmovf( a, na, b ); Set b = a.
- * poladdf( a, na, b, nb, c ); c = b + a, nc = max(na,nb)
- * polsubf( a, na, b, nb, c ); c = b - a, nc = max(na,nb)
- * polmulf( a, na, b, nb, c ); c = b * a, nc = na+nb
- *
- *
- * Division:
- *
- * i = poldivf( a, na, b, nb, c ); c = b / a, nc = MAXPOL
- *
- * returns i = the degree of the first nonzero coefficient of a.
- * The computed quotient c must be divided by x^i. An error message
- * is printed if a is identically zero.
- *
- *
- * Change of variables:
- * If a and b are polynomials, and t = a(x), then
- * c(t) = b(a(x))
- * is a polynomial found by substituting a(x) for t. The
- * subroutine call for this is
- *
- * polsbtf( a, na, b, nb, c );
- *
- *
- * Notes:
- * poldivf() is an integer routine; polevaf() is float.
- * Any of the arguments a, b, c may refer to the same array.
- *
- */
-
-/* powf.c
- *
- * Power function
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, z, powf();
- *
- * z = powf( x, y );
- *
- *
- *
- * DESCRIPTION:
- *
- * Computes x raised to the yth power. Analytically,
- *
- * x**y = exp( y log(x) ).
- *
- * Following Cody and Waite, this program uses a lookup table
- * of 2**-i/16 and pseudo extended precision arithmetic to
- * obtain an extra three bits of accuracy in both the logarithm
- * and the exponential.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -10,10 100,000 1.4e-7 3.6e-8
- * 1/10 < x < 10, x uniformly distributed.
- * -10 < y < 10, y uniformly distributed.
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * powf overflow x**y > MAXNUMF MAXNUMF
- * powf underflow x**y < 1/MAXNUMF 0.0
- * powf domain x<0 and y noninteger 0.0
- *
- */
-
-/* powif.c
- *
- * Real raised to integer power
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, powif();
- * int n;
- *
- * y = powif( x, n );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns argument x raised to the nth power.
- * The routine efficiently decomposes n as a sum of powers of
- * two. The desired power is a product of two-to-the-kth
- * powers of x. Thus to compute the 32767 power of x requires
- * 28 multiplications instead of 32767 multiplications.
- *
- *
- *
- * ACCURACY:
- *
- *
- * Relative error:
- * arithmetic x domain n domain # trials peak rms
- * IEEE .04,26 -26,26 100000 1.1e-6 2.0e-7
- * IEEE 1,2 -128,128 100000 1.1e-5 1.0e-6
- *
- * Returns MAXNUMF on overflow, zero on underflow.
- *
- */
-
-/* psif.c
- *
- * Psi (digamma) function
- *
- *
- * SYNOPSIS:
- *
- * float x, y, psif();
- *
- * y = psif( x );
- *
- *
- * DESCRIPTION:
- *
- * d -
- * psi(x) = -- ln | (x)
- * dx
- *
- * is the logarithmic derivative of the gamma function.
- * For integer x,
- * n-1
- * -
- * psi(n) = -EUL + > 1/k.
- * -
- * k=1
- *
- * This formula is used for 0 < n <= 10. If x is negative, it
- * is transformed to a positive argument by the reflection
- * formula psi(1-x) = psi(x) + pi cot(pi x).
- * For general positive x, the argument is made greater than 10
- * using the recurrence psi(x+1) = psi(x) + 1/x.
- * Then the following asymptotic expansion is applied:
- *
- * inf. B
- * - 2k
- * psi(x) = log(x) - 1/2x - > -------
- * - 2k
- * k=1 2k x
- *
- * where the B2k are Bernoulli numbers.
- *
- * ACCURACY:
- * Absolute error, relative when |psi| > 1 :
- * arithmetic domain # trials peak rms
- * IEEE -33,0 30000 8.2e-7 1.2e-7
- * IEEE 0,33 100000 7.3e-7 7.7e-8
- *
- * ERROR MESSAGES:
- * message condition value returned
- * psi singularity x integer <=0 MAXNUMF
- */
-
-/* rgammaf.c
- *
- * Reciprocal gamma function
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, rgammaf();
- *
- * y = rgammaf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns one divided by the gamma function of the argument.
- *
- * The function is approximated by a Chebyshev expansion in
- * the interval [0,1]. Range reduction is by recurrence
- * for arguments between -34.034 and +34.84425627277176174.
- * 1/MAXNUMF is returned for positive arguments outside this
- * range.
- *
- * The reciprocal gamma function has no singularities,
- * but overflow and underflow may occur for large arguments.
- * These conditions return either MAXNUMF or 1/MAXNUMF with
- * appropriate sign.
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -34,+34 100000 8.9e-7 1.1e-7
- */
-
-/* shichif.c
- *
- * Hyperbolic sine and cosine integrals
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, Chi, Shi;
- *
- * shichi( x, &Chi, &Shi );
- *
- *
- * DESCRIPTION:
- *
- * Approximates the integrals
- *
- * x
- * -
- * | | cosh t - 1
- * Chi(x) = eul + ln x + | ----------- dt,
- * | | t
- * -
- * 0
- *
- * x
- * -
- * | | sinh t
- * Shi(x) = | ------ dt
- * | | t
- * -
- * 0
- *
- * where eul = 0.57721566490153286061 is Euler's constant.
- * The integrals are evaluated by power series for x < 8
- * and by Chebyshev expansions for x between 8 and 88.
- * For large x, both functions approach exp(x)/2x.
- * Arguments greater than 88 in magnitude return MAXNUM.
- *
- *
- * ACCURACY:
- *
- * Test interval 0 to 88.
- * Relative error:
- * arithmetic function # trials peak rms
- * IEEE Shi 20000 3.5e-7 7.0e-8
- * Absolute error, except relative when |Chi| > 1:
- * IEEE Chi 20000 3.8e-7 7.6e-8
- */
-
-/* sicif.c
- *
- * Sine and cosine integrals
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, Ci, Si;
- *
- * sicif( x, &Si, &Ci );
- *
- *
- * DESCRIPTION:
- *
- * Evaluates the integrals
- *
- * x
- * -
- * | cos t - 1
- * Ci(x) = eul + ln x + | --------- dt,
- * | t
- * -
- * 0
- * x
- * -
- * | sin t
- * Si(x) = | ----- dt
- * | t
- * -
- * 0
- *
- * where eul = 0.57721566490153286061 is Euler's constant.
- * The integrals are approximated by rational functions.
- * For x > 8 auxiliary functions f(x) and g(x) are employed
- * such that
- *
- * Ci(x) = f(x) sin(x) - g(x) cos(x)
- * Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x)
- *
- *
- * ACCURACY:
- * Test interval = [0,50].
- * Absolute error, except relative when > 1:
- * arithmetic function # trials peak rms
- * IEEE Si 30000 2.1e-7 4.3e-8
- * IEEE Ci 30000 3.9e-7 2.2e-8
- */
-
-/* sindgf.c
- *
- * Circular sine of angle in degrees
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, sindgf();
- *
- * y = sindgf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Range reduction is into intervals of 45 degrees.
- *
- * Two polynomial approximating functions are employed.
- * Between 0 and pi/4 the sine is approximated by
- * x + x**3 P(x**2).
- * Between pi/4 and pi/2 the cosine is represented as
- * 1 - x**2 Q(x**2).
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE +-3600 100,000 1.2e-7 3.0e-8
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * sin total loss x > 2^24 0.0
- *
- */
-
-/* cosdgf.c
- *
- * Circular cosine of angle in degrees
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, cosdgf();
- *
- * y = cosdgf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Range reduction is into intervals of 45 degrees.
- *
- * Two polynomial approximating functions are employed.
- * Between 0 and pi/4 the cosine is approximated by
- * 1 - x**2 Q(x**2).
- * Between pi/4 and pi/2 the sine is represented as
- * x + x**3 P(x**2).
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -8192,+8192 100,000 3.0e-7 3.0e-8
- */
-
-/* sinf.c
- *
- * Circular sine
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, sinf();
- *
- * y = sinf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Range reduction is into intervals of pi/4. The reduction
- * error is nearly eliminated by contriving an extended precision
- * modular arithmetic.
- *
- * Two polynomial approximating functions are employed.
- * Between 0 and pi/4 the sine is approximated by
- * x + x**3 P(x**2).
- * Between pi/4 and pi/2 the cosine is represented as
- * 1 - x**2 Q(x**2).
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -4096,+4096 100,000 1.2e-7 3.0e-8
- * IEEE -8192,+8192 100,000 3.0e-7 3.0e-8
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * sin total loss x > 2^24 0.0
- *
- * Partial loss of accuracy begins to occur at x = 2^13
- * = 8192. Results may be meaningless for x >= 2^24
- * The routine as implemented flags a TLOSS error
- * for x >= 2^24 and returns 0.0.
- */
-
-/* cosf.c
- *
- * Circular cosine
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, cosf();
- *
- * y = cosf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Range reduction is into intervals of pi/4. The reduction
- * error is nearly eliminated by contriving an extended precision
- * modular arithmetic.
- *
- * Two polynomial approximating functions are employed.
- * Between 0 and pi/4 the cosine is approximated by
- * 1 - x**2 Q(x**2).
- * Between pi/4 and pi/2 the sine is represented as
- * x + x**3 P(x**2).
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -8192,+8192 100,000 3.0e-7 3.0e-8
- */
-
-/* sinhf.c
- *
- * Hyperbolic sine
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, sinhf();
- *
- * y = sinhf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns hyperbolic sine of argument in the range MINLOGF to
- * MAXLOGF.
- *
- * The range is partitioned into two segments. If |x| <= 1, a
- * polynomial approximation is used.
- * Otherwise the calculation is sinh(x) = ( exp(x) - exp(-x) )/2.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE +-MAXLOG 100000 1.1e-7 2.9e-8
- *
- */
-
-/* spencef.c
- *
- * Dilogarithm
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, spencef();
- *
- * y = spencef( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Computes the integral
- *
- * x
- * -
- * | | log t
- * spence(x) = - | ----- dt
- * | | t - 1
- * -
- * 1
- *
- * for x >= 0. A rational approximation gives the integral in
- * the interval (0.5, 1.5). Transformation formulas for 1/x
- * and 1-x are employed outside the basic expansion range.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,4 30000 4.4e-7 6.3e-8
- *
- *
- */
-
-/* sqrtf.c
- *
- * Square root
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, sqrtf();
- *
- * y = sqrtf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the square root of x.
- *
- * Range reduction involves isolating the power of two of the
- * argument and using a polynomial approximation to obtain
- * a rough value for the square root. Then Heron's iteration
- * is used three times to converge to an accurate value.
- *
- *
- *
- * ACCURACY:
- *
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,1.e38 100000 8.7e-8 2.9e-8
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * sqrtf domain x < 0 0.0
- *
- */
-
-/* stdtrf.c
- *
- * Student's t distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * float t, stdtrf();
- * short k;
- *
- * y = stdtrf( k, t );
- *
- *
- * DESCRIPTION:
- *
- * Computes the integral from minus infinity to t of the Student
- * t distribution with integer k > 0 degrees of freedom:
- *
- * t
- * -
- * | |
- * - | 2 -(k+1)/2
- * | ( (k+1)/2 ) | ( x )
- * ---------------------- | ( 1 + --- ) dx
- * - | ( k )
- * sqrt( k pi ) | ( k/2 ) |
- * | |
- * -
- * -inf.
- *
- * Relation to incomplete beta integral:
- *
- * 1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z )
- * where
- * z = k/(k + t**2).
- *
- * For t < -1, this is the method of computation. For higher t,
- * a direct method is derived from integration by parts.
- * Since the function is symmetric about t=0, the area under the
- * right tail of the density is found by calling the function
- * with -t instead of t.
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE +/- 100 5000 2.3e-5 2.9e-6
- */
-
-/* struvef.c
- *
- * Struve function
- *
- *
- *
- * SYNOPSIS:
- *
- * float v, x, y, struvef();
- *
- * y = struvef( v, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Computes the Struve function Hv(x) of order v, argument x.
- * Negative x is rejected unless v is an integer.
- *
- * This module also contains the hypergeometric functions 1F2
- * and 3F0 and a routine for the Bessel function Yv(x) with
- * noninteger v.
- *
- *
- *
- * ACCURACY:
- *
- * v varies from 0 to 10.
- * Absolute error (relative error when |Hv(x)| > 1):
- * arithmetic domain # trials peak rms
- * IEEE -10,10 100000 9.0e-5 4.0e-6
- *
- */
-
-/* tandgf.c
- *
- * Circular tangent of angle in degrees
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, tandgf();
- *
- * y = tandgf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the circular tangent of the radian argument x.
- *
- * Range reduction is into intervals of 45 degrees.
- *
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE +-2^24 50000 2.4e-7 4.8e-8
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * tanf total loss x > 2^24 0.0
- *
- */
- /* cotdgf.c
- *
- * Circular cotangent of angle in degrees
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, cotdgf();
- *
- * y = cotdgf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Range reduction is into intervals of 45 degrees.
- * A common routine computes either the tangent or cotangent.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE +-2^24 50000 2.4e-7 4.8e-8
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * cot total loss x > 2^24 0.0
- * cot singularity x = 0 MAXNUMF
- *
- */
-
-/* tanf.c
- *
- * Circular tangent
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, tanf();
- *
- * y = tanf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the circular tangent of the radian argument x.
- *
- * Range reduction is modulo pi/4. A polynomial approximation
- * is employed in the basic interval [0, pi/4].
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE +-4096 100000 3.3e-7 4.5e-8
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * tanf total loss x > 2^24 0.0
- *
- */
- /* cotf.c
- *
- * Circular cotangent
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, cotf();
- *
- * y = cotf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the circular cotangent of the radian argument x.
- * A common routine computes either the tangent or cotangent.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE +-4096 100000 3.0e-7 4.5e-8
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * cot total loss x > 2^24 0.0
- * cot singularity x = 0 MAXNUMF
- *
- */
-
-/* tanhf.c
- *
- * Hyperbolic tangent
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, tanhf();
- *
- * y = tanhf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns hyperbolic tangent of argument in the range MINLOG to
- * MAXLOG.
- *
- * A polynomial approximation is used for |x| < 0.625.
- * Otherwise,
- *
- * tanh(x) = sinh(x)/cosh(x) = 1 - 2/(exp(2x) + 1).
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE -2,2 100000 1.3e-7 2.6e-8
- *
- */
-
-/* ynf.c
- *
- * Bessel function of second kind of integer order
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, ynf();
- * int n;
- *
- * y = ynf( n, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of order n, where n is a
- * (possibly negative) integer.
- *
- * The function is evaluated by forward recurrence on
- * n, starting with values computed by the routines
- * y0() and y1().
- *
- * If n = 0 or 1 the routine for y0 or y1 is called
- * directly.
- *
- *
- *
- * ACCURACY:
- *
- *
- * Absolute error, except relative when y > 1:
- *
- * arithmetic domain # trials peak rms
- * IEEE 0, 30 10000 2.3e-6 3.4e-7
- *
- *
- * ERROR MESSAGES:
- *
- * message condition value returned
- * yn singularity x = 0 MAXNUMF
- * yn overflow MAXNUMF
- *
- * Spot checked against tables for x, n between 0 and 100.
- *
- */
-
- /* zetacf.c
- *
- * Riemann zeta function
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, y, zetacf();
- *
- * y = zetacf( x );
- *
- *
- *
- * DESCRIPTION:
- *
- *
- *
- * inf.
- * - -x
- * zetac(x) = > k , x > 1,
- * -
- * k=2
- *
- * is related to the Riemann zeta function by
- *
- * Riemann zeta(x) = zetac(x) + 1.
- *
- * Extension of the function definition for x < 1 is implemented.
- * Zero is returned for x > log2(MAXNUM).
- *
- * An overflow error may occur for large negative x, due to the
- * gamma function in the reflection formula.
- *
- * ACCURACY:
- *
- * Tabulated values have full machine accuracy.
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 1,50 30000 5.5e-7 7.5e-8
- *
- *
- */
-
-/* zetaf.c
- *
- * Riemann zeta function of two arguments
- *
- *
- *
- * SYNOPSIS:
- *
- * float x, q, y, zetaf();
- *
- * y = zetaf( x, q );
- *
- *
- *
- * DESCRIPTION:
- *
- *
- *
- * inf.
- * - -x
- * zeta(x,q) = > (k+q)
- * -
- * k=0
- *
- * where x > 1 and q is not a negative integer or zero.
- * The Euler-Maclaurin summation formula is used to obtain
- * the expansion
- *
- * n
- * - -x
- * zeta(x,q) = > (k+q)
- * -
- * k=1
- *
- * 1-x inf. B x(x+1)...(x+2j)
- * (n+q) 1 - 2j
- * + --------- - ------- + > --------------------
- * x-1 x - x+2j+1
- * 2(n+q) j=1 (2j)! (n+q)
- *
- * where the B2j are Bernoulli numbers. Note that (see zetac.c)
- * zeta(x,1) = zetac(x) + 1.
- *
- *
- *
- * ACCURACY:
- *
- * Relative error:
- * arithmetic domain # trials peak rms
- * IEEE 0,25 10000 6.9e-7 1.0e-7
- *
- * Large arguments may produce underflow in powf(), in which
- * case the results are inaccurate.
- *
- * REFERENCE:
- *
- * Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals,
- * Series, and Products, p. 1073; Academic Press, 1980.
- *
- */
generated by cgit v1.2.3 (git 2.25.1) at 2024-11-23 00:30:25 +0000