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-/* 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
- *
- *
- */
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