uClibc now has a math library. muahahahaha!

 -Erik

1 files changed, 3502 insertions, 0 deletions

diff --git a/libm/ldouble/README.txt b/libm/ldouble/README.txt
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+/*							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.
+ *
+ */