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authorEric Andersen <andersen@codepoet.org>2001-05-10 00:40:28 +0000
committerEric Andersen <andersen@codepoet.org>2001-05-10 00:40:28 +0000
commit1077fa4d772832f77a677ce7fb7c2d513b959e3f (patch)
tree579bee13fb0b58d2800206366ec2caecbb15f3fc /libm/double/README.txt
parent22358dd7ce7bb49792204b698f01a6f69b9c8e08 (diff)
uClibc now has a math library. muahahahaha!
-Erik
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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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