Totally rework the math library, this time based on the MacOs X

math library (which is itself based on the math lib from FreeBSD).
 -Erik

1 files changed, 0 insertions, 3502 deletions

diff --git a/libm/ldouble/README.txt b/libm/ldouble/README.txt
deleted file mode 100644
index 30fcaad36..000000000
--- a/libm/ldouble/README.txt
+++ /dev/null
@@ -1,3502 +0,0 @@
-/*							acoshl.c
- *
- *	Inverse hyperbolic cosine, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, acoshl();
- *
- * y = acoshl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns inverse hyperbolic cosine of argument.
- *
- * If 1 <= x < 1.5, a rational approximation
- *
- *	sqrt(2z) * 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
- *    IEEE      1,3         30000       2.0e-19     3.9e-20
- *
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * acoshl domain      |x| < 1            0.0
- *
- */
-
-/*							asinhl.c
- *
- *	Inverse hyperbolic sine, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, asinhl();
- *
- * y = asinhl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns inverse hyperbolic sine of argument.
- *
- * If |x| < 0.5, the function is approximated by a rational
- * form  x + x**3 P(x)/Q(x).  Otherwise,
- *
- *     asinh(x) = log( x + sqrt(1 + x*x) ).
- *
- *
- *
- * ACCURACY:
- *
- *                      Relative error:
- * arithmetic   domain     # trials      peak         rms
- *    IEEE     -3,3         30000       1.7e-19     3.5e-20
- *
- */
-
-/*							asinl.c
- *
- *	Inverse circular sine, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * double x, y, asinl();
- *
- * y = asinl( 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
- *    IEEE     -1, 1        30000       2.7e-19     4.8e-20
- *
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * asin domain        |x| > 1           0.0
- *
- */
-/*							acosl()
- *
- *	Inverse circular cosine, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * double x, y, acosl();
- *
- * y = acosl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns radian angle between -pi/2 and +pi/2 whose cosine
- * is x.
- *
- * Analytically, acos(x) = pi/2 - asin(x).  However if |x| is
- * near 1, there is cancellation error in subtracting asin(x)
- * from pi/2.  Hence if x < -0.5,
- *
- *    acos(x) =	 pi - 2.0 * asin( sqrt((1+x)/2) );
- *
- * or if x > +0.5,
- *
- *    acos(x) =	 2.0 * asin(  sqrt((1-x)/2) ).
- *
- *
- * ACCURACY:
- *
- *                      Relative error:
- * arithmetic   domain     # trials      peak         rms
- *    IEEE      -1, 1       30000       1.4e-19     3.5e-20
- *
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * asin domain        |x| > 1           0.0
- */
-
-/*							atanhl.c
- *
- *	Inverse hyperbolic tangent, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, atanhl();
- *
- * y = atanhl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns inverse hyperbolic tangent of argument in the range
- * MINLOGL to MAXLOGL.
- *
- * 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
- *    IEEE      -1,1        30000       1.1e-19     3.3e-20
- *
- */
-
-/*							atanl.c
- *
- *	Inverse circular tangent, long double precision
- *      (arctangent)
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, atanl();
- *
- * y = atanl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns radian angle between -pi/2 and +pi/2 whose tangent
- * is x.
- *
- * Range reduction is from four intervals into the interval
- * from zero to  tan( pi/8 ).  The approximant uses a rational
- * function of degree 3/4 of the form x + x**3 P(x)/Q(x).
- *
- *
- *
- * ACCURACY:
- *
- *                      Relative error:
- * arithmetic   domain     # trials      peak         rms
- *    IEEE      -10, 10    150000       1.3e-19     3.0e-20
- *
- */
-/*							atan2l()
- *
- *	Quadrant correct inverse circular tangent,
- *	long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, z, atan2l();
- *
- * z = atan2l( 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     60000       1.7e-19     3.2e-20
- * See atan.c.
- *
- */
-
-/*							bdtrl.c
- *
- *	Binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * long double p, y, bdtrl();
- *
- * y = bdtrl( 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 (k,n,p) with a and b between 0
- * and 10000 and p between 0 and 1.
- *    Relative error:
- * arithmetic   domain     # trials      peak         rms
- *    IEEE      0,10000      3000       1.6e-14     2.2e-15
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * bdtrl domain        k < 0            0.0
- *                     n < k
- *                     x < 0, x > 1
- *
- */
-/*							bdtrcl()
- *
- *	Complemented binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * long double p, y, bdtrcl();
- *
- * y = bdtrcl( 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:
- *
- * See incbet.c.
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * bdtrcl domain     x<0, x>1, n<k       0.0
- */
-/*							bdtril()
- *
- *	Inverse binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * long double p, y, bdtril();
- *
- * p = bdtril( 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:
- *
- * See incbi.c.
- * Tested at random k, n between 1 and 10000.  The "domain" refers to p:
- *                      Relative error:
- * arithmetic   domain     # trials      peak         rms
- *    IEEE       0,1        3500       2.0e-15     8.2e-17
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * bdtril domain     k < 0, n <= k         0.0
- *                  x < 0, x > 1
- */
-
-
-/*							btdtrl.c
- *
- *	Beta distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * long double a, b, x, y, btdtrl();
- *
- * y = btdtrl( 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
- *
- *
- * The mean value of this distribution is a/(a+b).  The variance
- * is ab/[(a+b)^2 (a+b+1)].
- *
- * This function is identical to the incomplete beta integral
- * function, incbetl(a, b, x).
- *
- * The complemented function is
- *
- * 1 - P(1-x)  =  incbetl( b, a, x );
- *
- *
- * ACCURACY:
- *
- * See incbetl.c.
- *
- */
-
-/*							cbrtl.c
- *
- *	Cube root, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, cbrtl();
- *
- * y = cbrtl( 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
- *    IEEE     .125,8        80000      7.0e-20     2.2e-20
- *    IEEE    exp(+-707)    100000      7.0e-20     2.4e-20
- *
- */
-
-/*							chdtrl.c
- *
- *	Chi-square distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * long double df, x, y, chdtrl();
- *
- * y = chdtrl( 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
- */
-/*							chdtrcl()
- *
- *	Complemented Chi-square distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * long double v, x, y, chdtrcl();
- *
- * y = chdtrcl( 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
- */
-/*							chdtril()
- *
- *	Inverse of complemented Chi-square distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * long double df, x, y, chdtril();
- *
- * x = chdtril( 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
- *
- */
-
-/*							clogl.c
- *
- *	Complex natural logarithm
- *
- *
- *
- * SYNOPSIS:
- *
- * void clogl();
- * cmplxl z, w;
- *
- * clogl( &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.
- */
-
-/*							cexpl()
- *
- *	Complex exponential function
- *
- *
- *
- * SYNOPSIS:
- *
- * void cexpl();
- * cmplxl z, w;
- *
- * cexpl( &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
- *
- */
-/*							csinl()
- *
- *	Complex circular sine
- *
- *
- *
- * SYNOPSIS:
- *
- * void csinl();
- * cmplxl z, w;
- *
- * csinl( &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.
- *
- */
-/*							ccosl()
- *
- *	Complex circular cosine
- *
- *
- *
- * SYNOPSIS:
- *
- * void ccosl();
- * cmplxl z, w;
- *
- * ccosl( &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
- */
-/*							ctanl()
- *
- *	Complex circular tangent
- *
- *
- *
- * SYNOPSIS:
- *
- * void ctanl();
- * cmplxl z, w;
- *
- * ctanl( &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.
- */
-/*							ccotl()
- *
- *	Complex circular cotangent
- *
- *
- *
- * SYNOPSIS:
- *
- * void ccotl();
- * cmplxl z, w;
- *
- * ccotl( &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.
- */
-
-/*							casinl()
- *
- *	Complex circular arc sine
- *
- *
- *
- * SYNOPSIS:
- *
- * void casinl();
- * cmplxl z, w;
- *
- * casinl( &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.
- */
-/*							cacosl()
- *
- *	Complex circular arc cosine
- *
- *
- *
- * SYNOPSIS:
- *
- * void cacosl();
- * cmplxl z, w;
- *
- * cacosl( &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
- */
-
-/*							catanl()
- *
- *	Complex circular arc tangent
- *
- *
- *
- * SYNOPSIS:
- *
- * void catanl();
- * cmplxl z, w;
- *
- * catanl( &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().
- */
-
-/*							cmplxl.c
- *
- *	Complex number arithmetic
- *
- *
- *
- * SYNOPSIS:
- *
- * typedef struct {
- *      long double r;     real part
- *      long double i;     imaginary part
- *     }cmplxl;
- *
- * cmplxl *a, *b, *c;
- *
- * caddl( a, b, c );     c = b + a
- * csubl( a, b, c );     c = b - a
- * cmull( a, b, c );     c = b * a
- * cdivl( a, b, c );     c = b / a
- * cnegl( c );           c = -c
- * cmovl( 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
- */
-
-/*							cabsl()
- *
- *	Complex absolute value
- *
- *
- *
- * SYNOPSIS:
- *
- * long double cabsl();
- * cmplxl z;
- * long 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
- */
-/*							csqrtl()
- *
- *	Complex square root
- *
- *
- *
- * SYNOPSIS:
- *
- * void csqrtl();
- * cmplxl z, w;
- *
- * csqrtl( &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.
- */
-
-/*							coshl.c
- *
- *	Hyperbolic cosine, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, coshl();
- *
- * y = coshl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns hyperbolic cosine of argument in the range MINLOGL to
- * MAXLOGL.
- *
- * cosh(x)  =  ( exp(x) + exp(-x) )/2.
- *
- *
- *
- * ACCURACY:
- *
- *                      Relative error:
- * arithmetic   domain     # trials      peak         rms
- *    IEEE     +-10000      30000       1.1e-19     2.8e-20
- *
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * cosh overflow    |x| > MAXLOGL       MAXNUML
- *
- *
- */
-
-/*							elliel.c
- *
- *	Incomplete elliptic integral of the second kind
- *
- *
- *
- * SYNOPSIS:
- *
- * long double phi, m, y, elliel();
- *
- * y = elliel( 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
- *    IEEE     -10,10       50000       2.7e-18     2.3e-19
- *
- *
- */
-
-/*							ellikl.c
- *
- *	Incomplete elliptic integral of the first kind
- *
- *
- *
- * SYNOPSIS:
- *
- * long double phi, m, y, ellikl();
- *
- * y = ellikl( 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        30000      3.6e-18     4.1e-19
- *
- *
- */
-
-/*							ellpel.c
- *
- *	Complete elliptic integral of the second kind
- *
- *
- *
- * SYNOPSIS:
- *
- * long double m1, y, ellpel();
- *
- * y = ellpel( m1 );
- *
- *
- *
- * DESCRIPTION:
- *
- * Approximates the integral
- *
- *
- *            pi/2
- *             -
- *            | |                 2
- * E(m)  =    |    sqrt( 1 - m sin t ) dt
- *          | |    
- *           -
- *            0
- *
- * Where m = 1 - m1, using the approximation
- *
- *      P(x)  -  x log x Q(x).
- *
- * Though there are no singularities, the argument m1 is used
- * rather than m for compatibility with ellpk().
- *
- * E(1) = 1; E(0) = pi/2.
- *
- *
- * ACCURACY:
- *
- *                      Relative error:
- * arithmetic   domain     # trials      peak         rms
- *    IEEE       0, 1       10000       1.1e-19     3.5e-20
- *
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * ellpel domain     x<0, x>1            0.0
- *
- */
-
-/*							ellpjl.c
- *
- *	Jacobian Elliptic Functions
- *
- *
- *
- * SYNOPSIS:
- *
- * long double u, m, sn, cn, dn, phi;
- * int ellpjl();
- *
- * ellpjl( 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-12 of 0 or 1.  In the
- * latter case with m close to 1, the approximation applies
- * only for phi < pi/2.
- *
- * ACCURACY:
- *
- * Tested at random points with u between 0 and 10, m between
- * 0 and 1.
- *
- *            Absolute error (* = relative error):
- * arithmetic   function   # trials      peak         rms
- *    IEEE      sn          10000       1.7e-18     2.3e-19
- *    IEEE      cn          20000       1.6e-18     2.2e-19
- *    IEEE      dn          10000       4.7e-15     2.7e-17
- *    IEEE      phi         10000       4.0e-19*    6.6e-20*
- *
- * Accuracy deteriorates when u is large.
- *
- */
-
-/*							ellpkl.c
- *
- *	Complete elliptic integral of the first kind
- *
- *
- *
- * SYNOPSIS:
- *
- * long double m1, y, ellpkl();
- *
- * y = ellpkl( m1 );
- *
- *
- *
- * DESCRIPTION:
- *
- * Approximates the integral
- *
- *
- *
- *            pi/2
- *             -
- *            | |
- *            |           dt
- * K(m)  =    |    ------------------
- *            |                   2
- *          | |    sqrt( 1 - m sin t )
- *           -
- *            0
- *
- * where m = 1 - m1, using the approximation
- *
- *     P(x)  -  log x Q(x).
- *
- * The argument m1 is used rather than m so that the logarithmic
- * singularity at m = 1 will be shifted to the origin; this
- * preserves maximum accuracy.
- *
- * K(0) = pi/2.
- *
- * ACCURACY:
- *
- *                      Relative error:
- * arithmetic   domain     # trials      peak         rms
- *    IEEE       0,1        10000       1.1e-19     3.3e-20
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * ellpkl domain      x<0, x>1           0.0
- *
- */
-
-/*							exp10l.c
- *
- *	Base 10 exponential function, long double precision
- *      (Common antilogarithm)
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, exp10l()
- *
- * y = exp10l( 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      +-4900      30000       1.0e-19     2.7e-20
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * exp10l underflow    x < -MAXL10        0.0
- * exp10l overflow     x > MAXL10       MAXNUM
- *
- * IEEE arithmetic: MAXL10 = 4932.0754489586679023819
- *
- */
-
-/*							exp2l.c
- *
- *	Base 2 exponential function, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, exp2l();
- *
- * y = exp2l( 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      +-16300     300000      9.1e-20     2.6e-20
- *
- *
- * See exp.c for comments on error amplification.
- *
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * exp2l underflow   x < -16382        0.0
- * exp2l overflow    x >= 16384       MAXNUM
- *
- */
-
-/*							expl.c
- *
- *	Exponential function, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, expl();
- *
- * y = expl( 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 of degree 2/3 is used to approximate exp(f) - 1
- * in the basic range [-0.5 ln 2, 0.5 ln 2].
- *
- *
- * ACCURACY:
- *
- *                      Relative error:
- * arithmetic   domain     # trials      peak         rms
- *    IEEE      +-10000     50000       1.12e-19    2.81e-20
- *
- *
- * 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 long 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         MAXNUM
- *
- */
-
-/*							fabsl.c
- *
- *		Absolute value
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y;
- *
- * y = fabsl( x );
- *
- *
- *
- * DESCRIPTION:
- * 
- * Returns the absolute value of the argument.
- *
- */
-
-/*							fdtrl.c
- *
- *	F distribution, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * int df1, df2;
- * long double x, y, fdtrl();
- *
- * y = fdtrl( 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) = incbetl( df1/2, df2/2, (df1*x/(df2 + df1*x) ).
- *
- *
- * The arguments a and b are greater than zero, and x
- * x is nonnegative.
- *
- * 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       10000       9.3e-18     2.9e-19
- *    IEEE      0,1    1,10000     10000       1.9e-14     2.9e-15
- *    IEEE      1,5    1,10000     10000       5.8e-15     1.4e-16
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * fdtrl domain     a<0, b<0, x<0         0.0
- *
- */
-/*							fdtrcl()
- *
- *	Complemented F distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int df1, df2;
- * long double x, y, fdtrcl();
- *
- * y = fdtrcl( df1, df2, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the area from x to infinity under the F density
- * function (also known as Snedcor's density or the
- * variance ratio density).
- *
- *
- *                      inf.
- *                       -
- *              1       | |  a-1      b-1
- * 1-P(x)  =  ------    |   t    (1-t)    dt
- *            B(a,b)  | |
- *                     -
- *                      x
- *
- * (See fdtr.c.)
- *
- * The incomplete beta integral is used, according to the
- * formula
- *
- *	P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ).
- *
- *
- * ACCURACY:
- *
- * See incbet.c.
- * Tested at random points (a,b,x).
- *
- *                x     a,b                     Relative error:
- * arithmetic  domain  domain     # trials      peak         rms
- *    IEEE      0,1    0,100       10000       4.2e-18     3.3e-19
- *    IEEE      0,1    1,10000     10000       7.2e-15     2.6e-16
- *    IEEE      1,5    1,10000     10000       1.7e-14     3.0e-15
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * fdtrcl domain    a<0, b<0, x<0         0.0
- *
- */
-/*							fdtril()
- *
- *	Inverse of complemented F distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int df1, df2;
- * long double x, p, fdtril();
- *
- * x = fdtril( 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:
- *
- * See incbi.c.
- * 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       40000       4.6e-18     2.7e-19
- *    IEEE     1,10000     30000       1.7e-14     1.4e-16
- *  For p between 10^-6 and .001:
- *    IEEE     1,100       20000       1.9e-15     3.9e-17
- *    IEEE     1,10000     30000       2.7e-15     4.0e-17
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * fdtril domain   p <= 0 or p > 1       0.0
- *                     v < 1
- */
-
-/*							ceill()
- *							floorl()
- *							frexpl()
- *							ldexpl()
- *							fabsl()
- *
- *	Floating point numeric utilities
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y;
- * long double ceill(), floorl(), frexpl(), ldexpl(), fabsl();
- * int expnt, n;
- *
- * y = floorl(x);
- * y = ceill(x);
- * y = frexpl( x, &expnt );
- * y = ldexpl( x, n );
- * y = fabsl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * All four routines return a long double precision floating point
- * result.
- *
- * floorl() returns the largest integer less than or equal to x.
- * It truncates toward minus infinity.
- *
- * ceill() returns the smallest integer greater than or equal
- * to x.  It truncates toward plus infinity.
- *
- * frexpl() 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.
- *
- * ldexpl() multiplies x by 2**n.
- *
- * fabsl() returns the absolute value of its argument.
- *
- * These functions are part of the standard C run time library
- * for some but not all C compilers.  The ones supplied are
- * written in C for 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.
- */
-
-/*							gammal.c
- *
- *	Gamma function
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, gammal();
- * extern int sgngam;
- *
- * y = gammal( 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| <= 13 are reduced by recurrence and the function
- * approximated by a rational function of degree 7/8 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
- *    IEEE     -40,+40      10000       3.6e-19     7.9e-20
- *    IEEE    -1755,+1755   10000       4.8e-18     6.5e-19
- *
- * Accuracy for large arguments is dominated by error in powl().
- *
- */
-/*							lgaml()
- *
- *	Natural logarithm of gamma function
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, lgaml();
- * extern int sgngam;
- *
- * y = lgaml( 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 33, 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 MAXLGML (10^4928) return MAXNUML.
- *
- *
- *
- * ACCURACY:
- *
- *
- * arithmetic      domain        # trials     peak         rms
- *    IEEE         -40, 40        100000     2.2e-19     4.6e-20
- *    IEEE    10^-2000,10^+2000    20000     1.6e-19     3.3e-20
- * The error criterion was relative when the function magnitude
- * was greater than one but absolute when it was less than one.
- *
- */
-
-/*							gdtrl.c
- *
- *	Gamma distribution function
- *
- *
- *
- * SYNOPSIS:
- *
- * long double a, b, x, y, gdtrl();
- *
- * y = gdtrl( 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
- * gdtrl domain        x < 0            0.0
- *
- */
-/*							gdtrcl.c
- *
- *	Complemented gamma distribution function
- *
- *
- *
- * SYNOPSIS:
- *
- * long double a, b, x, y, gdtrcl();
- *
- * y = gdtrcl( 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
- * gdtrcl 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
-*/
-
-/*							igamil()
- *
- *      Inverse of complemented imcomplete gamma integral
- *
- *
- *
- * SYNOPSIS:
- *
- * long double a, x, y, igamil();
- *
- * x = igamil( a, y );
- *
- *
- *
- * DESCRIPTION:
- *
- * Given y, the function finds x such that
- *
- *  igamc( a, x ) = y.
- *
- * Starting with the approximate value
- *
- *         3
- *  x = a t
- *
- *  where
- *
- *  t = 1 - d - ndtri(y) sqrt(d)
- * 
- * and
- *
- *  d = 1/9a,
- *
- * the routine performs up to 10 Newton iterations to find the
- * root of igamc(a,x) - y = 0.
- *
- *
- * ACCURACY:
- *
- * Tested for a ranging from 0.5 to 30 and x from 0 to 0.5.
- *
- *                      Relative error:
- * arithmetic   domain     # trials      peak         rms
- *    DEC       0,0.5         3400       8.8e-16     1.3e-16
- *    IEEE      0,0.5        10000       1.1e-14     1.0e-15
- *
- */
-
-/*							igaml.c
- *
- *	Incomplete gamma integral
- *
- *
- *
- * SYNOPSIS:
- *
- * long double a, x, y, igaml();
- *
- * y = igaml( 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
- *    DEC       0,30         4000       4.4e-15     6.3e-16
- *    IEEE      0,30        10000       3.6e-14     5.1e-15
- *
- */
-/*							igamcl()
- *
- *	Complemented incomplete gamma integral
- *
- *
- *
- * SYNOPSIS:
- *
- * long double a, x, y, igamcl();
- *
- * y = igamcl( a, x );
- *
- *
- *
- * DESCRIPTION:
- *
- * The function is defined by
- *
- *
- *  igamc(a,x)   =   1 - igam(a,x)
- *
- *                            inf.
- *                              -
- *                     1       | |  -t  a-1
- *               =   -----     |   e   t   dt.
- *                    -      | |
- *                   | (a)    -
- *                             x
- *
- *
- * In this implementation both arguments must be positive.
- * The integral is evaluated by either a power series or
- * continued fraction expansion, depending on the relative
- * values of a and x.
- *
- *
- *
- * ACCURACY:
- *
- *                      Relative error:
- * arithmetic   domain     # trials      peak         rms
- *    DEC       0,30         2000       2.7e-15     4.0e-16
- *    IEEE      0,30        60000       1.4e-12     6.3e-15
- *
- */
-
-/*							incbetl.c
- *
- *	Incomplete beta integral
- *
- *
- * SYNOPSIS:
- *
- * long double a, b, x, y, incbetl();
- *
- * y = incbetl( 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 random points (a,b,x) with x between 0 and 1.
- * arithmetic   domain     # trials      peak         rms
- *    IEEE       0,5       20000        4.5e-18     2.4e-19
- *    IEEE       0,100    100000        3.9e-17     1.0e-17
- * Half-integer a, b:
- *    IEEE      .5,10000  100000        3.9e-14     4.4e-15
- * Outputs smaller than the IEEE gradual underflow threshold
- * were excluded from these statistics.
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * incbetl domain     x<0, x>1          0.0
- */
-
-/*							incbil()
- *
- *      Inverse of imcomplete beta integral
- *
- *
- *
- * SYNOPSIS:
- *
- * long double a, b, x, y, incbil();
- *
- * x = incbil( a, b, y );
- *
- *
- *
- * DESCRIPTION:
- *
- * Given y, the function finds x such that
- *
- *  incbet( a, b, x ) = y.
- *
- * the routine performs up to 10 Newton iterations to find the
- * root of incbet(a,b,x) - y = 0.
- *
- *
- * ACCURACY:
- *
- *                      Relative error:
- *                x       a,b
- * arithmetic   domain   domain   # trials    peak       rms
- *    IEEE      0,1    .5,10000    10000    1.1e-14   1.4e-16
- */
-
-/*							j0l.c
- *
- *	Bessel function of order zero
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, j0l();
- *
- * y = j0l( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of first kind, order zero of the argument.
- *
- * The domain is divided into the intervals [0, 9] and
- * (9, infinity). In the first interval the rational approximation
- * is (x^2 - r^2) (x^2 - s^2) (x^2 - t^2) P7(x^2) / Q8(x^2),
- * where r, s, t are the first three zeros of the function.
- * In the second interval the expansion is in terms of the
- * modulus M0(x) = sqrt(J0(x)^2 + Y0(x)^2) and phase  P0(x)
- * = atan(Y0(x)/J0(x)).  M0 is approximated by sqrt(1/x)P7(1/x)/Q7(1/x).
- * The approximation to J0 is M0 * cos(x -  pi/4 + 1/x P5(1/x^2)/Q6(1/x^2)).
- *
- *
- * ACCURACY:
- *
- *                      Absolute error:
- * arithmetic   domain      # trials      peak         rms
- *    IEEE      0, 30       100000      2.8e-19      7.4e-20
- *
- *
- */
-/*							y0l.c
- *
- *	Bessel function of the second kind, order zero
- *
- *
- *
- * SYNOPSIS:
- *
- * double x, y, y0l();
- *
- * y = y0l( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of the second kind, of order
- * zero, of the argument.
- *
- * The domain is divided into the intervals [0, 5>, [5,9> and
- * [9, infinity). In the first interval a rational approximation
- * R(x) is employed to compute y0(x)  = R(x) + 2/pi * log(x) * j0(x).
- *
- * In the second interval, the approximation is
- *     (x - p)(x - q)(x - r)(x - s)P7(x)/Q7(x)
- * where p, q, r, s are zeros of y0(x).
- *
- * The third interval uses the same approximations to modulus
- * and phase as j0(x), whence y0(x) = modulus * sin(phase).
- *
- * ACCURACY:
- *
- *  Absolute error, when y0(x) < 1; else relative error:
- *
- * arithmetic   domain     # trials      peak         rms
- *    IEEE      0, 30       100000      3.4e-19     7.6e-20
- *
- */
-
-/*							j1l.c
- *
- *	Bessel function of order one
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, j1l();
- *
- * y = j1l( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of order one of the argument.
- *
- * The domain is divided into the intervals [0, 9] and
- * (9, infinity). In the first interval the rational approximation
- * is (x^2 - r^2) (x^2 - s^2) (x^2 - t^2) x P8(x^2) / Q8(x^2),
- * where r, s, t are the first three zeros of the function.
- * In the second interval the expansion is in terms of the
- * modulus M1(x) = sqrt(J1(x)^2 + Y1(x)^2) and phase  P1(x)
- * = atan(Y1(x)/J1(x)).  M1 is approximated by sqrt(1/x)P7(1/x)/Q8(1/x).
- * The approximation to j1 is M1 * cos(x -  3 pi/4 + 1/x P5(1/x^2)/Q6(1/x^2)).
- *
- *
- * ACCURACY:
- *
- *                      Absolute error:
- * arithmetic   domain      # trials      peak         rms
- *    IEEE      0, 30        40000      1.8e-19      5.0e-20
- *
- *
- */
-/*							y1l.c
- *
- *	Bessel function of the second kind, order zero
- *
- *
- *
- * SYNOPSIS:
- *
- * double x, y, y1l();
- *
- * y = y1l( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns Bessel function of the second kind, of order
- * zero, of the argument.
- *
- * The domain is divided into the intervals [0, 4.5>, [4.5,9> and
- * [9, infinity). In the first interval a rational approximation
- * R(x) is employed to compute y0(x)  = R(x) + 2/pi * log(x) * j0(x).
- *
- * In the second interval, the approximation is
- *     (x - p)(x - q)(x - r)(x - s)P9(x)/Q10(x)
- * where p, q, r, s are zeros of y1(x).
- *
- * The third interval uses the same approximations to modulus
- * and phase as j1(x), whence y1(x) = modulus * sin(phase).
- *
- * ACCURACY:
- *
- *  Absolute error, when y0(x) < 1; else relative error:
- *
- * arithmetic   domain     # trials      peak         rms
- *    IEEE      0, 30       36000       2.7e-19     5.3e-20
- *
- */
-
-/*							jnl.c
- *
- *	Bessel function of integer order
- *
- *
- *
- * SYNOPSIS:
- *
- * int n;
- * long double x, y, jnl();
- *
- * y = jnl( 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   domain      # trials      peak         rms
- *    IEEE     -30, 30        5000       3.3e-19     4.7e-20
- *
- *
- * Not suitable for large n or x.
- *
- */
-
-/*							ldrand.c
- *
- *	Pseudorandom number generator
- *
- *
- *
- * SYNOPSIS:
- *
- * double y;
- * int ldrand();
- *
- * ldrand( &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.
- *
- * Versions invoked by the different arithmetic compile
- * time options IBMPC, and MIEEE, produce the same sequences.
- *
- */
-
-/*							log10l.c
- *
- *	Common logarithm, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, log10l();
- *
- * y = log10l( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns the base 10 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     30000      9.0e-20     2.6e-20
- *    IEEE     exp(+-10000)  30000      6.0e-20     2.3e-20
- *
- * In the tests over the interval exp(+-10000), the logarithms
- * of the random arguments were uniformly distributed over
- * [-10000, +10000].
- *
- * ERROR MESSAGES:
- *
- * log singularity:  x = 0; returns MINLOG
- * log domain:       x < 0; returns MINLOG
- */
-
-/*							log2l.c
- *
- *	Base 2 logarithm, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, log2l();
- *
- * y = log2l( 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 (natural)
- * 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      9.8e-20     2.7e-20
- *    IEEE     exp(+-10000)  70000      5.4e-20     2.3e-20
- *
- * In the tests over the interval exp(+-10000), the logarithms
- * of the random arguments were uniformly distributed over
- * [-10000, +10000].
- *
- * ERROR MESSAGES:
- *
- * log singularity:  x = 0; returns MINLOG
- * log domain:       x < 0; returns MINLOG
- */
-
-/*							logl.c
- *
- *	Natural logarithm, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, logl();
- *
- * y = logl( 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      8.71e-20    2.75e-20
- *    IEEE     exp(+-10000) 100000      5.39e-20    2.34e-20
- *
- * In the tests over the interval exp(+-10000), the logarithms
- * of the random arguments were uniformly distributed over
- * [-10000, +10000].
- *
- * ERROR MESSAGES:
- *
- * log singularity:  x = 0; returns MINLOG
- * log domain:       x < 0; returns MINLOG
- */
-
-/*							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 mconf.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:
- *
- * mconf.h
- *
- */
-
-/*							nbdtrl.c
- *
- *	Negative binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * long double p, y, nbdtrl();
- *
- * y = nbdtrl( 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 (k,n,p) with k and n between 1 and 10,000
- * and p between 0 and 1.
- *
- * arithmetic   domain     # trials      peak         rms
- *    Absolute error:
- *    IEEE      0,10000     10000       9.8e-15     2.1e-16
- *
- */
-/*							nbdtrcl.c
- *
- *	Complemented negative binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * long double p, y, nbdtrcl();
- *
- * y = nbdtrcl( 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 incbetl.c.
- *
- */
-/*							nbdtril
- *
- *	Functional inverse of negative binomial distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k, n;
- * long double p, y, nbdtril();
- *
- * p = nbdtril( 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
- * See also incbil.c.
- */
-
-/*							ndtril.c
- *
- *	Inverse of Normal distribution function
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, ndtril();
- *
- * x = ndtril( 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 log(y) );  then the approximation is
- * x = z - log(z)/z  - (1/z) P(1/z) / Q(1/z) .
- * For larger arguments,  x/sqrt(2 pi) = w + w^3 R(w^2)/S(w^2)) ,
- * where w = y - 0.5 .
- *
- * ACCURACY:
- *
- *                      Relative error:
- * arithmetic   domain        # trials      peak         rms
- *  Arguments uniformly distributed:
- *    IEEE       0, 1           5000       7.8e-19     9.9e-20
- *  Arguments exponentially distributed:
- *    IEEE     exp(-11355),-1  30000       1.7e-19     4.3e-20
- *
- *
- * ERROR MESSAGES:
- *
- *   message         condition    value returned
- * ndtril domain      x <= 0        -MAXNUML
- * ndtril domain      x >= 1         MAXNUML
- *
- */
-
-/*							ndtril.c
- *
- *	Inverse of Normal distribution function
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, ndtril();
- *
- * x = ndtril( 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 log(y) );  then the approximation is
- * x = z - log(z)/z  - (1/z) P(1/z) / Q(1/z) .
- * For larger arguments,  x/sqrt(2 pi) = w + w^3 R(w^2)/S(w^2)) ,
- * where w = y - 0.5 .
- *
- * ACCURACY:
- *
- *                      Relative error:
- * arithmetic   domain        # trials      peak         rms
- *  Arguments uniformly distributed:
- *    IEEE       0, 1           5000       7.8e-19     9.9e-20
- *  Arguments exponentially distributed:
- *    IEEE     exp(-11355),-1  30000       1.7e-19     4.3e-20
- *
- *
- * ERROR MESSAGES:
- *
- *   message         condition    value returned
- * ndtril domain      x <= 0        -MAXNUML
- * ndtril domain      x >= 1         MAXNUML
- *
- */
-
-/*							pdtrl.c
- *
- *	Poisson distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k;
- * long double m, y, pdtrl();
- *
- * y = pdtrl( 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().
- *
- */
-/*							pdtrcl()
- *
- *	Complemented poisson distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k;
- * long double m, y, pdtrcl();
- *
- * y = pdtrcl( 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.
- *
- */
-/*							pdtril()
- *
- *	Inverse Poisson distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * int k;
- * long double m, y, pdtrl();
- *
- * m = pdtril( 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
- *
- */
-
-/*							polevll.c
- *							p1evll.c
- *
- *	Evaluate polynomial
- *
- *
- *
- * SYNOPSIS:
- *
- * int N;
- * long double x, y, coef[N+1], polevl[];
- *
- * y = polevll( 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 p1evll() assumes that coef[N] = 1.0 and is
- * omitted from the array.  Its calling arguments are
- * otherwise the same as polevll().
- *
- *  This module also contains the following globally declared constants:
- * MAXNUML = 1.189731495357231765021263853E4932L;
- * MACHEPL = 5.42101086242752217003726400434970855712890625E-20L;
- * MAXLOGL =  1.1356523406294143949492E4L;
- * MINLOGL = -1.1355137111933024058873E4L;
- * LOGE2L  = 6.9314718055994530941723E-1L;
- * LOG2EL  = 1.4426950408889634073599E0L;
- * PIL     = 3.1415926535897932384626L;
- * PIO2L   = 1.5707963267948966192313L;
- * PIO4L   = 7.8539816339744830961566E-1L;
- *
- * 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.
- *
- */
-
-/*							powil.c
- *
- *	Real raised to integer power, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, powil();
- * int n;
- *
- * y = powil( x, n );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns argument x raised to the nth power.
- * The routine efficiently decomposes n as a sum of powers of
- * two. The desired power is a product of two-to-the-kth
- * powers of x.  Thus to compute the 32767 power of x requires
- * 28 multiplications instead of 32767 multiplications.
- *
- *
- *
- * ACCURACY:
- *
- *
- *                      Relative error:
- * arithmetic   x domain   n domain  # trials      peak         rms
- *    IEEE     .001,1000  -1022,1023  50000       4.3e-17     7.8e-18
- *    IEEE        1,2     -1022,1023  20000       3.9e-17     7.6e-18
- *    IEEE     .99,1.01     0,8700    10000       3.6e-16     7.2e-17
- *
- * Returns MAXNUM on overflow, zero on underflow.
- *
- */
-
-/*							powl.c
- *
- *	Power function, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, z, powl();
- *
- * z = powl( 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/32 and pseudo extended precision arithmetic to
- * obtain several extra bits of accuracy in both the logarithm
- * and the exponential.
- *
- *
- *
- * ACCURACY:
- *
- * The relative error of pow(x,y) can be estimated
- * by   y dl ln(2),   where dl is the absolute error of
- * the internally computed base 2 logarithm.  At the ends
- * of the approximation interval the logarithm equal 1/32
- * and its relative error is about 1 lsb = 1.1e-19.  Hence
- * the predicted relative error in the result is 2.3e-21 y .
- *
- *                      Relative error:
- * arithmetic   domain     # trials      peak         rms
- *
- *    IEEE     +-1000       40000      2.8e-18      3.7e-19
- * .001 < x < 1000, with log(x) uniformly distributed.
- * -1000 < y < 1000, y uniformly distributed.
- *
- *    IEEE     0,8700       60000      6.5e-18      1.0e-18
- * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
- *
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * pow overflow     x**y > MAXNUM      MAXNUM
- * pow underflow   x**y < 1/MAXNUM       0.0
- * pow domain      x<0 and y noninteger  0.0
- *
- */
-
-/*							sinhl.c
- *
- *	Hyperbolic sine, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, sinhl();
- *
- * y = sinhl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns hyperbolic sine of argument in the range MINLOGL to
- * MAXLOGL.
- *
- * 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
- *    IEEE       -2,2       10000       1.5e-19     3.9e-20
- *    IEEE     +-10000      30000       1.1e-19     2.8e-20
- *
- */
-
-/*							sinl.c
- *
- *	Circular sine, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, sinl();
- *
- * y = sinl( 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 the Cody
- * and Waite polynomial form
- *      x + x**3 P(x**2) .
- * Between pi/4 and pi/2 the cosine is represented as
- *      1 - .5 x**2 + x**4 Q(x**2) .
- *
- *
- * ACCURACY:
- *
- *                      Relative error:
- * arithmetic   domain      # trials      peak         rms
- *    IEEE     +-5.5e11      200,000    1.2e-19     2.9e-20
- * 
- * ERROR MESSAGES:
- *
- *   message           condition        value returned
- * sin total loss   x > 2**39               0.0
- *
- * Loss of precision occurs for x > 2**39 = 5.49755813888e11.
- * The routine as implemented flags a TLOSS error for
- * x > 2**39 and returns 0.0.
- */
-/*							cosl.c
- *
- *	Circular cosine, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, cosl();
- *
- * y = cosl( 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 - .5 x**2 + x**4 Q(x**2) .
- * Between pi/4 and pi/2 the sine is represented by the Cody
- * and Waite polynomial form
- *      x  +  x**3 P(x**2) .
- *
- *
- * ACCURACY:
- *
- *                      Relative error:
- * arithmetic   domain      # trials      peak         rms
- *    IEEE     +-5.5e11       50000      1.2e-19     2.9e-20
- */
-
-/*							sqrtl.c
- *
- *	Square root, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, sqrtl();
- *
- * y = sqrtl( 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.
- *
- * Note, some arithmetic coprocessors such as the 8087 and
- * 68881 produce correctly rounded square roots, which this
- * routine will not.
- *
- * ACCURACY:
- *
- *
- *                      Relative error:
- * arithmetic   domain     # trials      peak         rms
- *    IEEE      0,10        30000       8.1e-20     3.1e-20
- *
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * sqrt domain        x < 0            0.0
- *
- */
-
-/*							stdtrl.c
- *
- *	Student's t distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * long double p, t, stdtrl();
- * int k;
- *
- * p = stdtrl( k, t );
- *
- *
- * DESCRIPTION:
- *
- * Computes the integral from minus infinity to t of the Student
- * t distribution with integer k > 0 degrees of freedom:
- *
- *                                      t
- *                                      -
- *                                     | |
- *              -                      |         2   -(k+1)/2
- *             | ( (k+1)/2 )           |  (     x   )
- *       ----------------------        |  ( 1 + --- )        dx
- *                     -               |  (      k  )
- *       sqrt( k pi ) | ( k/2 )        |
- *                                   | |
- *                                    -
- *                                   -inf.
- * 
- * Relation to incomplete beta integral:
- *
- *        1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z )
- * where
- *        z = k/(k + t**2).
- *
- * For t < -1.6, 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 <= 100.  The "domain" refers to t.
- *                      Relative error:
- * arithmetic   domain     # trials      peak         rms
- *    IEEE     -100,-1.6    10000       5.7e-18     9.8e-19
- *    IEEE     -1.6,100     10000       3.8e-18     1.0e-19
- */
-
-/*							stdtril.c
- *
- *	Functional inverse of Student's t distribution
- *
- *
- *
- * SYNOPSIS:
- *
- * long double p, t, stdtril();
- * int k;
- *
- * t = stdtril( k, p );
- *
- *
- * DESCRIPTION:
- *
- * Given probability p, finds the argument t such that stdtrl(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       0,1        3500       4.2e-17     4.1e-18
- */
-
-/*							tanhl.c
- *
- *	Hyperbolic tangent, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, tanhl();
- *
- * y = tanhl( x );
- *
- *
- *
- * DESCRIPTION:
- *
- * Returns hyperbolic tangent of argument in the range MINLOGL to
- * MAXLOGL.
- *
- * 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
- *    IEEE      -2,2        30000       1.3e-19     2.4e-20
- *
- */
-
-/*							tanl.c
- *
- *	Circular tangent, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, tanl();
- *
- * y = tanl( 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
- *    IEEE     +-1.07e9       30000     1.9e-19     4.8e-20
- *
- * ERROR MESSAGES:
- *
- *   message         condition          value returned
- * tan total loss   x > 2^39                0.0
- *
- */
-/*							cotl.c
- *
- *	Circular cotangent, long double precision
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, cotl();
- *
- * y = cotl( 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      1.9e-19     5.1e-20
- *
- *
- * ERROR MESSAGES:
- *
- *   message         condition          value returned
- * cot total loss   x > 2^39                0.0
- * cot singularity  x = 0                  MAXNUM
- *
- */
-
-/*							unityl.c
- *
- * Relative error approximations for function arguments near
- * unity.
- *
- *    log1p(x) = log(1+x)
- *    expm1(x) = exp(x) - 1
- *    cos1m(x) = cos(x) - 1
- *
- */
-
-/*							ynl.c
- *
- *	Bessel function of second kind of integer order
- *
- *
- *
- * SYNOPSIS:
- *
- * long double x, y, ynl();
- * int n;
- *
- * y = ynl( 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
- * y0l() and y1l().
- *
- * If n = 0 or 1 the routine for y0l or y1l is called
- * directly.
- *
- *
- *
- * ACCURACY:
- *
- *
- *       Absolute error, except relative error when y > 1.
- *       x >= 0,  -30 <= n <= +30.
- * arithmetic   domain     # trials      peak         rms
- *    IEEE     -30, 30       10000       1.3e-18     1.8e-19
- *
- *
- * ERROR MESSAGES:
- *
- *   message         condition      value returned
- * ynl singularity   x = 0              MAXNUML
- * ynl overflow                         MAXNUML
- *
- * Spot checked against tables for x, n between 0 and 100.
- *
- */