Cephes Mathematical Library

Source code archives


Documentation for single precision library.
Documentation for double precision library.
Documentation for 80-bit long double library.
Documentation for 128-bit long double library.
Documentation for extended precision library.

Double Precision Special Functions

Select function name for additional information. For other precisions, see the archives and descriptions listed above.
  • acosh, Inverse hyperbolic cosine
  • airy, Airy functions
  • asin, Inverse circular sine
  • acos, Inverse circular cosine
  • asinh, Inverse hyperbolic sine
  • atan, Inverse circular tangent
  • atan2, Quadrant correct inverse circular tangent
  • atanh, Inverse hyperbolic tangent
  • bdtr, Binomial distribution
  • bdtrc, Complemented binomial distribution
  • bdtri, Inverse binomial distribution
  • beta, Beta function
  • btdtr, Beta distribution
  • cbrt, Cube root
  • chbevl, Evaluate Chebyshev series
  • chdtr, Chi-square distribution
  • chdtrc, Complemented Chi-square distribution
  • chdtri, Inverse of complemented Chi-square distribution
  • cheby, Find Chebyshev coefficients
  • clog, Complex natural logarithm
  • cexp, Complex exponential function
  • csin, Complex circular sine
  • ccos, Complex circular cosine
  • ctan, Complex circular tangent
  • ccot, Complex circular cotangent
  • casin, Complex circular arc sine
  • cacos, Complex circular arc cosine
  • catan, Complex circular arc tangent
  • csinh, Complex hyperbolic sine
  • casinh, Complex inverse hyperbolic sine
  • ccosh, Complex hyperbolic cosine
  • cacosh, Complex inverse hyperbolic cosine
  • ctanh, Complex hyperbolic tangent
  • catanh, Complex inverse hyperbolic tangent
  • cpow, Complex power function
  • cmplx, Complex number arithmetic
  • cabs, Complex absolute value
  • csqrt, Complex square root
  • const, Globally declared constants
  • cosh, Hyperbolic cosine
  • dawsn, Dawson's Integral
  • drand, Pseudorandom number generator
  • ei, Exponential Integral
  • eigens, Eigenvalues and eigenvectors of a real symmetric matrix
  • ellie, Incomplete elliptic integral of the second kind
  • ellik, Incomplete elliptic integral of the first kind
  • ellpe, Complete elliptic integral of the second kind
  • ellpj, Jacobian elliptic functions
  • ellpk, Complete elliptic integral of the first kind
  • euclid, Rational arithmetic routines
  • exp, Exponential function
  • exp10, Base 10 exponential function
  • exp2, Base 2 exponential function
  • expn, Exponential integral En
  • expx2, Exponential of squared argument
  • fabs, Absolute value
  • fac, Factorial function
  • fdtr, F distribution
  • fdtrc, Complemented F distribution
  • fdtri, Inverse of complemented F distribution
  • fftr, Fast Fourier transform
  • floor, Floor function
  • ceil, Ceil function
  • frexp, Extract exponent
  • ldexp, Apply exponent
  • fresnl, Fresnel integral
  • gamma, Gamma function
  • lgam, Natural logarithm of gamma function
  • gdtr, Gamma distribution function
  • gdtrc, Complemented gamma distribution function
  • gels, Linear system with symmetric coefficient matrix
  • hyp2f1, Gauss hypergeometric function
  • hyperg, Confluent hypergeometric function
  • i0, Modified Bessel function of order zero
  • i0e, Exponentially scaled modified Bessel function of order zero
  • i1, Modified Bessel function of order one
  • i1e, Exponentially scaled modified Bessel function of order one
  • igam, Incomplete gamma integral
  • igamc, Complemented incomplete gamma integral
  • igami, Inverse of complemented imcomplete gamma integral
  • incbet, Incomplete beta integral
  • incbi, Inverse of imcomplete beta integral
  • isnan, Test for not a number
  • isfinite, Test for infinity
  • signbit, Extract sign
  • iv, Modified Bessel function of noninteger order
  • j0, Bessel function of order zero
  • y0, Bessel function of the second kind, order zero
  • j1, Bessel function of order one
  • y1, Bessel function of the second kind, order one
  • jn, Bessel function of integer order
  • jv, Bessel function of noninteger order
  • k0, Modified Bessel function, third kind, order zero
  • k0e, Modified Bessel function, third kind, order zero, exponentially scaled
  • k1, Modified Bessel function, third kind, order one
  • k1e, Modified Bessel function, third kind, order one, exponentially scaled
  • kn, Modified Bessel function, third kind, integer order
  • kolmogorov, Kolmogorov, Smirnov distributions
  • lmdif, Linear predictive coding
  • levnsn, Linear predictive coding
  • log, Natural logarithm
  • log10, Common logarithm
  • log2, Base 2 logarithm
  • lrand, Pseudorandom integer number generator
  • lsqrt, Integer square root
  • minv, Matrix inversion
  • mtransp, Matrix transpose
  • nbdtr, Negative binomial distribution
  • nbdtrc, Complemented negative binomial distribution
  • nbdtri, Functional inverse of negative binomial distribution
  • ndtr, Normal distribution function
  • erf, Error function
  • erfc, Complementary error function
  • ndtri, Inverse of normal distribution function
  • pdtr, Poisson distribution function
  • pdtrc, Complemented Poisson distribution function
  • pdtri, Inverse of Poisson distribution function
  • planck, Integral of Planck's black body radiation formula
  • polevl, Evaluate polynomial
  • p1evl, Evaluate polynomial
  • polmisc, Functions of a polynomial
  • polrt, Roots of a polynomial
  • polylog, Polylogarithms
  • polyn, Arithmetic operations on polynomials
  • polyr, Arithmetic operations on polynomials with rational coefficients
  • pow, Power function
  • powi, Integer power function
  • psi, Psi (digamma) function
  • revers, Reversion of power series
  • rgamma, Reciprocal gamma function
  • round, Round to nearest or even integer
  • shichi, Hyperbolic sine and cosine integrals
  • sici, Sine and cosine integrals
  • simpsn, Numerical integration of tabulated function
  • simq, Simultaneous linear equations
  • sin, Circular sine
  • cos, Circular cosine
  • sincos, Sine and cosine by interpolation
  • sindg, Circular sine of angle in degrees
  • cosdg, Circular cosine of angle in degrees
  • sinh, Hyperbolic sine
  • spence, Dilogarithm
  • sqrt, Square root
  • stdtr, Student's t distribution
  • stdtri, Functional inverse of Student's t distribution
  • struve, Struve function
  • tan, Circular tangent
  • cot, Circular cotangent
  • tandg,Circular tangent of argument in degrees
  • cotdg,Circular cotangent of argument in degrees
  • tanh, Hyperbolic tangent
  • log1p, Relative error logarithm
  • expm1, Relative error exponential
  • cosm1, Relative error cosine
  • yn, Bessel function of second kind of integer order
  • zeta, Zeta function of two arguments
  • zetac, Riemann zeta function of two arguments
  •  
    /*							acosh.c
     *
     *	Inverse hyperbolic cosine
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, acosh();
     *
     * y = acosh( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns inverse hyperbolic cosine of argument.
     *
     * If 1 <= x < 1.5, a rational approximation
     *
     *	sqrt(z) * P(z)/Q(z)
     *
     * where z = x-1, is used.  Otherwise,
     *
     * acosh(x)  =  log( x + sqrt( (x-1)(x+1) ).
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       1,3         30000       4.2e-17     1.1e-17
     *    IEEE      1,3         30000       4.6e-16     8.7e-17
     *
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * acosh domain       |x| < 1            NAN
     *
     */
    
     
    /*							airy.c
     *
     *	Airy function
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, ai, aip, bi, bip;
     * int airy();
     *
     * airy( x, &ai, &aip, &bi, &bip );
     *
     *
     *
     * DESCRIPTION:
     *
     * Solution of the differential equation
     *
     *	y"(x) = xy.
     *
     * The function returns the two independent solutions Ai, Bi
     * and their first derivatives Ai'(x), Bi'(x).
     *
     * Evaluation is by power series summation for small x,
     * by rational minimax approximations for large x.
     *
     *
     *
     * ACCURACY:
     * Error criterion is absolute when function <= 1, relative
     * when function > 1, except * denotes relative error criterion.
     * For large negative x, the absolute error increases as x^1.5.
     * For large positive x, the relative error increases as x^1.5.
     *
     * Arithmetic  domain   function  # trials      peak         rms
     * IEEE        -10, 0     Ai        10000       1.6e-15     2.7e-16
     * IEEE          0, 10    Ai        10000       2.3e-14*    1.8e-15*
     * IEEE        -10, 0     Ai'       10000       4.6e-15     7.6e-16
     * IEEE          0, 10    Ai'       10000       1.8e-14*    1.5e-15*
     * IEEE        -10, 10    Bi        30000       4.2e-15     5.3e-16
     * IEEE        -10, 10    Bi'       30000       4.9e-15     7.3e-16
     * DEC         -10, 0     Ai         5000       1.7e-16     2.8e-17
     * DEC           0, 10    Ai         5000       2.1e-15*    1.7e-16*
     * DEC         -10, 0     Ai'        5000       4.7e-16     7.8e-17
     * DEC           0, 10    Ai'       12000       1.8e-15*    1.5e-16*
     * DEC         -10, 10    Bi        10000       5.5e-16     6.8e-17
     * DEC         -10, 10    Bi'        7000       5.3e-16     8.7e-17
     *
     */
    
     
    /*							asin.c
     *
     *	Inverse circular sine
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, asin();
     *
     * y = asin( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns radian angle between -pi/2 and +pi/2 whose sine is x.
     *
     * A rational function of the form x + x**3 P(x**2)/Q(x**2)
     * is used for |x| in the interval [0, 0.5].  If |x| > 0.5 it is
     * transformed by the identity
     *
     *    asin(x) = pi/2 - 2 asin( sqrt( (1-x)/2 ) ).
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC      -1, 1        40000       2.6e-17     7.1e-18
     *    IEEE     -1, 1        10^6        1.9e-16     5.4e-17
     *
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * asin domain        |x| > 1           NAN
     *
     */
    
     
    /*							acos()
     *
     *	Inverse circular cosine
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, acos();
     *
     * y = acos( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns radian angle between 0 and pi whose cosine
     * is x.
     *
     * Analytically, acos(x) = pi/2 - asin(x).  However if |x| is
     * near 1, there is cancellation error in subtracting asin(x)
     * from pi/2.  Hence if x < -0.5,
     *
     *    acos(x) =	 pi - 2.0 * asin( sqrt((1+x)/2) );
     *
     * or if x > +0.5,
     *
     *    acos(x) =	 2.0 * asin(  sqrt((1-x)/2) ).
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       -1, 1       50000       3.3e-17     8.2e-18
     *    IEEE      -1, 1       10^6        2.2e-16     6.5e-17
     *
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * asin domain        |x| > 1           NAN
     */
    
     
    /*							asinh.c
     *
     *	Inverse hyperbolic sine
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, asinh();
     *
     * y = asinh( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns inverse hyperbolic sine of argument.
     *
     * If |x| < 0.5, the function is approximated by a rational
     * form  x + x**3 P(x)/Q(x).  Otherwise,
     *
     *     asinh(x) = log( x + sqrt(1 + x*x) ).
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC      -3,3         75000       4.6e-17     1.1e-17
     *    IEEE     -1,1         30000       3.7e-16     7.8e-17
     *    IEEE      1,3         30000       2.5e-16     6.7e-17
     *
     */
    
     
    /*							atan.c
     *
     *	Inverse circular tangent
     *      (arctangent)
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, atan();
     *
     * y = atan( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns radian angle between -pi/2 and +pi/2 whose tangent
     * is x.
     *
     * Range reduction is from three intervals into the interval
     * from zero to 0.66.  The approximant uses a rational
     * function of degree 4/5 of the form x + x**3 P(x)/Q(x).
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       -10, 10     50000       2.4e-17     8.3e-18
     *    IEEE      -10, 10      10^6       1.8e-16     5.0e-17
     *
     */
    
     
    /*							atan2()
     *
     *	Quadrant correct inverse circular tangent
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, z, atan2();
     *
     * z = atan2( y, x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns radian angle whose tangent is y/x.
     * Define compile time symbol ANSIC = 1 for ANSI standard,
     * range -PI < z <= +PI, args (y,x); else ANSIC = 0 for range
     * 0 to 2PI, args (x,y).
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      -10, 10      10^6       2.5e-16     6.9e-17
     * See atan.c.
     *
     */
    
     
    /*							atanh.c
     *
     *	Inverse hyperbolic tangent
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, atanh();
     *
     * y = atanh( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns inverse hyperbolic tangent of argument in the range
     * MINLOG to MAXLOG.
     *
     * If |x| < 0.5, the rational form x + x**3 P(x)/Q(x) is
     * employed.  Otherwise,
     *        atanh(x) = 0.5 * log( (1+x)/(1-x) ).
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       -1,1        50000       2.4e-17     6.4e-18
     *    IEEE      -1,1        30000       1.9e-16     5.2e-17
     *
     */
    
     
    /*							bdtr.c
     *
     *	Binomial distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * int k, n;
     * double p, y, bdtr();
     *
     * y = bdtr( k, n, p );
     *
     * DESCRIPTION:
     *
     * Returns the sum of the terms 0 through k of the Binomial
     * probability density:
     *
     *   k
     *   --  ( n )   j      n-j
     *   >   (   )  p  (1-p)
     *   --  ( j )
     *  j=0
     *
     * The terms are not summed directly; instead the incomplete
     * beta integral is employed, according to the formula
     *
     * y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ).
     *
     * The arguments must be positive, with p ranging from 0 to 1.
     *
     * ACCURACY:
     *
     * Tested at random points (a,b,p), with p between 0 and 1.
     *
     *               a,b                     Relative error:
     * arithmetic  domain     # trials      peak         rms
     *  For p between 0.001 and 1:
     *    IEEE     0,100       100000      4.3e-15     2.6e-16
     * See also incbet.c.
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * bdtr domain         k < 0            0.0
     *                     n < k
     *                     x < 0, x > 1
     */
    
     
    /*							bdtrc()
     *
     *	Complemented binomial distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * int k, n;
     * double p, y, bdtrc();
     *
     * y = bdtrc( k, n, p );
     *
     * DESCRIPTION:
     *
     * Returns the sum of the terms k+1 through n of the Binomial
     * probability density:
     *
     *   n
     *   --  ( n )   j      n-j
     *   >   (   )  p  (1-p)
     *   --  ( j )
     *  j=k+1
     *
     * The terms are not summed directly; instead the incomplete
     * beta integral is employed, according to the formula
     *
     * y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ).
     *
     * The arguments must be positive, with p ranging from 0 to 1.
     *
     * ACCURACY:
     *
     * Tested at random points (a,b,p).
     *
     *               a,b                     Relative error:
     * arithmetic  domain     # trials      peak         rms
     *  For p between 0.001 and 1:
     *    IEEE     0,100       100000      6.7e-15     8.2e-16
     *  For p between 0 and .001:
     *    IEEE     0,100       100000      1.5e-13     2.7e-15
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * bdtrc domain      x<0, x>1, n<k       0.0
     */
    
     
    /*							bdtri()
     *
     *	Inverse binomial distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * int k, n;
     * double p, y, bdtri();
     *
     * p = bdtr( k, n, y );
     *
     * DESCRIPTION:
     *
     * Finds the event probability p such that the sum of the
     * terms 0 through k of the Binomial probability density
     * is equal to the given cumulative probability y.
     *
     * This is accomplished using the inverse beta integral
     * function and the relation
     *
     * 1 - p = incbi( n-k, k+1, y ).
     *
     * ACCURACY:
     *
     * Tested at random points (a,b,p).
     *
     *               a,b                     Relative error:
     * arithmetic  domain     # trials      peak         rms
     *  For p between 0.001 and 1:
     *    IEEE     0,100       100000      2.3e-14     6.4e-16
     *    IEEE     0,10000     100000      6.6e-12     1.2e-13
     *  For p between 10^-6 and 0.001:
     *    IEEE     0,100       100000      2.0e-12     1.3e-14
     *    IEEE     0,10000     100000      1.5e-12     3.2e-14
     * See also incbi.c.
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * bdtri domain     k < 0, n <= k         0.0
     *                  x < 0, x > 1
     */
    
     
    /*							beta.c
     *
     *	Beta function
     *
     *
     *
     * SYNOPSIS:
     *
     * double a, b, y, beta();
     *
     * y = beta( a, b );
     *
     *
     *
     * DESCRIPTION:
     *
     *                   -     -
     *                  | (a) | (b)
     * beta( a, b )  =  -----------.
     *                     -
     *                    | (a+b)
     *
     * For large arguments the logarithm of the function is
     * evaluated using lgam(), then exponentiated.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC        0,30        1700       7.7e-15     1.5e-15
     *    IEEE       0,30       30000       8.1e-14     1.1e-14
     *
     * ERROR MESSAGES:
     *
     *   message         condition          value returned
     * beta overflow    log(beta) > MAXLOG       0.0
     *                  a or b <0 integer        0.0
     *
     */
    
     
    /*							btdtr.c
     *
     *	Beta distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * double a, b, x, y, btdtr();
     *
     * y = btdtr( a, b, x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the area from zero to x under the beta density
     * function:
     *
     *
     *                          x
     *            -             -
     *           | (a+b)       | |  a-1      b-1
     * P(x)  =  ----------     |   t    (1-t)    dt
     *           -     -     | |
     *          | (a) | (b)   -
     *                         0
     *
     *
     * This function is identical to the incomplete beta
     * integral function incbet(a, b, x).
     *
     * The complemented function is
     *
     * 1 - P(1-x)  =  incbet( b, a, x );
     *
     *
     * ACCURACY:
     *
     * See incbet.c.
     *
     */
    
     
    /*							cbrt.c
     *
     *	Cube root
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, cbrt();
     *
     * y = cbrt( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the cube root of the argument, which may be negative.
     *
     * Range reduction involves determining the power of 2 of
     * the argument.  A polynomial of degree 2 applied to the
     * mantissa, and multiplication by the cube root of 1, 2, or 4
     * approximates the root to within about 0.1%.  Then Newton's
     * iteration is used three times to converge to an accurate
     * result.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC        -10,10     200000      1.8e-17     6.2e-18
     *    IEEE       0,1e308     30000      1.5e-16     5.0e-17
     *
     */
    
     
    /*							chbevl.c
     *
     *	Evaluate Chebyshev series
     *
     *
     *
     * SYNOPSIS:
     *
     * int N;
     * double x, y, coef[N], chebevl();
     *
     * y = chbevl( x, coef, N );
     *
     *
     *
     * DESCRIPTION:
     *
     * Evaluates the series
     *
     *        N-1
     *         - '
     *  y  =   >   coef[i] T (x/2)
     *         -            i
     *        i=0
     *
     * of Chebyshev polynomials Ti at argument x/2.
     *
     * Coefficients are stored in reverse order, i.e. the zero
     * order term is last in the array.  Note N is the number of
     * coefficients, not the order.
     *
     * If coefficients are for the interval a to b, x must
     * have been transformed to x -> 2(2x - b - a)/(b-a) before
     * entering the routine.  This maps x from (a, b) to (-1, 1),
     * over which the Chebyshev polynomials are defined.
     *
     * If the coefficients are for the inverted interval, in
     * which (a, b) is mapped to (1/b, 1/a), the transformation
     * required is x -> 2(2ab/x - b - a)/(b-a).  If b is infinity,
     * this becomes x -> 4a/x - 1.
     *
     *
     *
     * SPEED:
     *
     * Taking advantage of the recurrence properties of the
     * Chebyshev polynomials, the routine requires one more
     * addition per loop than evaluating a nested polynomial of
     * the same degree.
     *
     */
    
     
    /*							chdtr.c
     *
     *	Chi-square distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * double df, x, y, chdtr();
     *
     * y = chdtr( df, x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the area under the left hand tail (from 0 to x)
     * of the Chi square probability density function with
     * v degrees of freedom.
     *
     *
     *                                  inf.
     *                                    -
     *                        1          | |  v/2-1  -t/2
     *  P( x | v )   =   -----------     |   t      e     dt
     *                    v/2  -       | |
     *                   2    | (v/2)   -
     *                                   x
     *
     * where x is the Chi-square variable.
     *
     * The incomplete gamma integral is used, according to the
     * formula
     *
     *	y = chdtr( v, x ) = igam( v/2.0, x/2.0 ).
     *
     *
     * The arguments must both be positive.
     *
     *
     *
     * ACCURACY:
     *
     * See igam().
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * chdtr domain   x < 0 or v < 1        0.0
     */
    
     
    /*							chdtrc()
     *
     *	Complemented Chi-square distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * double v, x, y, chdtrc();
     *
     * y = chdtrc( v, x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the area under the right hand tail (from x to
     * infinity) of the Chi square probability density function
     * with v degrees of freedom:
     *
     *
     *                                  inf.
     *                                    -
     *                        1          | |  v/2-1  -t/2
     *  P( x | v )   =   -----------     |   t      e     dt
     *                    v/2  -       | |
     *                   2    | (v/2)   -
     *                                   x
     *
     * where x is the Chi-square variable.
     *
     * The incomplete gamma integral is used, according to the
     * formula
     *
     *	y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ).
     *
     *
     * The arguments must both be positive.
     *
     *
     *
     * ACCURACY:
     *
     * See igamc().
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * chdtrc domain  x < 0 or v < 1        0.0
     */
    
     
    /*							chdtri()
     *
     *	Inverse of complemented Chi-square distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * double df, x, y, chdtri();
     *
     * x = chdtri( df, y );
     *
     *
     *
     *
     * DESCRIPTION:
     *
     * Finds the Chi-square argument x such that the integral
     * from x to infinity of the Chi-square density is equal
     * to the given cumulative probability y.
     *
     * This is accomplished using the inverse gamma integral
     * function and the relation
     *
     *    x/2 = igami( df/2, y );
     *
     *
     *
     *
     * ACCURACY:
     *
     * See igami.c.
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * chdtri domain   y < 0 or y > 1        0.0
     *                     v < 1
     *
     */
    
     
    /*	cheby.c
     *
     * Program to calculate coefficients of the Chebyshev polynomial
     * expansion of a given input function.  The algorithm computes
     * the discrete Fourier cosine transform of the function evaluated
     * at unevenly spaced points.  Library routine chbevl.c uses the
     * coefficients to calculate an approximate value of the original
     * function.
     */
    
     
    /*							clog.c
     *
     *	Complex natural logarithm
     *
     *
     *
     * SYNOPSIS:
     *
     * void clog();
     * cmplx z, w;
     *
     * clog( &z, &w );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns complex logarithm to the base e (2.718...) of
     * the complex argument x.
     *
     * If z = x + iy, r = sqrt( x**2 + y**2 ),
     * then
     *       w = log(r) + i arctan(y/x).
     * 
     * The arctangent ranges from -PI to +PI.
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       -10,+10      7000       8.5e-17     1.9e-17
     *    IEEE      -10,+10     30000       5.0e-15     1.1e-16
     *
     * Larger relative error can be observed for z near 1 +i0.
     * In IEEE arithmetic the peak absolute error is 5.2e-16, rms
     * absolute error 1.0e-16.
     */
    
     
    /*							cexp()
     *
     *	Complex exponential function
     *
     *
     *
     * SYNOPSIS:
     *
     * void cexp();
     * cmplx z, w;
     *
     * cexp( &z, &w );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the exponential of the complex argument z
     * into the complex result w.
     *
     * If
     *     z = x + iy,
     *     r = exp(x),
     *
     * then
     *
     *     w = r cos y + i r sin y.
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       -10,+10      8700       3.7e-17     1.1e-17
     *    IEEE      -10,+10     30000       3.0e-16     8.7e-17
     *
     */
    
     
    /*							csin()
     *
     *	Complex circular sine
     *
     *
     *
     * SYNOPSIS:
     *
     * void csin();
     * cmplx z, w;
     *
     * csin( &z, &w );
     *
     *
     *
     * DESCRIPTION:
     *
     * If
     *     z = x + iy,
     *
     * then
     *
     *     w = sin x  cosh y  +  i cos x sinh y.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       -10,+10      8400       5.3e-17     1.3e-17
     *    IEEE      -10,+10     30000       3.8e-16     1.0e-16
     * Also tested by csin(casin(z)) = z.
     *
     */
    
     
    /*							ccos()
     *
     *	Complex circular cosine
     *
     *
     *
     * SYNOPSIS:
     *
     * void ccos();
     * cmplx z, w;
     *
     * ccos( &z, &w );
     *
     *
     *
     * DESCRIPTION:
     *
     * If
     *     z = x + iy,
     *
     * then
     *
     *     w = cos x  cosh y  -  i sin x sinh y.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       -10,+10      8400       4.5e-17     1.3e-17
     *    IEEE      -10,+10     30000       3.8e-16     1.0e-16
     */
    
     
    /*							ctan()
     *
     *	Complex circular tangent
     *
     *
     *
     * SYNOPSIS:
     *
     * void ctan();
     * cmplx z, w;
     *
     * ctan( &z, &w );
     *
     *
     *
     * DESCRIPTION:
     *
     * If
     *     z = x + iy,
     *
     * then
     *
     *           sin 2x  +  i sinh 2y
     *     w  =  --------------------.
     *            cos 2x  +  cosh 2y
     *
     * On the real axis the denominator is zero at odd multiples
     * of PI/2.  The denominator is evaluated by its Taylor
     * series near these points.
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       -10,+10      5200       7.1e-17     1.6e-17
     *    IEEE      -10,+10     30000       7.2e-16     1.2e-16
     * Also tested by ctan * ccot = 1 and catan(ctan(z))  =  z.
     */
    
     
    /*							ccot()
     *
     *	Complex circular cotangent
     *
     *
     *
     * SYNOPSIS:
     *
     * void ccot();
     * cmplx z, w;
     *
     * ccot( &z, &w );
     *
     *
     *
     * DESCRIPTION:
     *
     * If
     *     z = x + iy,
     *
     * then
     *
     *           sin 2x  -  i sinh 2y
     *     w  =  --------------------.
     *            cosh 2y  -  cos 2x
     *
     * On the real axis, the denominator has zeros at even
     * multiples of PI/2.  Near these points it is evaluated
     * by a Taylor series.
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       -10,+10      3000       6.5e-17     1.6e-17
     *    IEEE      -10,+10     30000       9.2e-16     1.2e-16
     * Also tested by ctan * ccot = 1 + i0.
     */
    
     
    /*							casin()
     *
     *	Complex circular arc sine
     *
     *
     *
     * SYNOPSIS:
     *
     * void casin();
     * cmplx z, w;
     *
     * casin( &z, &w );
     *
     *
     *
     * DESCRIPTION:
     *
     * Inverse complex sine:
     *
     *                               2
     * w = -i clog( iz + csqrt( 1 - z ) ).
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       -10,+10     10100       2.1e-15     3.4e-16
     *    IEEE      -10,+10     30000       2.2e-14     2.7e-15
     * Larger relative error can be observed for z near zero.
     * Also tested by csin(casin(z)) = z.
     */
    
     
    /*							cacos()
     *
     *	Complex circular arc cosine
     *
     *
     *
     * SYNOPSIS:
     *
     * void cacos();
     * cmplx z, w;
     *
     * cacos( &z, &w );
     *
     *
     *
     * DESCRIPTION:
     *
     *
     * w = arccos z  =  PI/2 - arcsin z.
     *
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       -10,+10      5200      1.6e-15      2.8e-16
     *    IEEE      -10,+10     30000      1.8e-14      2.2e-15
     */
    
     
    /*							catan()
     *
     *	Complex circular arc tangent
     *
     *
     *
     * SYNOPSIS:
     *
     * void catan();
     * cmplx z, w;
     *
     * catan( &z, &w );
     *
     *
     *
     * DESCRIPTION:
     *
     * If
     *     z = x + iy,
     *
     * then
     *          1       (    2x     )
     * Re w  =  - arctan(-----------)  +  k PI
     *          2       (     2    2)
     *                  (1 - x  - y )
     *
     *               ( 2         2)
     *          1    (x  +  (y+1) )
     * Im w  =  - log(------------)
     *          4    ( 2         2)
     *               (x  +  (y-1) )
     *
     * Where k is an arbitrary integer.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       -10,+10      5900       1.3e-16     7.8e-18
     *    IEEE      -10,+10     30000       2.3e-15     8.5e-17
     * The check catan( ctan(z) )  =  z, with |x| and |y| < PI/2,
     * had peak relative error 1.5e-16, rms relative error
     * 2.9e-17.  See also clog().
     */
    
     
    /*							csinh
     *
     *	Complex hyperbolic sine
     *
     *
     *
     * SYNOPSIS:
     *
     * void csinh();
     * cmplx z, w;
     *
     * csinh( &z, &w );
     *
     *
     * DESCRIPTION:
     *
     * csinh z = (cexp(z) - cexp(-z))/2
     *         = sinh x * cos y  +  i cosh x * sin y .
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      -10,+10     30000       3.1e-16     8.2e-17
     *
     */
    
     
    /*							casinh
     *
     *	Complex inverse hyperbolic sine
     *
     *
     *
     * SYNOPSIS:
     *
     * void casinh();
     * cmplx z, w;
     *
     * casinh (&z, &w);
     *
     *
     *
     * DESCRIPTION:
     *
     * casinh z = -i casin iz .
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      -10,+10     30000       1.8e-14     2.6e-15
     *
     */
    
     
    /*							ccosh
     *
     *	Complex hyperbolic cosine
     *
     *
     *
     * SYNOPSIS:
     *
     * void ccosh();
     * cmplx z, w;
     *
     * ccosh (&z, &w);
     *
     *
     *
     * DESCRIPTION:
     *
     * ccosh(z) = cosh x  cos y + i sinh x sin y .
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      -10,+10     30000       2.9e-16     8.1e-17
     *
     */
    
     
    /*							cacosh
     *
     *	Complex inverse hyperbolic cosine
     *
     *
     *
     * SYNOPSIS:
     *
     * void cacosh();
     * cmplx z, w;
     *
     * cacosh (&z, &w);
     *
     *
     *
     * DESCRIPTION:
     *
     * acosh z = i acos z .
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      -10,+10     30000       1.6e-14     2.1e-15
     *
     */
    
     
    /*							ctanh
     *
     *	Complex hyperbolic tangent
     *
     *
     *
     * SYNOPSIS:
     *
     * void ctanh();
     * cmplx z, w;
     *
     * ctanh (&z, &w);
     *
     *
     *
     * DESCRIPTION:
     *
     * tanh z = (sinh 2x  +  i sin 2y) / (cosh 2x + cos 2y) .
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      -10,+10     30000       1.7e-14     2.4e-16
     *
     */
    
     
    /*							catanh
     *
     *	Complex inverse hyperbolic tangent
     *
     *
     *
     * SYNOPSIS:
     *
     * void catanh();
     * cmplx z, w;
     *
     * catanh (&z, &w);
     *
     *
     *
     * DESCRIPTION:
     *
     * Inverse tanh, equal to  -i catan (iz);
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      -10,+10     30000       2.3e-16     6.2e-17
     *
     */
    
     
    /*							cpow
     *
     *	Complex power function
     *
     *
     *
     * SYNOPSIS:
     *
     * void cpow();
     * cmplx a, z, w;
     *
     * cpow (&a, &z, &w);
     *
     *
     *
     * DESCRIPTION:
     *
     * Raises complex A to the complex Zth power.
     * Definition is per AMS55 # 4.2.8,
     * analytically equivalent to cpow(a,z) = cexp(z clog(a)).
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      -10,+10     30000       9.4e-15     1.5e-15
     *
     */
    
     
    /*							cmplx.c
     *
     *	Complex number arithmetic
     *
     *
     *
     * SYNOPSIS:
     *
     * typedef struct {
     *      double r;     real part
     *      double i;     imaginary part
     *     }cmplx;
     *
     * cmplx *a, *b, *c;
     *
     * cadd( a, b, c );     c = b + a
     * csub( a, b, c );     c = b - a
     * cmul( a, b, c );     c = b * a
     * cdiv( a, b, c );     c = b / a
     * cneg( c );           c = -c
     * cmov( b, c );        c = b
     *
     *
     *
     * DESCRIPTION:
     *
     * Addition:
     *    c.r  =  b.r + a.r
     *    c.i  =  b.i + a.i
     *
     * Subtraction:
     *    c.r  =  b.r - a.r
     *    c.i  =  b.i - a.i
     *
     * Multiplication:
     *    c.r  =  b.r * a.r  -  b.i * a.i
     *    c.i  =  b.r * a.i  +  b.i * a.r
     *
     * Division:
     *    d    =  a.r * a.r  +  a.i * a.i
     *    c.r  = (b.r * a.r  + b.i * a.i)/d
     *    c.i  = (b.i * a.r  -  b.r * a.i)/d
     * ACCURACY:
     *
     * In DEC arithmetic, the test (1/z) * z = 1 had peak relative
     * error 3.1e-17, rms 1.2e-17.  The test (y/z) * (z/y) = 1 had
     * peak relative error 8.3e-17, rms 2.1e-17.
     *
     * Tests in the rectangle {-10,+10}:
     *                      Relative error:
     * arithmetic   function  # trials      peak         rms
     *    DEC        cadd       10000       1.4e-17     3.4e-18
     *    IEEE       cadd      100000       1.1e-16     2.7e-17
     *    DEC        csub       10000       1.4e-17     4.5e-18
     *    IEEE       csub      100000       1.1e-16     3.4e-17
     *    DEC        cmul        3000       2.3e-17     8.7e-18
     *    IEEE       cmul      100000       2.1e-16     6.9e-17
     *    DEC        cdiv       18000       4.9e-17     1.3e-17
     *    IEEE       cdiv      100000       3.7e-16     1.1e-16
     */
    
     
    /*							cabs()
     *
     *	Complex absolute value
     *
     *
     *
     * SYNOPSIS:
     *
     * double cabs();
     * cmplx z;
     * double a;
     *
     * a = cabs( &z );
     *
     *
     *
     * DESCRIPTION:
     *
     *
     * If z = x + iy
     *
     * then
     *
     *       a = sqrt( x**2 + y**2 ).
     * 
     * Overflow and underflow are avoided by testing the magnitudes
     * of x and y before squaring.  If either is outside half of
     * the floating point full scale range, both are rescaled.
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       -30,+30     30000       3.2e-17     9.2e-18
     *    IEEE      -10,+10    100000       2.7e-16     6.9e-17
     */
    
     
    /*							csqrt()
     *
     *	Complex square root
     *
     *
     *
     * SYNOPSIS:
     *
     * void csqrt();
     * cmplx z, w;
     *
     * csqrt( &z, &w );
     *
     *
     *
     * DESCRIPTION:
     *
     *
     * If z = x + iy,  r = |z|, then
     *
     *                       1/2
     * Im w  =  [ (r - x)/2 ]   ,
     *
     * Re w  =  y / 2 Im w.
     *
     *
     * Note that -w is also a square root of z.  The root chosen
     * is always in the upper half plane.
     *
     * Because of the potential for cancellation error in r - x,
     * the result is sharpened by doing a Heron iteration
     * (see sqrt.c) in complex arithmetic.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       -10,+10     25000       3.2e-17     9.6e-18
     *    IEEE      -10,+10    100000       3.2e-16     7.7e-17
     *
     *                        2
     * Also tested by csqrt( z ) = z, and tested by arguments
     * close to the real axis.
     */
    
     
    /*							const.c
     *
     *	Globally declared constants
     *
     *
     *
     * SYNOPSIS:
     *
     * extern double nameofconstant;
     *
     *
     *
     *
     * DESCRIPTION:
     *
     * This file contains a number of mathematical constants and
     * also some needed size parameters of the computer arithmetic.
     * The values are supplied as arrays of hexadecimal integers
     * for IEEE arithmetic; arrays of octal constants for DEC
     * arithmetic; and in a normal decimal scientific notation for
     * other machines.  The particular notation used is determined
     * by a symbol (DEC, IBMPC, or UNK) defined in the include file
     * mconf.h.
     *
     * The default size parameters are as follows.
     *
     * For DEC and UNK modes:
     * MACHEP =  1.38777878078144567553E-17       2**-56
     * MAXLOG =  8.8029691931113054295988E1       log(2**127)
     * MINLOG = -8.872283911167299960540E1        log(2**-128)
     * MAXNUM =  1.701411834604692317316873e38    2**127
     *
     * For IEEE arithmetic (IBMPC):
     * MACHEP =  1.11022302462515654042E-16       2**-53
     * MAXLOG =  7.09782712893383996843E2         log(2**1024)
     * MINLOG = -7.08396418532264106224E2         log(2**-1022)
     * MAXNUM =  1.7976931348623158E308           2**1024
     *
     * The global symbols for mathematical constants are
     * PI     =  3.14159265358979323846           pi
     * PIO2   =  1.57079632679489661923           pi/2
     * PIO4   =  7.85398163397448309616E-1        pi/4
     * SQRT2  =  1.41421356237309504880           sqrt(2)
     * SQRTH  =  7.07106781186547524401E-1        sqrt(2)/2
     * LOG2E  =  1.4426950408889634073599         1/log(2)
     * SQ2OPI =  7.9788456080286535587989E-1      sqrt( 2/pi )
     * LOGE2  =  6.93147180559945309417E-1        log(2)
     * LOGSQ2 =  3.46573590279972654709E-1        log(2)/2
     * THPIO4 =  2.35619449019234492885           3*pi/4
     * TWOOPI =  6.36619772367581343075535E-1     2/pi
     *
     * These lists are subject to change.
     */
    
     
    /*							cosh.c
     *
     *	Hyperbolic cosine
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, cosh();
     *
     * y = cosh( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns hyperbolic cosine of argument in the range MINLOG to
     * MAXLOG.
     *
     * cosh(x)  =  ( exp(x) + exp(-x) )/2.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       +- 88       50000       4.0e-17     7.7e-18
     *    IEEE     +-MAXLOG     30000       2.6e-16     5.7e-17
     *
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * cosh overflow    |x| > MAXLOG       MAXNUM
     *
     *
     */
    
     
    /*							dawsn.c
     *
     *	Dawson's Integral
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, dawsn();
     *
     * y = dawsn( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Approximates the integral
     *
     *                             x
     *                             -
     *                      2     | |        2
     *  dawsn(x)  =  exp( -x  )   |    exp( t  ) dt
     *                          | |
     *                           -
     *                           0
     *
     * Three different rational approximations are employed, for
     * the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up.
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0,10        10000       6.9e-16     1.0e-16
     *    DEC       0,10         6000       7.4e-17     1.4e-17
     *
     *
     */
    
     
    /*							drand.c
     *
     *	Pseudorandom number generator
     *
     *
     *
     * SYNOPSIS:
     *
     * double y, drand();
     *
     * drand( &y );
     *
     *
     *
     * DESCRIPTION:
     *
     * Yields a random number 1.0 <= y < 2.0.
     *
     * The three-generator congruential algorithm by Brian
     * Wichmann and David Hill (BYTE magazine, March, 1987,
     * pp 127-8) is used. The period, given by them, is
     * 6953607871644.
     *
     * Versions invoked by the different arithmetic compile
     * time options DEC, IBMPC, and MIEEE, produce
     * approximately the same sequences, differing only in the
     * least significant bits of the numbers. The UNK option
     * implements the algorithm as recommended in the BYTE
     * article.  It may be used on all computers. However,
     * the low order bits of a double precision number may
     * not be adequately random, and may vary due to arithmetic
     * implementation details on different computers.
     *
     * The other compile options generate an additional random
     * integer that overwrites the low order bits of the double
     * precision number.  This reduces the period by a factor of
     * two but tends to overcome the problems mentioned.
     *
     */
    
     
    /*							ei.c
     *
     *	Exponential integral
     *
     *
     * SYNOPSIS:
     *
     * double x, y, ei();
     *
     * y = ei( x );
     *
     *
     *
     * DESCRIPTION:
     *
     *               x
     *                -     t
     *               | |   e
     *    Ei(x) =   -|-   ---  dt .
     *             | |     t
     *              -
     *             -inf
     * 
     * Not defined for x <= 0.
     * See also expn.c.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE       0,100       50000      8.6e-16     1.3e-16
     *
     */
    
     
    /*							eigens.c
     *
     *	Eigenvalues and eigenvectors of a real symmetric matrix
     *
     *
     *
     * SYNOPSIS:
     *
     * int n;
     * double A[n*(n+1)/2], EV[n*n], E[n];
     * void eigens( A, EV, E, n );
     *
     *
     *
     * DESCRIPTION:
     *
     * The algorithm is due to J. vonNeumann.
     *
     * A[] is a symmetric matrix stored in lower triangular form.
     * That is, A[ row, column ] = A[ (row*row+row)/2 + column ]
     * or equivalently with row and column interchanged.  The
     * indices row and column run from 0 through n-1.
     *
     * EV[] is the output matrix of eigenvectors stored columnwise.
     * That is, the elements of each eigenvector appear in sequential
     * memory order.  The jth element of the ith eigenvector is
     * EV[ n*i+j ] = EV[i][j].
     *
     * E[] is the output matrix of eigenvalues.  The ith element
     * of E corresponds to the ith eigenvector (the ith row of EV).
     *
     * On output, the matrix A will have been diagonalized and its
     * orginal contents are destroyed.
     *
     * ACCURACY:
     *
     * The error is controlled by an internal parameter called RANGE
     * which is set to 1e-10.  After diagonalization, the
     * off-diagonal elements of A will have been reduced by
     * this factor.
     *
     * ERROR MESSAGES:
     *
     * None.
     *
     */
    
     
    /*							ellie.c
     *
     *	Incomplete elliptic integral of the second kind
     *
     *
     *
     * SYNOPSIS:
     *
     * double phi, m, y, ellie();
     *
     * y = ellie( phi, m );
     *
     *
     *
     * DESCRIPTION:
     *
     * Approximates the integral
     *
     *
     *                phi
     *                 -
     *                | |
     *                |                   2
     * E(phi_\m)  =    |    sqrt( 1 - m sin t ) dt
     *                |
     *              | |    
     *               -
     *                0
     *
     * of amplitude phi and modulus m, using the arithmetic -
     * geometric mean algorithm.
     *
     *
     *
     * ACCURACY:
     *
     * Tested at random arguments with phi in [-10, 10] and m in
     * [0, 1].
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC        0,2         2000       1.9e-16     3.4e-17
     *    IEEE     -10,10      150000       3.3e-15     1.4e-16
     *
     *
     */
    
     
    /*							ellik.c
     *
     *	Incomplete elliptic integral of the first kind
     *
     *
     *
     * SYNOPSIS:
     *
     * double phi, m, y, ellik();
     *
     * y = ellik( phi, m );
     *
     *
     *
     * DESCRIPTION:
     *
     * Approximates the integral
     *
     *
     *
     *                phi
     *                 -
     *                | |
     *                |           dt
     * F(phi_\m)  =    |    ------------------
     *                |                   2
     *              | |    sqrt( 1 - m sin t )
     *               -
     *                0
     *
     * of amplitude phi and modulus m, using the arithmetic -
     * geometric mean algorithm.
     *
     *
     *
     *
     * ACCURACY:
     *
     * Tested at random points with m in [0, 1] and phi as indicated.
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE     -10,10       200000      7.4e-16     1.0e-16
     *
     *
     */
    
     
    /*							ellpe.c
     *
     *	Complete elliptic integral of the second kind
     *
     *
     *
     * SYNOPSIS:
     *
     * double m1, y, ellpe();
     *
     * y = ellpe( m1 );
     *
     *
     *
     * DESCRIPTION:
     *
     * Approximates the integral
     *
     *
     *            pi/2
     *             -
     *            | |                 2
     * E(m)  =    |    sqrt( 1 - m sin t ) dt
     *          | |    
     *           -
     *            0
     *
     * Where m = 1 - m1, using the approximation
     *
     *      P(x)  -  x log x Q(x).
     *
     * Though there are no singularities, the argument m1 is used
     * rather than m for compatibility with ellpk().
     *
     * E(1) = 1; E(0) = pi/2.
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC        0, 1       13000       3.1e-17     9.4e-18
     *    IEEE       0, 1       10000       2.1e-16     7.3e-17
     *
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * ellpe domain      x<0, x>1            0.0
     *
     */
    
     
    /*							ellpj.c
     *
     *	Jacobian Elliptic Functions
     *
     *
     *
     * SYNOPSIS:
     *
     * double u, m, sn, cn, dn, phi;
     * int ellpj();
     *
     * ellpj( u, m, &sn, &cn, &dn, &phi );
     *
     *
     *
     * DESCRIPTION:
     *
     *
     * Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m),
     * and dn(u|m) of parameter m between 0 and 1, and real
     * argument u.
     *
     * These functions are periodic, with quarter-period on the
     * real axis equal to the complete elliptic integral
     * ellpk(1.0-m).
     *
     * Relation to incomplete elliptic integral:
     * If u = ellik(phi,m), then sn(u|m) = sin(phi),
     * and cn(u|m) = cos(phi).  Phi is called the amplitude of u.
     *
     * Computation is by means of the arithmetic-geometric mean
     * algorithm, except when m is within 1e-9 of 0 or 1.  In the
     * latter case with m close to 1, the approximation applies
     * only for phi < pi/2.
     *
     * ACCURACY:
     *
     * Tested at random points with u between 0 and 10, m between
     * 0 and 1.
     *
     *            Absolute error (* = relative error):
     * arithmetic   function   # trials      peak         rms
     *    DEC       sn           1800       4.5e-16     8.7e-17
     *    IEEE      phi         10000       9.2e-16*    1.4e-16*
     *    IEEE      sn          50000       4.1e-15     4.6e-16
     *    IEEE      cn          40000       3.6e-15     4.4e-16
     *    IEEE      dn         100000       3.9e-15     1.7e-16
     *
     * Larger errors occur for m near 1.
     * Peak error observed in consistency check using addition
     * theorem for sn(u+v) was 4e-16 (absolute).  Also tested by
     * the above relation to the incomplete elliptic integral.
     * Accuracy deteriorates when u is large.
     *
     */
    
     
    /*							ellpk.c
     *
     *	Complete elliptic integral of the first kind
     *
     *
     *
     * SYNOPSIS:
     *
     * double m1, y, ellpk();
     *
     * y = ellpk( m1 );
     *
     *
     *
     * DESCRIPTION:
     *
     * Approximates the integral
     *
     *
     *
     *            pi/2
     *             -
     *            | |
     *            |           dt
     * K(m)  =    |    ------------------
     *            |                   2
     *          | |    sqrt( 1 - m sin t )
     *           -
     *            0
     *
     * where m = 1 - m1, using the approximation
     *
     *     P(x)  -  log x Q(x).
     *
     * The argument m1 is used rather than m so that the logarithmic
     * singularity at m = 1 will be shifted to the origin; this
     * preserves maximum accuracy.
     *
     * K(0) = pi/2.
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC        0,1        16000       3.5e-17     1.1e-17
     *    IEEE       0,1        30000       2.5e-16     6.8e-17
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * ellpk domain       x<0, x>1           0.0
     *
     */
    
     
    /*							euclid.c
     *
     *	Rational arithmetic routines
     *
     *
     *
     * SYNOPSIS:
     *
     * 
     * typedef struct
     *      {
     *      double n;  numerator
     *      double d;  denominator
     *      }fract;
     *
     * radd( a, b, c )      c = b + a
     * rsub( a, b, c )      c = b - a
     * rmul( a, b, c )      c = b * a
     * rdiv( a, b, c )      c = b / a
     * euclid( &n, &d )     Reduce n/d to lowest terms,
     *                      return greatest common divisor.
     *
     * Arguments of the routines are pointers to the structures.
     * The double precision numbers are assumed, without checking,
     * to be integer valued.  Overflow conditions are reported.
     */
    
     
    /*							exp.c
     *
     *	Exponential function
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, exp();
     *
     * y = exp( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns e (2.71828...) raised to the x power.
     *
     * Range reduction is accomplished by separating the argument
     * into an integer k and fraction f such that
     *
     *     x    k  f
     *    e  = 2  e.
     *
     * A Pade' form  1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
     * of degree 2/3 is used to approximate exp(f) in the basic
     * interval [-0.5, 0.5].
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       +- 88       50000       2.8e-17     7.0e-18
     *    IEEE      +- 708      40000       2.0e-16     5.6e-17
     *
     *
     * Error amplification in the exponential function can be
     * a serious matter.  The error propagation involves
     * exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ),
     * which shows that a 1 lsb error in representing X produces
     * a relative error of X times 1 lsb in the function.
     * While the routine gives an accurate result for arguments
     * that are exactly represented by a double precision
     * computer number, the result contains amplified roundoff
     * error for large arguments not exactly represented.
     *
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * exp underflow    x < MINLOG         0.0
     * exp overflow     x > MAXLOG         INFINITY
     *
     */
    
     
    /*							exp10.c
     *
     *	Base 10 exponential function
     *      (Common antilogarithm)
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, exp10();
     *
     * y = exp10( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns 10 raised to the x power.
     *
     * Range reduction is accomplished by expressing the argument
     * as 10**x = 2**n 10**f, with |f| < 0.5 log10(2).
     * The Pade' form
     *
     *    1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
     *
     * is used to approximate 10**f.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE     -307,+307    30000       2.2e-16     5.5e-17
     * Test result from an earlier version (2.1):
     *    DEC       -38,+38     70000       3.1e-17     7.0e-18
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * exp10 underflow    x < -MAXL10        0.0
     * exp10 overflow     x > MAXL10       MAXNUM
     *
     * DEC arithmetic: MAXL10 = 38.230809449325611792.
     * IEEE arithmetic: MAXL10 = 308.2547155599167.
     *
     */
    
     
    /*							exp2.c
     *
     *	Base 2 exponential function
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, exp2();
     *
     * y = exp2( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns 2 raised to the x power.
     *
     * Range reduction is accomplished by separating the argument
     * into an integer k and fraction f such that
     *     x    k  f
     *    2  = 2  2.
     *
     * A Pade' form
     *
     *   1 + 2x P(x**2) / (Q(x**2) - x P(x**2) )
     *
     * approximates 2**x in the basic range [-0.5, 0.5].
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE    -1022,+1024   30000       1.8e-16     5.4e-17
     *
     *
     * See exp.c for comments on error amplification.
     *
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * exp underflow    x < -MAXL2        0.0
     * exp overflow     x > MAXL2         MAXNUM
     *
     * For DEC arithmetic, MAXL2 = 127.
     * For IEEE arithmetic, MAXL2 = 1024.
     */
    
     
    /*							expn.c
     *
     *		Exponential integral En
     *
     *
     *
     * SYNOPSIS:
     *
     * int n;
     * double x, y, expn();
     *
     * y = expn( n, x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Evaluates the exponential integral
     *
     *                 inf.
     *                   -
     *                  | |   -xt
     *                  |    e
     *      E (x)  =    |    ----  dt.
     *       n          |      n
     *                | |     t
     *                 -
     *                  1
     *
     *
     * Both n and x must be nonnegative.
     *
     * The routine employs either a power series, a continued
     * fraction, or an asymptotic formula depending on the
     * relative values of n and x.
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       0, 30        5000       2.0e-16     4.6e-17
     *    IEEE      0, 30       10000       1.7e-15     3.6e-16
     *
     */
    
     
    /*							expx2.c
     *
     *	Exponential of squared argument
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, expx2();
     * int sign;
     *
     * y = expx2( x, sign );
     *
     *
     *
     * DESCRIPTION:
     *
     * Computes y = exp(x*x) while suppressing error amplification
     * that would ordinarily arise from the inexactness of the
     * exponential argument x*x.
     *
     * If sign < 0, the result is inverted; i.e., y = exp(-x*x) .
     * 
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic    domain     # trials      peak         rms
     *   IEEE      -26.6, 26.6    10^7       3.9e-16     8.9e-17
     *
     */
    
     
    /*							fabs.c
     *
     *		Absolute value
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y;
     *
     * y = fabs( x );
     *
     *
     *
     * DESCRIPTION:
     * 
     * Returns the absolute value of the argument.
     *
     */
    
     
    /*							fac.c
     *
     *	Factorial function
     *
     *
     *
     * SYNOPSIS:
     *
     * double y, fac();
     * int i;
     *
     * y = fac( i );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns factorial of i  =  1 * 2 * 3 * ... * i.
     * fac(0) = 1.0.
     *
     * Due to machine arithmetic bounds the largest value of
     * i accepted is 33 in DEC arithmetic or 170 in IEEE
     * arithmetic.  Greater values, or negative ones,
     * produce an error message and return MAXNUM.
     *
     *
     *
     * ACCURACY:
     *
     * For i < 34 the values are simply tabulated, and have
     * full machine accuracy.  If i > 55, fac(i) = gamma(i+1);
     * see gamma.c.
     *
     *                      Relative error:
     * arithmetic   domain      peak
     *    IEEE      0, 170    1.4e-15
     *    DEC       0, 33      1.4e-17
     *
     */
    
     
    /*							fdtr.c
     *
     *	F distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * int df1, df2;
     * double x, y, fdtr();
     *
     * y = fdtr( df1, df2, x );
     *
     * DESCRIPTION:
     *
     * Returns the area from zero to x under the F density
     * function (also known as Snedcor's density or the
     * variance ratio density).  This is the density
     * of x = (u1/df1)/(u2/df2), where u1 and u2 are random
     * variables having Chi square distributions with df1
     * and df2 degrees of freedom, respectively.
     *
     * The incomplete beta integral is used, according to the
     * formula
     *
     *	P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ).
     *
     *
     * The arguments a and b are greater than zero, and x is
     * nonnegative.
     *
     * ACCURACY:
     *
     * Tested at random points (a,b,x).
     *
     *                x     a,b                     Relative error:
     * arithmetic  domain  domain     # trials      peak         rms
     *    IEEE      0,1    0,100       100000      9.8e-15     1.7e-15
     *    IEEE      1,5    0,100       100000      6.5e-15     3.5e-16
     *    IEEE      0,1    1,10000     100000      2.2e-11     3.3e-12
     *    IEEE      1,5    1,10000     100000      1.1e-11     1.7e-13
     * See also incbet.c.
     *
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * fdtr domain     a<0, b<0, x<0         0.0
     *
     */
    
     
    /*							fdtrc()
     *
     *	Complemented F distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * int df1, df2;
     * double x, y, fdtrc();
     *
     * y = fdtrc( df1, df2, x );
     *
     * DESCRIPTION:
     *
     * Returns the area from x to infinity under the F density
     * function (also known as Snedcor's density or the
     * variance ratio density).
     *
     *
     *                      inf.
     *                       -
     *              1       | |  a-1      b-1
     * 1-P(x)  =  ------    |   t    (1-t)    dt
     *            B(a,b)  | |
     *                     -
     *                      x
     *
     *
     * The incomplete beta integral is used, according to the
     * formula
     *
     *	P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ).
     *
     *
     * ACCURACY:
     *
     * Tested at random points (a,b,x) in the indicated intervals.
     *                x     a,b                     Relative error:
     * arithmetic  domain  domain     # trials      peak         rms
     *    IEEE      0,1    1,100       100000      3.7e-14     5.9e-16
     *    IEEE      1,5    1,100       100000      8.0e-15     1.6e-15
     *    IEEE      0,1    1,10000     100000      1.8e-11     3.5e-13
     *    IEEE      1,5    1,10000     100000      2.0e-11     3.0e-12
     * See also incbet.c.
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * fdtrc domain    a<0, b<0, x<0         0.0
     *
     */
    
     
    /*							fdtri()
     *
     *	Inverse of complemented F distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * int df1, df2;
     * double x, p, fdtri();
     *
     * x = fdtri( df1, df2, p );
     *
     * DESCRIPTION:
     *
     * Finds the F density argument x such that the integral
     * from x to infinity of the F density is equal to the
     * given probability p.
     *
     * This is accomplished using the inverse beta integral
     * function and the relations
     *
     *      z = incbi( df2/2, df1/2, p )
     *      x = df2 (1-z) / (df1 z).
     *
     * Note: the following relations hold for the inverse of
     * the uncomplemented F distribution:
     *
     *      z = incbi( df1/2, df2/2, p )
     *      x = df2 z / (df1 (1-z)).
     *
     * ACCURACY:
     *
     * Tested at random points (a,b,p).
     *
     *              a,b                     Relative error:
     * arithmetic  domain     # trials      peak         rms
     *  For p between .001 and 1:
     *    IEEE     1,100       100000      8.3e-15     4.7e-16
     *    IEEE     1,10000     100000      2.1e-11     1.4e-13
     *  For p between 10^-6 and 10^-3:
     *    IEEE     1,100        50000      1.3e-12     8.4e-15
     *    IEEE     1,10000      50000      3.0e-12     4.8e-14
     * See also fdtrc.c.
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * fdtri domain   p <= 0 or p > 1       0.0
     *                     v < 1
     *
     */
    
     
    /*							fftr.c
     *
     *	FFT of Real Valued Sequence
     *
     *
     *
     * SYNOPSIS:
     *
     * double x[], sine[];
     * int m;
     *
     * fftr( x, m, sine );
     *
     *
     *
     * DESCRIPTION:
     *
     * Computes the (complex valued) discrete Fourier transform of
     * the real valued sequence x[].  The input sequence x[] contains
     * n = 2**m samples.  The program fills array sine[k] with
     * n/4 + 1 values of sin( 2 PI k / n ).
     *
     * Data format for complex valued output is real part followed
     * by imaginary part.  The output is developed in the input
     * array x[].
     *
     * The algorithm takes advantage of the fact that the FFT of an
     * n point real sequence can be obtained from an n/2 point
     * complex FFT.
     *
     * A radix 2 FFT algorithm is used.
     *
     * Execution time on an LSI-11/23 with floating point chip
     * is 1.0 sec for n = 256.
     *
     *
     *
     * REFERENCE:
     *
     * E. Oran Brigham, The Fast Fourier Transform;
     * Prentice-Hall, Inc., 1974
     *
     */
    
           
    /*							ceil()
     *							floor()
     *							frexp()
     *							ldexp()
     *
     *	Floating point numeric utilities
     *
     *
     *
     * SYNOPSIS:
     *
     * double ceil(), floor(), frexp(), ldexp();
     * double x, y;
     * int expnt, n;
     *
     * y = floor(x);
     * y = ceil(x);
     * y = frexp( x, &expnt );
     * y = ldexp( x, n );
     *
     *
     *
     * DESCRIPTION:
     *
     * All four routines return a double precision floating point
     * result.
     *
     * floor() returns the largest integer less than or equal to x.
     * It truncates toward minus infinity.
     *
     * ceil() returns the smallest integer greater than or equal
     * to x.  It truncates toward plus infinity.
     *
     * frexp() extracts the exponent from x.  It returns an integer
     * power of two to expnt and the significand between 0.5 and 1
     * to y.  Thus  x = y * 2**expn.
     *
     * ldexp() multiplies x by 2**n.
     *
     * These functions are part of the standard C run time library
     * for many but not all C compilers.  The ones supplied are
     * written in C for either DEC or IEEE arithmetic.  They should
     * be used only if your compiler library does not already have
     * them.
     *
     * The IEEE versions assume that denormal numbers are implemented
     * in the arithmetic.  Some modifications will be required if
     * the arithmetic has abrupt rather than gradual underflow.
     */
    
     
    /*							fresnl.c
     *
     *	Fresnel integral
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, S, C;
     * void fresnl();
     *
     * fresnl( x, &S, &C );
     *
     *
     * DESCRIPTION:
     *
     * Evaluates the Fresnel integrals
     *
     *           x
     *           -
     *          | |
     * C(x) =   |   cos(pi/2 t**2) dt,
     *        | |
     *         -
     *          0
     *
     *           x
     *           -
     *          | |
     * S(x) =   |   sin(pi/2 t**2) dt.
     *        | |
     *         -
     *          0
     *
     *
     * The integrals are evaluated by a power series for x < 1.
     * For x >= 1 auxiliary functions f(x) and g(x) are employed
     * such that
     *
     * C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 )
     * S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 )
     *
     *
     *
     * ACCURACY:
     *
     *  Relative error.
     *
     * Arithmetic  function   domain     # trials      peak         rms
     *   IEEE       S(x)      0, 10       10000       2.0e-15     3.2e-16
     *   IEEE       C(x)      0, 10       10000       1.8e-15     3.3e-16
     *   DEC        S(x)      0, 10        6000       2.2e-16     3.9e-17
     *   DEC        C(x)      0, 10        5000       2.3e-16     3.9e-17
     */
    
     
    /*							gamma.c
     *
     *	Gamma function
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, gamma();
     * extern int sgngam;
     *
     * y = gamma( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns gamma function of the argument.  The result is
     * correctly signed, and the sign (+1 or -1) is also
     * returned in a global (extern) variable named sgngam.
     * This variable is also filled in by the logarithmic gamma
     * function lgam().
     *
     * Arguments |x| <= 34 are reduced by recurrence and the function
     * approximated by a rational function of degree 6/7 in the
     * interval (2,3).  Large arguments are handled by Stirling's
     * formula. Large negative arguments are made positive using
     * a reflection formula.  
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC      -34, 34      10000       1.3e-16     2.5e-17
     *    IEEE    -170,-33      20000       2.3e-15     3.3e-16
     *    IEEE     -33,  33     20000       9.4e-16     2.2e-16
     *    IEEE      33, 171.6   20000       2.3e-15     3.2e-16
     *
     * Error for arguments outside the test range will be larger
     * owing to error amplification by the exponential function.
     *
     */
    
     
    /*							lgam()
     *
     *	Natural logarithm of gamma function
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, lgam();
     * extern int sgngam;
     *
     * y = lgam( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the base e (2.718...) logarithm of the absolute
     * value of the gamma function of the argument.
     * The sign (+1 or -1) of the gamma function is returned in a
     * global (extern) variable named sgngam.
     *
     * For arguments greater than 13, the logarithm of the gamma
     * function is approximated by the logarithmic version of
     * Stirling's formula using a polynomial approximation of
     * degree 4. Arguments between -33 and +33 are reduced by
     * recurrence to the interval [2,3] of a rational approximation.
     * The cosecant reflection formula is employed for arguments
     * less than -33.
     *
     * Arguments greater than MAXLGM return MAXNUM and an error
     * message.  MAXLGM = 2.035093e36 for DEC
     * arithmetic or 2.556348e305 for IEEE arithmetic.
     *
     *
     *
     * ACCURACY:
     *
     *
     * arithmetic      domain        # trials     peak         rms
     *    DEC     0, 3                  7000     5.2e-17     1.3e-17
     *    DEC     2.718, 2.035e36       5000     3.9e-17     9.9e-18
     *    IEEE    0, 3                 28000     5.4e-16     1.1e-16
     *    IEEE    2.718, 2.556e305     40000     3.5e-16     8.3e-17
     * The error criterion was relative when the function magnitude
     * was greater than one but absolute when it was less than one.
     *
     * The following test used the relative error criterion, though
     * at certain points the relative error could be much higher than
     * indicated.
     *    IEEE    -200, -4             10000     4.8e-16     1.3e-16
     *
     */
    
     
    /*							gdtr.c
     *
     *	Gamma distribution function
     *
     *
     *
     * SYNOPSIS:
     *
     * double a, b, x, y, gdtr();
     *
     * y = gdtr( a, b, x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the integral from zero to x of the gamma probability
     * density function:
     *
     *
     *                x
     *        b       -
     *       a       | |   b-1  -at
     * y =  -----    |    t    e    dt
     *       -     | |
     *      | (b)   -
     *               0
     *
     *  The incomplete gamma integral is used, according to the
     * relation
     *
     * y = igam( b, ax ).
     *
     *
     * ACCURACY:
     *
     * See igam().
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * gdtr domain         x < 0            0.0
     *
     */
    
     
    /*							gdtrc.c
     *
     *	Complemented gamma distribution function
     *
     *
     *
     * SYNOPSIS:
     *
     * double a, b, x, y, gdtrc();
     *
     * y = gdtrc( a, b, x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the integral from x to infinity of the gamma
     * probability density function:
     *
     *
     *               inf.
     *        b       -
     *       a       | |   b-1  -at
     * y =  -----    |    t    e    dt
     *       -     | |
     *      | (b)   -
     *               x
     *
     *  The incomplete gamma integral is used, according to the
     * relation
     *
     * y = igamc( b, ax ).
     *
     *
     * ACCURACY:
     *
     * See igamc().
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * gdtrc domain         x < 0            0.0
     *
     */
    
     
    /*
    C
    C     ..................................................................
    C
    C        SUBROUTINE GELS
    C
    C        PURPOSE
    C           TO SOLVE A SYSTEM OF SIMULTANEOUS LINEAR EQUATIONS WITH
    C           SYMMETRIC COEFFICIENT MATRIX UPPER TRIANGULAR PART OF WHICH
    C           IS ASSUMED TO BE STORED COLUMNWISE.
    C
    C        USAGE
    C           CALL GELS(R,A,M,N,EPS,IER,AUX)
    C
    C        DESCRIPTION OF PARAMETERS
    C           R      - M BY N RIGHT HAND SIDE MATRIX.  (DESTROYED)
    C                    ON RETURN R CONTAINS THE SOLUTION OF THE EQUATIONS.
    C           A      - UPPER TRIANGULAR PART OF THE SYMMETRIC
    C                    M BY M COEFFICIENT MATRIX.  (DESTROYED)
    C           M      - THE NUMBER OF EQUATIONS IN THE SYSTEM.
    C           N      - THE NUMBER OF RIGHT HAND SIDE VECTORS.
    C           EPS    - AN INPUT CONSTANT WHICH IS USED AS RELATIVE
    C                    TOLERANCE FOR TEST ON LOSS OF SIGNIFICANCE.
    C           IER    - RESULTING ERROR PARAMETER CODED AS FOLLOWS
    C                    IER=0  - NO ERROR,
    C                    IER=-1 - NO RESULT BECAUSE OF M LESS THAN 1 OR
    C                             PIVOT ELEMENT AT ANY ELIMINATION STEP
    C                             EQUAL TO 0,
    C                    IER=K  - WARNING DUE TO POSSIBLE LOSS OF SIGNIFI-
    C                             CANCE INDICATED AT ELIMINATION STEP K+1,
    C                             WHERE PIVOT ELEMENT WAS LESS THAN OR
    C                             EQUAL TO THE INTERNAL TOLERANCE EPS TIMES
    C                             ABSOLUTELY GREATEST MAIN DIAGONAL
    C                             ELEMENT OF MATRIX A.
    C           AUX    - AN AUXILIARY STORAGE ARRAY WITH DIMENSION M-1.
    C
    C        REMARKS
    C           UPPER TRIANGULAR PART OF MATRIX A IS ASSUMED TO BE STORED
    C           COLUMNWISE IN M*(M+1)/2 SUCCESSIVE STORAGE LOCATIONS, RIGHT
    C           HAND SIDE MATRIX R COLUMNWISE IN N*M SUCCESSIVE STORAGE
    C           LOCATIONS. ON RETURN SOLUTION MATRIX R IS STORED COLUMNWISE
    C           TOO.
    C           THE PROCEDURE GIVES RESULTS IF THE NUMBER OF EQUATIONS M IS
    C           GREATER THAN 0 AND PIVOT ELEMENTS AT ALL ELIMINATION STEPS
    C           ARE DIFFERENT FROM 0. HOWEVER WARNING IER=K - IF GIVEN -
    C           INDICATES POSSIBLE LOSS OF SIGNIFICANCE. IN CASE OF A WELL
    C           SCALED MATRIX A AND APPROPRIATE TOLERANCE EPS, IER=K MAY BE
    C           INTERPRETED THAT MATRIX A HAS THE RANK K. NO WARNING IS
    C           GIVEN IN CASE M=1.
    C           ERROR PARAMETER IER=-1 DOES NOT NECESSARILY MEAN THAT
    C           MATRIX A IS SINGULAR, AS ONLY MAIN DIAGONAL ELEMENTS
    C           ARE USED AS PIVOT ELEMENTS. POSSIBLY SUBROUTINE GELG (WHICH
    C           WORKS WITH TOTAL PIVOTING) WOULD BE ABLE TO FIND A SOLUTION.
    C
    C        SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
    C           NONE
    C
    C        METHOD
    C           SOLUTION IS DONE BY MEANS OF GAUSS-ELIMINATION WITH
    C           PIVOTING IN MAIN DIAGONAL, IN ORDER TO PRESERVE
    C           SYMMETRY IN REMAINING COEFFICIENT MATRICES.
    C
    C     ..................................................................
    C
    */
    
     
    /*							hyp2f1.c
     *
     *	Gauss hypergeometric function   F
     *	                               2 1
     *
     *
     * SYNOPSIS:
     *
     * double a, b, c, x, y, hyp2f1();
     *
     * y = hyp2f1( a, b, c, x );
     *
     *
     * DESCRIPTION:
     *
     *
     *  hyp2f1( a, b, c, x )  =   F ( a, b; c; x )
     *                           2 1
     *
     *           inf.
     *            -   a(a+1)...(a+k) b(b+1)...(b+k)   k+1
     *   =  1 +   >   -----------------------------  x   .
     *            -         c(c+1)...(c+k) (k+1)!
     *          k = 0
     *
     *  Cases addressed are
     *	Tests and escapes for negative integer a, b, or c
     *	Linear transformation if c - a or c - b negative integer
     *	Special case c = a or c = b
     *	Linear transformation for  x near +1
     *	Transformation for x < -0.5
     *	Psi function expansion if x > 0.5 and c - a - b integer
     *      Conditionally, a recurrence on c to make c-a-b > 0
     *
     * |x| > 1 is rejected.
     *
     * The parameters a, b, c are considered to be integer
     * valued if they are within 1.0e-14 of the nearest integer
     * (1.0e-13 for IEEE arithmetic).
     *
     * ACCURACY:
     *
     *
     *               Relative error (-1 < x < 1):
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      -1,7        230000      1.2e-11     5.2e-14
     *
     * Several special cases also tested with a, b, c in
     * the range -7 to 7.
     *
     * ERROR MESSAGES:
     *
     * A "partial loss of precision" message is printed if
     * the internally estimated relative error exceeds 1^-12.
     * A "singularity" message is printed on overflow or
     * in cases not addressed (such as x < -1).
     */
    
     
    /*							hyperg.c
     *
     *	Confluent hypergeometric function
     *
     *
     *
     * SYNOPSIS:
     *
     * double a, b, x, y, hyperg();
     *
     * y = hyperg( a, b, x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Computes the confluent hypergeometric function
     *
     *                          1           2
     *                       a x    a(a+1) x
     *   F ( a,b;x )  =  1 + ---- + --------- + ...
     *  1 1                  b 1!   b(b+1) 2!
     *
     * Many higher transcendental functions are special cases of
     * this power series.
     *
     * As is evident from the formula, b must not be a negative
     * integer or zero unless a is an integer with 0 >= a > b.
     *
     * The routine attempts both a direct summation of the series
     * and an asymptotic expansion.  In each case error due to
     * roundoff, cancellation, and nonconvergence is estimated.
     * The result with smaller estimated error is returned.
     *
     *
     *
     * ACCURACY:
     *
     * Tested at random points (a, b, x), all three variables
     * ranging from 0 to 30.
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       0,30         2000       1.2e-15     1.3e-16
     qtst1:
     21800   max =  1.4200E-14   rms =  1.0841E-15  ave = -5.3640E-17 
     ltstd:
     25500   max = 1.2759e-14   rms = 3.7155e-16  ave = 1.5384e-18 
     *    IEEE      0,30        30000       1.8e-14     1.1e-15
     *
     * Larger errors can be observed when b is near a negative
     * integer or zero.  Certain combinations of arguments yield
     * serious cancellation error in the power series summation
     * and also are not in the region of near convergence of the
     * asymptotic series.  An error message is printed if the
     * self-estimated relative error is greater than 1.0e-12.
     *
     */
    
     
    /*							i0.c
     *
     *	Modified Bessel function of order zero
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, i0();
     *
     * y = i0( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns modified Bessel function of order zero of the
     * argument.
     *
     * The function is defined as i0(x) = j0( ix ).
     *
     * The range is partitioned into the two intervals [0,8] and
     * (8, infinity).  Chebyshev polynomial expansions are employed
     * in each interval.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       0,30         6000       8.2e-17     1.9e-17
     *    IEEE      0,30        30000       5.8e-16     1.4e-16
     *
     */
    
     
    /*							i0e.c
     *
     *	Modified Bessel function of order zero,
     *	exponentially scaled
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, i0e();
     *
     * y = i0e( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns exponentially scaled modified Bessel function
     * of order zero of the argument.
     *
     * The function is defined as i0e(x) = exp(-|x|) j0( ix ).
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0,30        30000       5.4e-16     1.2e-16
     * See i0().
     *
     */
    
     
    /*							i1.c
     *
     *	Modified Bessel function of order one
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, i1();
     *
     * y = i1( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns modified Bessel function of order one of the
     * argument.
     *
     * The function is defined as i1(x) = -i j1( ix ).
     *
     * The range is partitioned into the two intervals [0,8] and
     * (8, infinity).  Chebyshev polynomial expansions are employed
     * in each interval.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       0, 30        3400       1.2e-16     2.3e-17
     *    IEEE      0, 30       30000       1.9e-15     2.1e-16
     *
     *
     */
    
     
    /*							i1e.c
     *
     *	Modified Bessel function of order one,
     *	exponentially scaled
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, i1e();
     *
     * y = i1e( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns exponentially scaled modified Bessel function
     * of order one of the argument.
     *
     * The function is defined as i1(x) = -i exp(-|x|) j1( ix ).
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0, 30       30000       2.0e-15     2.0e-16
     * See i1().
     *
     */
    
     
    /*							igam.c
     *
     *	Incomplete gamma integral
     *
     *
     *
     * SYNOPSIS:
     *
     * double a, x, y, igam();
     *
     * y = igam( a, x );
     *
     * DESCRIPTION:
     *
     * The function is defined by
     *
     *                           x
     *                            -
     *                   1       | |  -t  a-1
     *  igam(a,x)  =   -----     |   e   t   dt.
     *                  -      | |
     *                 | (a)    -
     *                           0
     *
     *
     * In this implementation both arguments must be positive.
     * The integral is evaluated by either a power series or
     * continued fraction expansion, depending on the relative
     * values of a and x.
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0,30       200000       3.6e-14     2.9e-15
     *    IEEE      0,100      300000       9.9e-14     1.5e-14
     */
    
     
    /*							igamc()
     *
     *	Complemented incomplete gamma integral
     *
     *
     *
     * SYNOPSIS:
     *
     * double a, x, y, igamc();
     *
     * y = igamc( a, x );
     *
     * DESCRIPTION:
     *
     * The function is defined by
     *
     *
     *  igamc(a,x)   =   1 - igam(a,x)
     *
     *                            inf.
     *                              -
     *                     1       | |  -t  a-1
     *               =   -----     |   e   t   dt.
     *                    -      | |
     *                   | (a)    -
     *                             x
     *
     *
     * In this implementation both arguments must be positive.
     * The integral is evaluated by either a power series or
     * continued fraction expansion, depending on the relative
     * values of a and x.
     *
     * ACCURACY:
     *
     * Tested at random a, x.
     *                a         x                      Relative error:
     * arithmetic   domain   domain     # trials      peak         rms
     *    IEEE     0.5,100   0,100      200000       1.9e-14     1.7e-15
     *    IEEE     0.01,0.5  0,100      200000       1.4e-13     1.6e-15
     */
    
     
    /*							igami()
     *
     *      Inverse of complemented imcomplete gamma integral
     *
     *
     *
     * SYNOPSIS:
     *
     * double a, x, p, igami();
     *
     * x = igami( a, p );
     *
     * DESCRIPTION:
     *
     * Given p, the function finds x such that
     *
     *  igamc( a, x ) = p.
     *
     * It is valid in the right-hand tail of the distribution, p < 0.5.
     * Starting with the approximate value
     *
     *         3
     *  x = a t
     *
     *  where
     *
     *  t = 1 - d - ndtri(p) sqrt(d)
     * 
     * and
     *
     *  d = 1/9a,
     *
     * the routine performs up to 10 Newton iterations to find the
     * root of igamc(a,x) - p = 0.
     *
     * ACCURACY:
     *
     * Tested at random a, p in the intervals indicated.
     *
     *                a        p                      Relative error:
     * arithmetic   domain   domain     # trials      peak         rms
     *    IEEE     0.5,100   0,0.5       100000       1.0e-14     1.7e-15
     *    IEEE     0.01,0.5  0,0.5       100000       9.0e-14     3.4e-15
     *    IEEE    0.5,10000  0,0.5        20000       2.3e-13     3.8e-14
     */
    
     
    /*							incbet.c
     *
     *	Incomplete beta integral
     *
     *
     * SYNOPSIS:
     *
     * double a, b, x, y, incbet();
     *
     * y = incbet( a, b, x );
     *
     *
     * DESCRIPTION:
     *
     * Returns incomplete beta integral of the arguments, evaluated
     * from zero to x.  The function is defined as
     *
     *                  x
     *     -            -
     *    | (a+b)      | |  a-1     b-1
     *  -----------    |   t   (1-t)   dt.
     *   -     -     | |
     *  | (a) | (b)   -
     *                 0
     *
     * The domain of definition is 0 <= x <= 1.  In this
     * implementation a and b are restricted to positive values.
     * The integral from x to 1 may be obtained by the symmetry
     * relation
     *
     *    1 - incbet( a, b, x )  =  incbet( b, a, 1-x ).
     *
     * The integral is evaluated by a continued fraction expansion
     * or, when b*x is small, by a power series.
     *
     * ACCURACY:
     *
     * Tested at uniformly distributed random points (a,b,x) with a and b
     * in "domain" and x between 0 and 1.
     *                                        Relative error
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0,5         10000       6.9e-15     4.5e-16
     *    IEEE      0,85       250000       2.2e-13     1.7e-14
     *    IEEE      0,1000      30000       5.3e-12     6.3e-13
     *    IEEE      0,10000    250000       9.3e-11     7.1e-12
     *    IEEE      0,100000    10000       8.7e-10     4.8e-11
     * Outputs smaller than the IEEE gradual underflow threshold
     * were excluded from these statistics.
     *
     * ERROR MESSAGES:
     *   message         condition      value returned
     * incbet domain      x<0, x>1          0.0
     * incbet underflow                     0.0
     */
    
     
    /*							incbi()
     *
     *      Inverse of imcomplete beta integral
     *
     *
     *
     * SYNOPSIS:
     *
     * double a, b, x, y, incbi();
     *
     * x = incbi( a, b, y );
     *
     *
     *
     * DESCRIPTION:
     *
     * Given y, the function finds x such that
     *
     *  incbet( a, b, x ) = y .
     *
     * The routine performs interval halving or Newton iterations to find the
     * root of incbet(a,b,x) - y = 0.
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     *                x     a,b
     * arithmetic   domain  domain  # trials    peak       rms
     *    IEEE      0,1    .5,10000   50000    5.8e-12   1.3e-13
     *    IEEE      0,1   .25,100    100000    1.8e-13   3.9e-15
     *    IEEE      0,1     0,5       50000    1.1e-12   5.5e-15
     *    VAX       0,1    .5,100     25000    3.5e-14   1.1e-15
     * With a and b constrained to half-integer or integer values:
     *    IEEE      0,1    .5,10000   50000    5.8e-12   1.1e-13
     *    IEEE      0,1    .5,100    100000    1.7e-14   7.9e-16
     * With a = .5, b constrained to half-integer or integer values:
     *    IEEE      0,1    .5,10000   10000    8.3e-11   1.0e-11
     */
    
         
    /*							isnan()
     *							signbit()
     *							isfinite()
     *
     *	Floating point numeric utilities
     *
     *
     *
     * SYNOPSIS:
     *
     * double ceil(), floor(), frexp(), ldexp();
     * int signbit(), isnan(), isfinite();
     * double x, y;
     * int expnt, n;
     *
     * y = floor(x);
     * y = ceil(x);
     * y = frexp( x, &expnt );
     * y = ldexp( x, n );
     * n = signbit(x);
     * n = isnan(x);
     * n = isfinite(x);
     *
     *
     *
     * DESCRIPTION:
     *
     * All four routines return a double precision floating point
     * result.
     *
     * floor() returns the largest integer less than or equal to x.
     * It truncates toward minus infinity.
     *
     * ceil() returns the smallest integer greater than or equal
     * to x.  It truncates toward plus infinity.
     *
     * frexp() extracts the exponent from x.  It returns an integer
     * power of two to expnt and the significand between 0.5 and 1
     * to y.  Thus  x = y * 2**expn.
     *
     * ldexp() multiplies x by 2**n.
     *
     * signbit(x) returns 1 if the sign bit of x is 1, else 0.
     *
     * These functions are part of the standard C run time library
     * for many but not all C compilers.  The ones supplied are
     * written in C for either DEC or IEEE arithmetic.  They should
     * be used only if your compiler library does not already have
     * them.
     *
     * The IEEE versions assume that denormal numbers are implemented
     * in the arithmetic.  Some modifications will be required if
     * the arithmetic has abrupt rather than gradual underflow.
     */
    
     
    /*							iv.c
     *
     *	Modified Bessel function of noninteger order
     *
     *
     *
     * SYNOPSIS:
     *
     * double v, x, y, iv();
     *
     * y = iv( v, x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns modified Bessel function of order v of the
     * argument.  If x is negative, v must be integer valued.
     *
     * The function is defined as Iv(x) = Jv( ix ).  It is
     * here computed in terms of the confluent hypergeometric
     * function, according to the formula
     *
     *              v  -x
     * Iv(x) = (x/2)  e   hyperg( v+0.5, 2v+1, 2x ) / gamma(v+1)
     *
     * If v is a negative integer, then v is replaced by -v.
     *
     *
     * ACCURACY:
     *
     * Tested at random points (v, x), with v between 0 and
     * 30, x between 0 and 28.
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       0,30          2000      3.1e-15     5.4e-16
     *    IEEE      0,30         10000      1.7e-14     2.7e-15
     *
     * Accuracy is diminished if v is near a negative integer.
     *
     * See also hyperg.c.
     *
     */
    
     
    /*							j0.c
     *
     *	Bessel function of order zero
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, j0();
     *
     * y = j0( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns Bessel function of order zero of the argument.
     *
     * The domain is divided into the intervals [0, 5] and
     * (5, infinity). In the first interval the following rational
     * approximation is used:
     *
     *
     *        2         2
     * (w - r  ) (w - r  ) P (w) / Q (w)
     *       1         2    3       8
     *
     *            2
     * where w = x  and the two r's are zeros of the function.
     *
     * In the second interval, the Hankel asymptotic expansion
     * is employed with two rational functions of degree 6/6
     * and 7/7.
     *
     *
     *
     * ACCURACY:
     *
     *                      Absolute error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       0, 30       10000       4.4e-17     6.3e-18
     *    IEEE      0, 30       60000       4.2e-16     1.1e-16
     *
     */
    
     
    /*							y0.c
     *
     *	Bessel function of the second kind, order zero
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, y0();
     *
     * y = y0( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns Bessel function of the second kind, of order
     * zero, of the argument.
     *
     * The domain is divided into the intervals [0, 5] and
     * (5, infinity). In the first interval a rational approximation
     * R(x) is employed to compute
     *   y0(x)  = R(x)  +   2 * log(x) * j0(x) / PI.
     * Thus a call to j0() is required.
     *
     * In the second interval, the Hankel asymptotic expansion
     * is employed with two rational functions of degree 6/6
     * and 7/7.
     *
     *
     *
     * ACCURACY:
     *
     *  Absolute error, when y0(x) < 1; else relative error:
     *
     * arithmetic   domain     # trials      peak         rms
     *    DEC       0, 30        9400       7.0e-17     7.9e-18
     *    IEEE      0, 30       30000       1.3e-15     1.6e-16
     *
     */
    
     
    /*							j1.c
     *
     *	Bessel function of order one
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, j1();
     *
     * y = j1( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns Bessel function of order one of the argument.
     *
     * The domain is divided into the intervals [0, 8] and
     * (8, infinity). In the first interval a 24 term Chebyshev
     * expansion is used. In the second, the asymptotic
     * trigonometric representation is employed using two
     * rational functions of degree 5/5.
     *
     *
     *
     * ACCURACY:
     *
     *                      Absolute error:
     * arithmetic   domain      # trials      peak         rms
     *    DEC       0, 30       10000       4.0e-17     1.1e-17
     *    IEEE      0, 30       30000       2.6e-16     1.1e-16
     *
     *
     */
    
     
    /*							y1.c
     *
     *	Bessel function of second kind of order one
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, y1();
     *
     * y = y1( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns Bessel function of the second kind of order one
     * of the argument.
     *
     * The domain is divided into the intervals [0, 8] and
     * (8, infinity). In the first interval a 25 term Chebyshev
     * expansion is used, and a call to j1() is required.
     * In the second, the asymptotic trigonometric representation
     * is employed using two rational functions of degree 5/5.
     *
     *
     *
     * ACCURACY:
     *
     *                      Absolute error:
     * arithmetic   domain      # trials      peak         rms
     *    DEC       0, 30       10000       8.6e-17     1.3e-17
     *    IEEE      0, 30       30000       1.0e-15     1.3e-16
     *
     * (error criterion relative when |y1| > 1).
     *
     */
    
     
    /*							jn.c
     *
     *	Bessel function of integer order
     *
     *
     *
     * SYNOPSIS:
     *
     * int n;
     * double x, y, jn();
     *
     * y = jn( n, x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns Bessel function of order n, where n is a
     * (possibly negative) integer.
     *
     * The ratio of jn(x) to j0(x) is computed by backward
     * recurrence.  First the ratio jn/jn-1 is found by a
     * continued fraction expansion.  Then the recurrence
     * relating successive orders is applied until j0 or j1 is
     * reached.
     *
     * If n = 0 or 1 the routine for j0 or j1 is called
     * directly.
     *
     *
     *
     * ACCURACY:
     *
     *                      Absolute error:
     * arithmetic   range      # trials      peak         rms
     *    DEC       0, 30        5500       6.9e-17     9.3e-18
     *    IEEE      0, 30        5000       4.4e-16     7.9e-17
     *
     *
     * Not suitable for large n or x. Use jv() instead.
     *
     */
    
     
    /*							jv.c
     *
     *	Bessel function of noninteger order
     *
     *
     *
     * SYNOPSIS:
     *
     * double v, x, y, jv();
     *
     * y = jv( v, x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns Bessel function of order v of the argument,
     * where v is real.  Negative x is allowed if v is an integer.
     *
     * Several expansions are included: the ascending power
     * series, the Hankel expansion, and two transitional
     * expansions for large v.  If v is not too large, it
     * is reduced by recurrence to a region of best accuracy.
     * The transitional expansions give 12D accuracy for v > 500.
     *
     *
     *
     * ACCURACY:
     * Results for integer v are indicated by *, where x and v
     * both vary from -125 to +125.  Otherwise,
     * x ranges from 0 to 125, v ranges as indicated by "domain."
     * Error criterion is absolute, except relative when |jv()| > 1.
     *
     * arithmetic  v domain  x domain    # trials      peak       rms
     *    IEEE      0,125     0,125      100000      4.6e-15    2.2e-16
     *    IEEE   -125,0       0,125       40000      5.4e-11    3.7e-13
     *    IEEE      0,500     0,500       20000      4.4e-15    4.0e-16
     * Integer v:
     *    IEEE   -125,125   -125,125      50000      3.5e-15*   1.9e-16*
     *
     */
    
     
    /*							k0.c
     *
     *	Modified Bessel function, third kind, order zero
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, k0();
     *
     * y = k0( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns modified Bessel function of the third kind
     * of order zero of the argument.
     *
     * The range is partitioned into the two intervals [0,8] and
     * (8, infinity).  Chebyshev polynomial expansions are employed
     * in each interval.
     *
     *
     *
     * ACCURACY:
     *
     * Tested at 2000 random points between 0 and 8.  Peak absolute
     * error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15.
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       0, 30        3100       1.3e-16     2.1e-17
     *    IEEE      0, 30       30000       1.2e-15     1.6e-16
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     *  K0 domain          x <= 0          MAXNUM
     *
     */
    
     
    /*							k0e()
     *
     *	Modified Bessel function, third kind, order zero,
     *	exponentially scaled
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, k0e();
     *
     * y = k0e( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns exponentially scaled modified Bessel function
     * of the third kind of order zero of the argument.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0, 30       30000       1.4e-15     1.4e-16
     * See k0().
     *
     */
    
     
    /*							k1.c
     *
     *	Modified Bessel function, third kind, order one
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, k1();
     *
     * y = k1( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Computes the modified Bessel function of the third kind
     * of order one of the argument.
     *
     * The range is partitioned into the two intervals [0,2] and
     * (2, infinity).  Chebyshev polynomial expansions are employed
     * in each interval.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       0, 30        3300       8.9e-17     2.2e-17
     *    IEEE      0, 30       30000       1.2e-15     1.6e-16
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * k1 domain          x <= 0          MAXNUM
     *
     */
    
     
    /*							k1e.c
     *
     *	Modified Bessel function, third kind, order one,
     *	exponentially scaled
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, k1e();
     *
     * y = k1e( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns exponentially scaled modified Bessel function
     * of the third kind of order one of the argument:
     *
     *      k1e(x) = exp(x) * k1(x).
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0, 30       30000       7.8e-16     1.2e-16
     * See k1().
     *
     */
    
     
    /*							kn.c
     *
     *	Modified Bessel function, third kind, integer order
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, kn();
     * int n;
     *
     * y = kn( n, x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns modified Bessel function of the third kind
     * of order n of the argument.
     *
     * The range is partitioned into the two intervals [0,9.55] and
     * (9.55, infinity).  An ascending power series is used in the
     * low range, and an asymptotic expansion in the high range.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       0,30         3000       1.3e-9      5.8e-11
     *    IEEE      0,30        90000       1.8e-8      3.0e-10
     *
     *  Error is high only near the crossover point x = 9.55
     * between the two expansions used.
     */
    
     
    /* Re Kolmogorov statistics, here is Birnbaum and Tingey's formula for the
       distribution of D+, the maximum of all positive deviations between a
       theoretical distribution function P(x) and an empirical one Sn(x)
       from n samples.
    
         +
        D  =         sup     [P(x) - S (x)]
         n     -inf < x < inf         n
    
    
                      [n(1-e)]
            +            -                    v-1              n-v
        Pr{D   > e} =    >    C    e (e + v/n)    (1 - e - v/n)
            n            -   n v
                        v=0
    
        [n(1-e)] is the largest integer not exceeding n(1-e).
        nCv is the number of combinations of n things taken v at a time.  */
    
     
    /*							lmdif.c
    *
    *     The purpose of lmdif is to minimize the sum of the squares of
    *     M nonlinear functions in N variables by a modification of
    *     the Levenberg-Marquardt algorithm. The user must provide a
    *     subroutine that calculates the functions.  The Jacobian is
    *     then calculated numerically by a forward-difference approximation.
    *
    *     Refer to the source code for information on the use of the routine.
    *
    *     This is a C language translation of the Fortran version of
    *     the corresponding routine from Argonne National Laboratories
    *     MINPACK subroutine suite.
    *
    */
    
     
    /*		Levnsn.c		*/
    /* Levinson-Durbin LPC
     * linear predictive coding
     *
     * | R0 R1 R2 ... RN-1 |   | A1 |       | -R1 |
     * | R1 R0 R1 ... RN-2 |   | A2 |       | -R2 |
     * | R2 R1 R0 ... RN-3 |   | A3 |   =   | -R3 |
     * |          ...      |   | ...|       | ... |
     * | RN-1 RN-2... R0   |   | AN |       | -RN |
     *
     * Ref: John Makhoul, "Linear Prediction, A Tutorial Review"
     * Proc. IEEE Vol. 63, PP 561-580 April, 1975.
     *
     * R is the input autocorrelation function.  R0 is the zero lag
     * term.  A is the output array of predictor coefficients.  Note
     * that a filter impulse response has a coefficient of 1.0 preceding
     * A1.  E is an array of mean square error for each prediction order
     * 1 to N.  REFL is an output array of the reflection coefficients.
     */
    
     
    /*							log.c
     *
     *	Natural logarithm
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, log();
     *
     * y = log( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the base e (2.718...) logarithm of x.
     *
     * The argument is separated into its exponent and fractional
     * parts.  If the exponent is between -1 and +1, the logarithm
     * of the fraction is approximated by
     *
     *     log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
     *
     * Otherwise, setting  z = 2(x-1)/x+1),
     * 
     *     log(x) = z + z**3 P(z)/Q(z).
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0.5, 2.0    150000      1.44e-16    5.06e-17
     *    IEEE      +-MAXNUM    30000       1.20e-16    4.78e-17
     *    DEC       0, 10       170000      1.8e-17     6.3e-18
     *
     * In the tests over the interval [+-MAXNUM], the logarithms
     * of the random arguments were uniformly distributed over
     * [0, MAXLOG].
     *
     * ERROR MESSAGES:
     *
     * log singularity:  x = 0; returns -INFINITY
     * log domain:       x < 0; returns NAN
     */
    
     
    /*							log10.c
     *
     *	Common logarithm
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, log10();
     *
     * y = log10( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns logarithm to the base 10 of x.
     *
     * The argument is separated into its exponent and fractional
     * parts.  The logarithm of the fraction is approximated by
     *
     *     log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0.5, 2.0     30000      1.5e-16     5.0e-17
     *    IEEE      0, MAXNUM    30000      1.4e-16     4.8e-17
     *    DEC       1, MAXNUM    50000      2.5e-17     6.0e-18
     *
     * In the tests over the interval [1, MAXNUM], the logarithms
     * of the random arguments were uniformly distributed over
     * [0, MAXLOG].
     *
     * ERROR MESSAGES:
     *
     * log10 singularity:  x = 0; returns -INFINITY
     * log10 domain:       x < 0; returns NAN
     */
    
     
    /*							log2.c
     *
     *	Base 2 logarithm
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, log2();
     *
     * y = log2( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the base 2 logarithm of x.
     *
     * The argument is separated into its exponent and fractional
     * parts.  If the exponent is between -1 and +1, the base e
     * logarithm of the fraction is approximated by
     *
     *     log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
     *
     * Otherwise, setting  z = 2(x-1)/x+1),
     * 
     *     log(x) = z + z**3 P(z)/Q(z).
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0.5, 2.0    30000       2.0e-16     5.5e-17
     *    IEEE      exp(+-700)  40000       1.3e-16     4.6e-17
     *
     * In the tests over the interval [exp(+-700)], the logarithms
     * of the random arguments were uniformly distributed.
     *
     * ERROR MESSAGES:
     *
     * log2 singularity:  x = 0; returns -INFINITY
     * log2 domain:       x < 0; returns NAN
     */
    
     
    /*							lrand.c
     *
     *	Pseudorandom number generator
     *
     *
     *
     * SYNOPSIS:
     *
     * long y, drand();
     *
     * drand( &y );
     *
     *
     *
     * DESCRIPTION:
     *
     * Yields a long integer random number.
     *
     * The three-generator congruential algorithm by Brian
     * Wichmann and David Hill (BYTE magazine, March, 1987,
     * pp 127-8) is used. The period, given by them, is
     * 6953607871644.
     *
     *
     */
    
     
    /*							lsqrt.c
     *
     *	Integer square root
     *
     *
     *
     * SYNOPSIS:
     *
     * long x, y;
     * long lsqrt();
     *
     * y = lsqrt( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns a long integer square root of the long integer
     * argument.  The computation is by binary long division.
     *
     * The largest possible result is lsqrt(2,147,483,647)
     * = 46341.
     *
     * If x < 0, the square root of |x| is returned, and an
     * error message is printed.
     *
     *
     * ACCURACY:
     *
     * An extra, roundoff, bit is computed; hence the result
     * is the nearest integer to the actual square root.
     * NOTE: only DEC arithmetic is currently supported.
     *
     */
    
     
    /*							minv.c
     *
     *	Matrix inversion
     *
     *
     *
     * SYNOPSIS:
     *
     * int n, errcod;
     * double A[n*n], X[n*n];
     * double B[n];
     * int IPS[n];
     * int minv();
     *
     * errcod = minv( A, X, n, B, IPS );
     *
     *
     *
     * DESCRIPTION:
     *
     * Finds the inverse of the n by n matrix A.  The result goes
     * to X.   B and IPS are scratch pad arrays of length n.
     * The contents of matrix A are destroyed.
     *
     * The routine returns nonzero on error; error messages are printed
     * by subroutine simq().
     *
     */
    
     
    /*							mtransp.c
     *
     *	Matrix transpose
     *
     *
     *
     * SYNOPSIS:
     *
     * int n;
     * double A[n*n], T[n*n];
     *
     * mtransp( n, A, T );
     *
     *
     *
     * DESCRIPTION:
     *
     *
     * T[r][c] = A[c][r]
     *
     *
     * Transposes the n by n square matrix A and puts the result in T.
     * The output, T, may occupy the same storage as A.
     *
     *
     *
     */
    
     
    /*							nbdtr.c
     *
     *	Negative binomial distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * int k, n;
     * double p, y, nbdtr();
     *
     * y = nbdtr( k, n, p );
     *
     * DESCRIPTION:
     *
     * Returns the sum of the terms 0 through k of the negative
     * binomial distribution:
     *
     *   k
     *   --  ( n+j-1 )   n      j
     *   >   (       )  p  (1-p)
     *   --  (   j   )
     *  j=0
     *
     * In a sequence of Bernoulli trials, this is the probability
     * that k or fewer failures precede the nth success.
     *
     * The terms are not computed individually; instead the incomplete
     * beta integral is employed, according to the formula
     *
     * y = nbdtr( k, n, p ) = incbet( n, k+1, p ).
     *
     * The arguments must be positive, with p ranging from 0 to 1.
     *
     * ACCURACY:
     *
     * Tested at random points (a,b,p), with p between 0 and 1.
     *
     *               a,b                     Relative error:
     * arithmetic  domain     # trials      peak         rms
     *    IEEE     0,100       100000      1.7e-13     8.8e-15
     * See also incbet.c.
     *
     */
    
     
    /*							nbdtr.c
     *
     *	Complemented negative binomial distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * int k, n;
     * double p, y, nbdtrc();
     *
     * y = nbdtrc( k, n, p );
     *
     * DESCRIPTION:
     *
     * Returns the sum of the terms k+1 to infinity of the negative
     * binomial distribution:
     *
     *   inf
     *   --  ( n+j-1 )   n      j
     *   >   (       )  p  (1-p)
     *   --  (   j   )
     *  j=k+1
     *
     * The terms are not computed individually; instead the incomplete
     * beta integral is employed, according to the formula
     *
     * y = nbdtrc( k, n, p ) = incbet( k+1, n, 1-p ).
     *
     * The arguments must be positive, with p ranging from 0 to 1.
     *
     * ACCURACY:
     *
     * Tested at random points (a,b,p), with p between 0 and 1.
     *
     *               a,b                     Relative error:
     * arithmetic  domain     # trials      peak         rms
     *    IEEE     0,100       100000      1.7e-13     8.8e-15
     * See also incbet.c.
     */
    
     
    /*							nbdtr.c
     *
     *	Functional inverse of negative binomial distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * int k, n;
     * double p, y, nbdtri();
     *
     * p = nbdtri( k, n, y );
     *
     * DESCRIPTION:
     *
     * Finds the argument p such that nbdtr(k,n,p) is equal to y.
     *
     * ACCURACY:
     *
     * Tested at random points (a,b,y), with y between 0 and 1.
     *
     *               a,b                     Relative error:
     * arithmetic  domain     # trials      peak         rms
     *    IEEE     0,100       100000      1.5e-14     8.5e-16
     * See also incbi.c.
     */
    
     
    /*							ndtr.c
     *
     *	Normal distribution function
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, ndtr();
     *
     * y = ndtr( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the area under the Gaussian probability density
     * function, integrated from minus infinity to x:
     *
     *                            x
     *                             -
     *                   1        | |          2
     *    ndtr(x)  = ---------    |    exp( - t /2 ) dt
     *               sqrt(2pi)  | |
     *                           -
     *                          -inf.
     *
     *             =  ( 1 + erf(z) ) / 2
     *             =  erfc(z) / 2
     *
     * where z = x/sqrt(2). Computation is via the functions
     * erf and erfc with care to avoid error amplification in computing exp(-x^2).
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE     -13,0        30000       1.3e-15     2.2e-16
     *
     *
     * ERROR MESSAGES:
     *
     *   message         condition         value returned
     * erfc underflow    x > 37.519379347       0.0
     *
     */
    
     
    /*							ndtr.c
     *
     *	Error function
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, erf();
     *
     * y = erf( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * The integral is
     *
     *                           x 
     *                            -
     *                 2         | |          2
     *   erf(x)  =  --------     |    exp( - t  ) dt.
     *              sqrt(pi)   | |
     *                          -
     *                           0
     *
     * The magnitude of x is limited to 9.231948545 for DEC
     * arithmetic; 1 or -1 is returned outside this range.
     *
     * For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise
     * erf(x) = 1 - erfc(x).
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       0,1         14000       4.7e-17     1.5e-17
     *    IEEE      0,1         30000       3.7e-16     1.0e-16
     *
     */
    
     
    /*							ndtr.c
     *
     *	Complementary error function
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, erfc();
     *
     * y = erfc( x );
     *
     *
     *
     * DESCRIPTION:
     *
     *
     *  1 - erf(x) =
     *
     *                           inf. 
     *                             -
     *                  2         | |          2
     *   erfc(x)  =  --------     |    exp( - t  ) dt
     *               sqrt(pi)   | |
     *                           -
     *                            x
     *
     *
     * For small x, erfc(x) = 1 - erf(x); otherwise rational
     * approximations are computed.
     *
     * A special function expx2.c is used to suppress error amplification
     * in computing exp(-x^2).
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0,26.6417   30000       1.3e-15     2.2e-16
     *
     *
     * ERROR MESSAGES:
     *
     *   message         condition              value returned
     * erfc underflow    x > 9.231948545 (DEC)       0.0
     *
     *
     */
    
     
    /*							ndtri.c
     *
     *	Inverse of Normal distribution function
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, ndtri();
     *
     * x = ndtri( y );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the argument, x, for which the area under the
     * Gaussian probability density function (integrated from
     * minus infinity to x) is equal to y.
     *
     *
     * For small arguments 0 < y < exp(-2), the program computes
     * z = sqrt( -2.0 * log(y) );  then the approximation is
     * x = z - log(z)/z  - (1/z) P(1/z) / Q(1/z).
     * There are two rational functions P/Q, one for 0 < y < exp(-32)
     * and the other for y up to exp(-2).  For larger arguments,
     * w = y - 0.5, and  x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain        # trials      peak         rms
     *    DEC      0.125, 1         5500       9.5e-17     2.1e-17
     *    DEC      6e-39, 0.135     3500       5.7e-17     1.3e-17
     *    IEEE     0.125, 1        20000       7.2e-16     1.3e-16
     *    IEEE     3e-308, 0.135   50000       4.6e-16     9.8e-17
     *
     *
     * ERROR MESSAGES:
     *
     *   message         condition    value returned
     * ndtri domain       x <= 0        -MAXNUM
     * ndtri domain       x >= 1         MAXNUM
     *
     */
    
     
    /*							pdtr.c
     *
     *	Poisson distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * int k;
     * double m, y, pdtr();
     *
     * y = pdtr( k, m );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the sum of the first k terms of the Poisson
     * distribution:
     *
     *   k         j
     *   --   -m  m
     *   >   e    --
     *   --       j!
     *  j=0
     *
     * The terms are not summed directly; instead the incomplete
     * gamma integral is employed, according to the relation
     *
     * y = pdtr( k, m ) = igamc( k+1, m ).
     *
     * The arguments must both be positive.
     *
     *
     *
     * ACCURACY:
     *
     * See igamc().
     *
     */
    
     
    /*							pdtrc()
     *
     *	Complemented poisson distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * int k;
     * double m, y, pdtrc();
     *
     * y = pdtrc( k, m );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the sum of the terms k+1 to infinity of the Poisson
     * distribution:
     *
     *  inf.       j
     *   --   -m  m
     *   >   e    --
     *   --       j!
     *  j=k+1
     *
     * The terms are not summed directly; instead the incomplete
     * gamma integral is employed, according to the formula
     *
     * y = pdtrc( k, m ) = igam( k+1, m ).
     *
     * The arguments must both be positive.
     *
     *
     *
     * ACCURACY:
     *
     * See igam.c.
     *
     */
    
     
    /*							pdtri()
     *
     *	Inverse Poisson distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * int k;
     * double m, y, pdtr();
     *
     * m = pdtri( k, y );
     *
     *
     *
     *
     * DESCRIPTION:
     *
     * Finds the Poisson variable x such that the integral
     * from 0 to x of the Poisson density is equal to the
     * given probability y.
     *
     * This is accomplished using the inverse gamma integral
     * function and the relation
     *
     *    m = igami( k+1, y ).
     *
     *
     *
     *
     * ACCURACY:
     *
     * See igami.c.
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * pdtri domain    y < 0 or y >= 1       0.0
     *                     k < 0
     *
     */
    
     
    /*							planck.c
     *
     *	Integral of Planck's black body radiation formula
     *
     *
     *
     * SYNOPSIS:
     *
     * double lambda, T, y, plancki();
     *
     * y = plancki( lambda, T );
     *
     *
     *
     * DESCRIPTION:
     *
     *  Evaluates the definite integral, from wavelength 0 to lambda,
     *  of Planck's radiation formula
     *                      -5
     *            c1  lambda
     *     E =  ------------------
     *            c2/(lambda T)
     *           e             - 1
     *
     * Physical constants c1 and c2 (see below) are built in
     * to the function program.  They are scaled to provide a result
     * in watts per square meter.  Argument T represents temperature in degrees
     * Kelvin; lambda is wavelength in meters.
     *
     * The integral is expressed in closed form, in terms of polylogarithms
     * (see polylog.c).
     *
     * The total area under the curve is
     *      (-1/8) (42 zeta(4) - 12 pi^2 zeta(2) + pi^4 ) c1 (T/c2)^4
     *       = (pi^4 / 15)  c1 (T/c2)^4
     *       =  sigma T^4
     *
     *
     * CONSTANTS:
     *
     * First radiation constant c1 = 2 pi h c^2 = 3.741 771 53 (17) e-16 W m2
     * Second radiation constant c2 = h c / k  = 0.014 387 770 (13) m K
     * Stefan-Boltzmann constant sigma = 5.670 373 (21) e-8 W m^-2 K^-4
     * Wien wavelength displacement law constant  wien = 2.8977721 (26) e-3 m K
     * These are NIST values as of 2010.
     *
     *
     * ACCURACY:
     *
     * The left tail of the function experiences some relative error
     * amplification in computing the dominant term exp(-c2/(lambda T)).
     * For the right-hand tail see planckc, below.
     *
     *                      Relative error.
     *   The domain refers to lambda T / c2.
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0.1, 10      50000      7.1e-15     5.4e-16
     *
     */
    
       
    /*							polevl.c
     *							p1evl.c
     *
     *	Evaluate polynomial
     *
     *
     *
     * SYNOPSIS:
     *
     * int N;
     * double x, y, coef[N+1], polevl[];
     *
     * y = polevl( x, coef, N );
     *
     *
     *
     * DESCRIPTION:
     *
     * Evaluates polynomial of degree N:
     *
     *                     2          N
     * y  =  C  + C x + C x  +...+ C x
     *        0    1     2          N
     *
     * Coefficients are stored in reverse order:
     *
     * coef[0] = C  , ..., coef[N] = C  .
     *            N                   0
     *
     *  The function p1evl() assumes that coef[N] = 1.0 and is
     * omitted from the array.  Its calling arguments are
     * otherwise the same as polevl().
     *
     *
     * SPEED:
     *
     * In the interest of speed, there are no checks for out
     * of bounds arithmetic.  This routine is used by most of
     * the functions in the library.  Depending on available
     * equipment features, the user may wish to rewrite the
     * program in microcode or assembly language.
     *
     */
    
     
    /*                                                     polmisc.c
     * Square root, sine, cosine, and arctangent of polynomial.
     * See polyn.c for data structures and discussion.
     */
    
     
    /*							polrt.c
     *
     *	Find roots of a polynomial
     *
     *
     *
     * SYNOPSIS:
     *
     * typedef struct
     *	{
     *	double r;
     *	double i;
     *	}cmplx;
     *
     * double xcof[], cof[];
     * int m;
     * cmplx root[];
     *
     * polrt( xcof, cof, m, root )
     *
     *
     *
     * DESCRIPTION:
     *
     * Iterative determination of the roots of a polynomial of
     * degree m whose coefficient vector is xcof[].  The
     * coefficients are arranged in ascending order; i.e., the
     * coefficient of x**m is xcof[m].
     *
     * The array cof[] is working storage the same size as xcof[].
     * root[] is the output array containing the complex roots.
     *
     *
     * ACCURACY:
     *
     * Termination depends on evaluation of the polynomial at
     * the trial values of the roots.  The values of multiple roots
     * or of roots that are nearly equal may have poor relative
     * accuracy after the first root in the neighborhood has been
     * found.
     *
     */
    
     
    /*							polylog.c
     *
     *	Polylogarithms
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, polylog();
     * int n;
     *
     * y = polylog( n, x );
     *
     *
     * The polylogarithm of order n is defined by the series
     *
     *
     *              inf   k
     *               -   x
     *  Li (x)  =    >   ---  .
     *    n          -     n
     *              k=1   k
     *
     *
     *  For x = 1,
     *
     *               inf
     *                -    1
     *   Li (1)  =    >   ---   =  Riemann zeta function (n)  .
     *     n          -     n
     *               k=1   k
     *
     *
     *  When n = 2, the function is the dilogarithm, related to Spence's integral:
     *
     *                 x                      1-x
     *                 -                        -
     *                | |  -ln(1-t)            | |  ln t
     *   Li (x)  =    |    -------- dt    =    |    ------ dt    =   spence(1-x) .
     *     2        | |       t              | |    1 - t
     *               -                        -
     *                0                        1
     *
     *
     *  See also the program cpolylog.c for the complex polylogarithm,
     *  whose definition is extended to x > 1.
     *
     *  References:
     *
     *  Lewin, L., _Polylogarithms and Associated Functions_,
     *  North Holland, 1981.
     *
     *  Lewin, L., ed., _Structural Properties of Polylogarithms_,
     *  American Mathematical Society, 1991.
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain   n   # trials      peak         rms
     *    IEEE      0, 1     2     50000      6.2e-16     8.0e-17
     *    IEEE      0, 1     3    100000      2.5e-16     6.6e-17
     *    IEEE      0, 1     4     30000      1.7e-16     4.9e-17
     *    IEEE      0, 1     5     30000      5.1e-16     7.8e-17
     *
     */
    
     
    /*							polyn.c
     *							polyr.c
     * Arithmetic operations on polynomials
     *
     * In the following descriptions a, b, c are polynomials of degree
     * na, nb, nc respectively.  The degree of a polynomial cannot
     * exceed a run-time value MAXPOL.  An operation that attempts
     * to use or generate a polynomial of higher degree may produce a
     * result that suffers truncation at degree MAXPOL.  The value of
     * MAXPOL is set by calling the function
     *
     *     polini( maxpol );
     *
     * where maxpol is the desired maximum degree.  This must be
     * done prior to calling any of the other functions in this module.
     * Memory for internal temporary polynomial storage is allocated
     * by polini().
     *
     * Each polynomial is represented by an array containing its
     * coefficients, together with a separately declared integer equal
     * to the degree of the polynomial.  The coefficients appear in
     * ascending order; that is,
     *
     *                                        2                      na
     * a(x)  =  a[0]  +  a[1] * x  +  a[2] * x   +  ...  +  a[na] * x  .
     *
     *
     *
     * sum = poleva( a, na, x );	Evaluate polynomial a(t) at t = x.
     * polprt( a, na, D );		Print the coefficients of a to D digits.
     * polclr( a, na );		Set a identically equal to zero, up to a[na].
     * polmov( a, na, b );		Set b = a.
     * poladd( a, na, b, nb, c );	c = b + a, nc = max(na,nb)
     * polsub( a, na, b, nb, c );	c = b - a, nc = max(na,nb)
     * polmul( a, na, b, nb, c );	c = b * a, nc = na+nb
     *
     *
     * Division:
     *
     * i = poldiv( a, na, b, nb, c );	c = b / a, nc = MAXPOL
     *
     * returns i = the degree of the first nonzero coefficient of a.
     * The computed quotient c must be divided by x^i.  An error message
     * is printed if a is identically zero.
     *
     *
     * Change of variables:
     * If a and b are polynomials, and t = a(x), then
     *     c(t) = b(a(x))
     * is a polynomial found by substituting a(x) for t.  The
     * subroutine call for this is
     *
     * polsbt( a, na, b, nb, c );
     *
     *
     * Notes:
     * poldiv() is an integer routine; poleva() is double.
     * Any of the arguments a, b, c may refer to the same array.
     *
     */
    
     
    /* Arithmetic operations on polynomials with rational coefficients
     *
     * In the following descriptions a, b, c are polynomials of degree
     * na, nb, nc respectively.  The degree of a polynomial cannot
     * exceed a run-time value MAXPOL.  An operation that attempts
     * to use or generate a polynomial of higher degree may produce a
     * result that suffers truncation at degree MAXPOL.  The value of
     * MAXPOL is set by calling the function
     *
     *     polini( maxpol );
     *
     * where maxpol is the desired maximum degree.  This must be
     * done prior to calling any of the other functions in this module.
     * Memory for internal temporary polynomial storage is allocated
     * by polini().
     *
     * Each polynomial is represented by an array containing its
     * coefficients, together with a separately declared integer equal
     * to the degree of the polynomial.  The coefficients appear in
     * ascending order; that is,
     *
     *                                        2                      na
     * a(x)  =  a[0]  +  a[1] * x  +  a[2] * x   +  ...  +  a[na] * x  .
     *
     *
     *
     * `a', `b', `c' are arrays of fracts.
     * poleva( a, na, &x, &sum );	Evaluate polynomial a(t) at t = x.
     * polprt( a, na, D );		Print the coefficients of a to D digits.
     * polclr( a, na );		Set a identically equal to zero, up to a[na].
     * polmov( a, na, b );		Set b = a.
     * poladd( a, na, b, nb, c );	c = b + a, nc = max(na,nb)
     * polsub( a, na, b, nb, c );	c = b - a, nc = max(na,nb)
     * polmul( a, na, b, nb, c );	c = b * a, nc = na+nb
     *
     *
     * Division:
     *
     * i = poldiv( a, na, b, nb, c );	c = b / a, nc = MAXPOL
     *
     * returns i = the degree of the first nonzero coefficient of a.
     * The computed quotient c must be divided by x^i.  An error message
     * is printed if a is identically zero.
     *
     *
     * Change of variables:
     * If a and b are polynomials, and t = a(x), then
     *     c(t) = b(a(x))
     * is a polynomial found by substituting a(x) for t.  The
     * subroutine call for this is
     *
     * polsbt( a, na, b, nb, c );
     *
     *
     * Notes:
     * poldiv() is an integer routine; poleva() is double.
     * Any of the arguments a, b, c may refer to the same array.
     *
     */
    
     
    /*							pow.c
     *
     *	Power function
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, z, pow();
     *
     * z = pow( x, y );
     *
     *
     *
     * DESCRIPTION:
     *
     * Computes x raised to the yth power.  Analytically,
     *
     *      x**y  =  exp( y log(x) ).
     *
     * Following Cody and Waite, this program uses a lookup table
     * of 2**-i/16 and pseudo extended precision arithmetic to
     * obtain an extra three bits of accuracy in both the logarithm
     * and the exponential.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE     -26,26       30000      4.2e-16      7.7e-17
     *    DEC      -26,26       60000      4.8e-17      9.1e-18
     * 1/26 < x < 26, with log(x) uniformly distributed.
     * -26 < y < 26, y uniformly distributed.
     *    IEEE     0,8700       30000      1.5e-14      2.1e-15
     * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
     *
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * pow overflow     x**y > MAXNUM      INFINITY
     * pow underflow   x**y < 1/MAXNUM       0.0
     * pow domain      x<0 and y noninteger  0.0
     *
     */
    
     
    /*							powi.c
     *
     *	Real raised to integer power
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, powi();
     * int n;
     *
     * y = powi( x, n );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns argument x raised to the nth power.
     * The routine efficiently decomposes n as a sum of powers of
     * two. The desired power is a product of two-to-the-kth
     * powers of x.  Thus to compute the 32767 power of x requires
     * 28 multiplications instead of 32767 multiplications.
     *
     *
     *
     * ACCURACY:
     *
     *
     *                      Relative error:
     * arithmetic   x domain   n domain  # trials      peak         rms
     *    DEC       .04,26     -26,26    100000       2.7e-16     4.3e-17
     *    IEEE      .04,26     -26,26     50000       2.0e-15     3.8e-16
     *    IEEE        1,2    -1022,1023   50000       8.6e-14     1.6e-14
     *
     * Returns MAXNUM on overflow, zero on underflow.
     *
     */
    
     
    /*							psi.c
     *
     *	Psi (digamma) function
     *
     *
     * SYNOPSIS:
     *
     * double x, y, psi();
     *
     * y = psi( x );
     *
     *
     * DESCRIPTION:
     *
     *              d      -
     *   psi(x)  =  -- ln | (x)
     *              dx
     *
     * is the logarithmic derivative of the gamma function.
     * For integer x,
     *                   n-1
     *                    -
     * psi(n) = -EUL  +   >  1/k.
     *                    -
     *                   k=1
     *
     * This formula is used for 0 < n <= 10.  If x is negative, it
     * is transformed to a positive argument by the reflection
     * formula  psi(1-x) = psi(x) + pi cot(pi x).
     * For general positive x, the argument is made greater than 10
     * using the recurrence  psi(x+1) = psi(x) + 1/x.
     * Then the following asymptotic expansion is applied:
     *
     *                           inf.   B
     *                            -      2k
     * psi(x) = log(x) - 1/2x -   >   -------
     *                            -        2k
     *                           k=1   2k x
     *
     * where the B2k are Bernoulli numbers.
     *
     * ACCURACY:
     *    Relative error (except absolute when |psi| < 1):
     * arithmetic   domain     # trials      peak         rms
     *    DEC       0,30         2500       1.7e-16     2.0e-17
     *    IEEE      0,30        30000       1.3e-15     1.4e-16
     *    IEEE      -30,0       40000       1.5e-15     2.2e-16
     *
     * ERROR MESSAGES:
     *     message         condition      value returned
     * psi singularity    x integer <=0      MAXNUM
     */
    
     
    /*							revers.c
     *
     *	Reversion of power series
     *
     *
     *
     * SYNOPSIS:
     *
     * extern int MAXPOL;
     * int n;
     * double x[n+1], y[n+1];
     *
     * polini(n);
     * revers( y, x, n );
     *
     *  Note, polini() initializes the polynomial arithmetic subroutines;
     *  see polyn.c.
     *
     *
     * DESCRIPTION:
     *
     * If
     *
     *          inf
     *           -       i
     *  y(x)  =  >   a  x
     *           -    i
     *          i=1
     *
     * then
     *
     *          inf
     *           -       j
     *  x(y)  =  >   A  y    ,
     *           -    j
     *          j=1
     *
     * where
     *                   1
     *         A    =   ---
     *          1        a
     *                    1
     *
     * etc.  The coefficients of x(y) are found by expanding
     *
     *          inf      inf
     *           -        -      i
     *  x(y)  =  >   A    >  a  x
     *           -    j   -   i
     *          j=1      i=1
     *
     *  and setting each coefficient of x , higher than the first,
     *  to zero.
     *
     *
     *
     * RESTRICTIONS:
     *
     *  y[0] must be zero, and y[1] must be nonzero.
     *
     */
    
     
    /*						rgamma.c
     *
     *	Reciprocal gamma function
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, rgamma();
     *
     * y = rgamma( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns one divided by the gamma function of the argument.
     *
     * The function is approximated by a Chebyshev expansion in
     * the interval [0,1].  Range reduction is by recurrence
     * for arguments between -34.034 and +34.84425627277176174.
     * 1/MAXNUM is returned for positive arguments outside this
     * range.  For arguments less than -34.034 the cosecant
     * reflection formula is applied; lograrithms are employed
     * to avoid unnecessary overflow.
     *
     * The reciprocal gamma function has no singularities,
     * but overflow and underflow may occur for large arguments.
     * These conditions return either MAXNUM or 1/MAXNUM with
     * appropriate sign.
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC      -30,+30       4000       1.2e-16     1.8e-17
     *    IEEE     -30,+30      30000       1.1e-15     2.0e-16
     * For arguments less than -34.034 the peak error is on the
     * order of 5e-15 (DEC), excepting overflow or underflow.
     */
    
     
    /*							round.c
     *
     *	Round double to nearest or even integer valued double
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, round();
     *
     * y = round(x);
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the nearest integer to x as a double precision
     * floating point result.  If x ends in 0.5 exactly, the
     * nearest even integer is chosen.
     * 
     *
     *
     * ACCURACY:
     *
     * If x is greater than 1/(2*MACHEP), its closest machine
     * representation is already an integer, so rounding does
     * not change it.
     */
    
     
    /*							shichi.c
     *
     *	Hyperbolic sine and cosine integrals
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, Chi, Shi, shichi();
     *
     * shichi( x, &Chi, &Shi );
     *
     *
     * DESCRIPTION:
     *
     * Approximates the integrals
     *
     *                            x
     *                            -
     *                           | |   cosh t - 1
     *   Chi(x) = eul + ln x +   |    -----------  dt,
     *                         | |          t
     *                          -
     *                          0
     *
     *               x
     *               -
     *              | |  sinh t
     *   Shi(x) =   |    ------  dt
     *            | |       t
     *             -
     *             0
     *
     * where eul = 0.57721566490153286061 is Euler's constant.
     * The integrals are evaluated by power series for x < 8
     * and by Chebyshev expansions for x between 8 and 88.
     * For large x, both functions approach exp(x)/2x.
     * Arguments greater than 88 in magnitude return MAXNUM.
     *
     *
     * ACCURACY:
     *
     * Test interval 0 to 88.
     *                      Relative error:
     * arithmetic   function  # trials      peak         rms
     *    DEC          Shi       3000       9.1e-17
     *    IEEE         Shi      30000       6.9e-16     1.6e-16
     *        Absolute error, except relative when |Chi| > 1:
     *    DEC          Chi       2500       9.3e-17
     *    IEEE         Chi      30000       8.4e-16     1.4e-16
     */
    
     
    /*							sici.c
     *
     *	Sine and cosine integrals
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, Ci, Si, sici();
     *
     * sici( x, &Si, &Ci );
     *
     *
     * DESCRIPTION:
     *
     * Evaluates the integrals
     *
     *                          x
     *                          -
     *                         |  cos t - 1
     *   Ci(x) = eul + ln x +  |  --------- dt,
     *                         |      t
     *                        -
     *                         0
     *             x
     *             -
     *            |  sin t
     *   Si(x) =  |  ----- dt
     *            |    t
     *           -
     *            0
     *
     * where eul = 0.57721566490153286061 is Euler's constant.
     * The integrals are approximated by rational functions.
     * For x > 8 auxiliary functions f(x) and g(x) are employed
     * such that
     *
     * Ci(x) = f(x) sin(x) - g(x) cos(x)
     * Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x)
     *
     *
     * ACCURACY:
     *    Test interval = [0,50].
     * Absolute error, except relative when > 1:
     * arithmetic   function   # trials      peak         rms
     *    IEEE        Si        30000       4.4e-16     7.3e-17
     *    IEEE        Ci        30000       6.9e-16     5.1e-17
     *    DEC         Si         5000       4.4e-17     9.0e-18
     *    DEC         Ci         5300       7.9e-17     5.2e-18
     */
    
     
    /*							simpsn.c	*/
    /* simpsn.c
     * Numerical integration of function tabulated
     * at equally spaced arguments
     */
    
     
    /*							simq.c
     *
     *	Solution of simultaneous linear equations AX = B
     *	by Gaussian elimination with partial pivoting
     *
     *
     *
     * SYNOPSIS:
     *
     * double A[n*n], B[n], X[n];
     * int n, flag;
     * int IPS[];
     * int simq();
     *
     * ercode = simq( A, B, X, n, flag, IPS );
     *
     *
     *
     * DESCRIPTION:
     *
     * B, X, IPS are vectors of length n.
     * A is an n x n matrix (i.e., a vector of length n*n),
     * stored row-wise: that is, A(i,j) = A[ij],
     * where ij = i*n + j, which is the transpose of the normal
     * column-wise storage.
     *
     * The contents of matrix A are destroyed.
     *
     * Set flag=0 to solve.
     * Set flag=-1 to do a new back substitution for different B vector
     * using the same A matrix previously reduced when flag=0.
     *
     * The routine returns nonzero on error; messages are printed.
     *
     *
     * ACCURACY:
     *
     * Depends on the conditioning (range of eigenvalues) of matrix A.
     *
     *
     * REFERENCE:
     *
     * Computer Solution of Linear Algebraic Systems,
     * by George E. Forsythe and Cleve B. Moler; Prentice-Hall, 1967.
     *
     */
    
     
    /*							sin.c
     *
     *	Circular sine
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, sin();
     *
     * y = sin( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Range reduction is into intervals of pi/4.  The reduction
     * error is nearly eliminated by contriving an extended precision
     * modular arithmetic.
     *
     * Two polynomial approximating functions are employed.
     * Between 0 and pi/4 the sine is approximated by
     *      x  +  x**3 P(x**2).
     * Between pi/4 and pi/2 the cosine is represented as
     *      1  -  x**2 Q(x**2).
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain      # trials      peak         rms
     *    DEC       0, 10       150000       3.0e-17     7.8e-18
     *    IEEE -1.07e9,+1.07e9  130000       2.1e-16     5.4e-17
     * 
     * ERROR MESSAGES:
     *
     *   message           condition        value returned
     * sin total loss   x > 1.073741824e9      0.0
     *
     * Partial loss of accuracy begins to occur at x = 2**30
     * = 1.074e9.  The loss is not gradual, but jumps suddenly to
     * about 1 part in 10e7.  Results may be meaningless for
     * x > 2**49 = 5.6e14.  The routine as implemented flags a
     * TLOSS error for x > 2**30 and returns 0.0.
     */
    
     
    /*							cos.c
     *
     *	Circular cosine
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, cos();
     *
     * y = cos( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Range reduction is into intervals of pi/4.  The reduction
     * error is nearly eliminated by contriving an extended precision
     * modular arithmetic.
     *
     * Two polynomial approximating functions are employed.
     * Between 0 and pi/4 the cosine is approximated by
     *      1  -  x**2 Q(x**2).
     * Between pi/4 and pi/2 the sine is represented as
     *      x  +  x**3 P(x**2).
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain      # trials      peak         rms
     *    IEEE -1.07e9,+1.07e9  130000       2.1e-16     5.4e-17
     *    DEC        0,+1.07e9   17000       3.0e-17     7.2e-18
     */
    
     
    /*							sincos.c
     *
     *	Circular sine and cosine of argument in degrees
     *	Table lookup and interpolation algorithm
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, sine, cosine, flg, sincos();
     *
     * sincos( x, &sine, &cosine, flg );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns both the sine and the cosine of the argument x.
     * Several different compile time options and minimax
     * approximations are supplied to permit tailoring the
     * tradeoff between computation speed and accuracy.
     * 
     * Since range reduction is time consuming, the reduction
     * of x modulo 360 degrees is also made optional.
     *
     * sin(i) is internally tabulated for 0 <= i <= 90 degrees.
     * Approximation polynomials, ranging from linear interpolation
     * to cubics in (x-i)**2, compute the sine and cosine
     * of the residual x-i which is between -0.5 and +0.5 degree.
     * In the case of the high accuracy options, the residual
     * and the tabulated values are combined using the trigonometry
     * formulas for sin(A+B) and cos(A+B).
     *
     * Compile time options are supplied for 5, 11, or 17 decimal
     * relative accuracy (ACC5, ACC11, ACC17 respectively).
     * A subroutine flag argument "flg" chooses betwen this
     * accuracy and table lookup only (peak absolute error
     * = 0.0087).
     *
     * If the argument flg = 1, then the tabulated value is
     * returned for the nearest whole number of degrees. The
     * approximation polynomials are not computed.  At
     * x = 0.5 deg, the absolute error is then sin(0.5) = 0.0087.
     *
     * An intermediate speed and precision can be obtained using
     * the compile time option LINTERP and flg = 1.  This yields
     * a linear interpolation using a slope estimated from the sine
     * or cosine at the nearest integer argument.  The peak absolute
     * error with this option is 3.8e-5.  Relative error at small
     * angles is about 1e-5.
     *
     * If flg = 0, then the approximation polynomials are computed
     * and applied.
     *
     *
     *
     * SPEED:
     *
     * Relative speed comparisons follow for 6MHz IBM AT clone
     * and Microsoft C version 4.0.  These figures include
     * software overhead of do loop and function calls.
     * Since system hardware and software vary widely, the
     * numbers should be taken as representative only.
     *
     *			flg=0	flg=0	flg=1	flg=1
     *			ACC11	ACC5	LINTERP	Lookup only
     * In-line 8087 (/FPi)
     * sin(), cos()		1.0	1.0	1.0	1.0
     *
     * In-line 8087 (/FPi)
     * sincos()		1.1	1.4	1.9	3.0
     *
     * Software (/FPa)
     * sin(), cos()		0.19	0.19	0.19	0.19
     *
     * Software (/FPa)
     * sincos()		0.39	0.50	0.73	1.7
     *
     *
     *
     * ACCURACY:
     *
     * The accurate approximations are designed with a relative error
     * criterion.  The absolute error is greatest at x = 0.5 degree.
     * It decreases from a local maximum at i+0.5 degrees to full
     * machine precision at each integer i degrees.  With the
     * ACC5 option, the relative error of 6.3e-6 is equivalent to
     * an absolute angular error of 0.01 arc second in the argument
     * at x = i+0.5 degrees.  For small angles < 0.5 deg, the ACC5
     * accuracy is 6.3e-6 (.00063%) of reading; i.e., the absolute
     * error decreases in proportion to the argument.  This is true
     * for both the sine and cosine approximations, since the latter
     * is for the function 1 - cos(x).
     *
     * If absolute error is of most concern, use the compile time
     * option ABSERR to obtain an absolute error of 2.7e-8 for ACC5
     * precision.  This is about half the absolute error of the
     * relative precision option.  In this case the relative error
     * for small angles will increase to 9.5e-6 -- a reasonable
     * tradeoff.
     */
    
     
    /*							sindg.c
     *
     *	Circular sine of angle in degrees
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, sindg();
     *
     * y = sindg( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Range reduction is into intervals of 45 degrees.
     *
     * Two polynomial approximating functions are employed.
     * Between 0 and pi/4 the sine is approximated by
     *      x  +  x**3 P(x**2).
     * Between pi/4 and pi/2 the cosine is represented as
     *      1  -  x**2 P(x**2).
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain      # trials      peak         rms
     *    DEC       +-1000        3100      3.3e-17      9.0e-18
     *    IEEE      +-1000       30000      2.3e-16      5.6e-17
     * 
     * ERROR MESSAGES:
     *
     *   message           condition        value returned
     * sindg total loss   x > 8.0e14 (DEC)      0.0
     *                    x > 1.0e14 (IEEE)
     *
     */
    
     
    /*							cosdg.c
     *
     *	Circular cosine of angle in degrees
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, cosdg();
     *
     * y = cosdg( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Range reduction is into intervals of 45 degrees.
     *
     * Two polynomial approximating functions are employed.
     * Between 0 and pi/4 the cosine is approximated by
     *      1  -  x**2 P(x**2).
     * Between pi/4 and pi/2 the sine is represented as
     *      x  +  x**3 P(x**2).
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain      # trials      peak         rms
     *    DEC      +-1000         3400       3.5e-17     9.1e-18
     *    IEEE     +-1000        30000       2.1e-16     5.7e-17
     *  See also sin().
     *
     */
    
     
    /*							sinh.c
     *
     *	Hyperbolic sine
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, sinh();
     *
     * y = sinh( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns hyperbolic sine of argument in the range MINLOG to
     * MAXLOG.
     *
     * The range is partitioned into two segments.  If |x| <= 1, a
     * rational function of the form x + x**3 P(x)/Q(x) is employed.
     * Otherwise the calculation is sinh(x) = ( exp(x) - exp(-x) )/2.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC      +- 88        50000       4.0e-17     7.7e-18
     *    IEEE     +-MAXLOG     30000       2.6e-16     5.7e-17
     *
     */
    
     
    /*							spence.c
     *
     *	Dilogarithm
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, spence();
     *
     * y = spence( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Computes the integral
     *
     *                    x
     *                    -
     *                   | | log t
     * spence(x)  =  -   |   ----- dt
     *                 | |   t - 1
     *                  -
     *                  1
     *
     * for x >= 0.  A rational approximation gives the integral in
     * the interval (0.5, 1.5).  Transformation formulas for 1/x
     * and 1-x are employed outside the basic expansion range.
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      0,4         30000       3.9e-15     5.4e-16
     *    DEC       0,4          3000       2.5e-16     4.5e-17
     *
     *
     */
    
     
    /*							sqrt.c
     *
     *	Square root
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, sqrt();
     *
     * y = sqrt( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the square root of x.
     *
     * Range reduction involves isolating the power of two of the
     * argument and using a polynomial approximation to obtain
     * a rough value for the square root.  Then Heron's iteration
     * is used three times to converge to an accurate value.
     *
     *
     *
     * ACCURACY:
     *
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       0, 10       60000       2.1e-17     7.9e-18
     *    IEEE      0,1.7e308   30000       1.7e-16     6.3e-17
     *
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * sqrt domain        x < 0            0.0
     *
     */
    
     
    /*							stdtr.c
     *
     *	Student's t distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * double t, stdtr();
     * short k;
     *
     * y = stdtr( k, t );
     *
     *
     * DESCRIPTION:
     *
     * Computes the integral from minus infinity to t of the Student
     * t distribution with integer k > 0 degrees of freedom:
     *
     *                                      t
     *                                      -
     *                                     | |
     *              -                      |         2   -(k+1)/2
     *             | ( (k+1)/2 )           |  (     x   )
     *       ----------------------        |  ( 1 + --- )        dx
     *                     -               |  (      k  )
     *       sqrt( k pi ) | ( k/2 )        |
     *                                   | |
     *                                    -
     *                                   -inf.
     * 
     * Relation to incomplete beta integral:
     *
     *        1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z )
     * where
     *        z = k/(k + t**2).
     *
     * For t < -2, this is the method of computation.  For higher t,
     * a direct method is derived from integration by parts.
     * Since the function is symmetric about t=0, the area under the
     * right tail of the density is found by calling the function
     * with -t instead of t.
     * 
     * ACCURACY:
     *
     * Tested at random 1 <= k <= 25.  The "domain" refers to t.
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE     -100,-2      50000       5.9e-15     1.4e-15
     *    IEEE     -2,100      500000       2.7e-15     4.9e-17
     */
    
     
    /*							stdtri.c
     *
     *	Functional inverse of Student's t distribution
     *
     *
     *
     * SYNOPSIS:
     *
     * double p, t, stdtri();
     * int k;
     *
     * t = stdtri( k, p );
     *
     *
     * DESCRIPTION:
     *
     * Given probability p, finds the argument t such that stdtr(k,t)
     * is equal to p.
     * 
     * ACCURACY:
     *
     * Tested at random 1 <= k <= 100.  The "domain" refers to p:
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE    .001,.999     25000       5.7e-15     8.0e-16
     *    IEEE    10^-6,.001    25000       2.0e-12     2.9e-14
     */
    
     
    /*							struve.c
     *
     *      Struve function
     *
     *
     *
     * SYNOPSIS:
     *
     * double v, x, y, struve();
     *
     * y = struve( v, x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Computes the Struve function Hv(x) of order v, argument x.
     * Negative x is rejected unless v is an integer.
     *
     * This module also contains the hypergeometric functions 1F2
     * and 3F0 and a routine for the Bessel function Yv(x) with
     * noninteger v.
     *
     *
     *
     * ACCURACY:
     *
     * Not accurately characterized, but spot checked against tables.
     *
     */
    
     
    /*							tan.c
     *
     *	Circular tangent
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, tan();
     *
     * y = tan( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the circular tangent of the radian argument x.
     *
     * Range reduction is modulo pi/4.  A rational function
     *       x + x**3 P(x**2)/Q(x**2)
     * is employed in the basic interval [0, pi/4].
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC      +-1.07e9      44000      4.1e-17     1.0e-17
     *    IEEE     +-1.07e9      30000      2.9e-16     8.1e-17
     *
     * ERROR MESSAGES:
     *
     *   message         condition          value returned
     * tan total loss   x > 1.073741824e9     0.0
     *
     */
    
     
    /*							cot.c
     *
     *	Circular cotangent
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, cot();
     *
     * y = cot( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the circular cotangent of the radian argument x.
     *
     * Range reduction is modulo pi/4.  A rational function
     *       x + x**3 P(x**2)/Q(x**2)
     * is employed in the basic interval [0, pi/4].
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE     +-1.07e9      30000      2.9e-16     8.2e-17
     *
     *
     * ERROR MESSAGES:
     *
     *   message         condition          value returned
     * cot total loss   x > 1.073741824e9       0.0
     * cot singularity  x = 0                  INFINITY
     *
     */
    
     
    /*							tandg.c
     *
     *	Circular tangent of argument in degrees
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, tandg();
     *
     * y = tandg( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the circular tangent of the argument x in degrees.
     *
     * Range reduction is modulo pi/4.  A rational function
     *       x + x**3 P(x**2)/Q(x**2)
     * is employed in the basic interval [0, pi/4].
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC      0,10          8000      3.4e-17      1.2e-17
     *    IEEE     0,10         30000      3.2e-16      8.4e-17
     *
     * ERROR MESSAGES:
     *
     *   message         condition          value returned
     * tandg total loss   x > 8.0e14 (DEC)      0.0
     *                    x > 1.0e14 (IEEE)
     * tandg singularity  x = 180 k  +  90     MAXNUM
     */
    
     
    /*							cotdg.c
     *
     *	Circular cotangent of argument in degrees
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, cotdg();
     *
     * y = cotdg( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns the circular cotangent of the argument x in degrees.
     *
     * Range reduction is modulo pi/4.  A rational function
     *       x + x**3 P(x**2)/Q(x**2)
     * is employed in the basic interval [0, pi/4].
     *
     *
     * ERROR MESSAGES:
     *
     *   message         condition          value returned
     * cotdg total loss   x > 8.0e14 (DEC)      0.0
     *                    x > 1.0e14 (IEEE)
     * cotdg singularity  x = 180 k            MAXNUM
     */
    
     
    /*							tanh.c
     *
     *	Hyperbolic tangent
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, tanh();
     *
     * y = tanh( x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns hyperbolic tangent of argument in the range MINLOG to
     * MAXLOG.
     *
     * A rational function is used for |x| < 0.625.  The form
     * x + x**3 P(x)/Q(x) of Cody & Waite is employed.
     * Otherwise,
     *    tanh(x) = sinh(x)/cosh(x) = 1  -  2/(exp(2x) + 1).
     *
     *
     *
     * ACCURACY:
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       -2,2        50000       3.3e-17     6.4e-18
     *    IEEE      -2,2        30000       2.5e-16     5.8e-17
     *
     */
    
         
    /*							unity.c
     *
     * Relative error approximations for function arguments near
     * unity.
     *
     *    log1p(x) = log(1+x)
     *    expm1(x) = exp(x) - 1
     *    cosm1(x) = cos(x) - 1
     *
     */
    
     
    /*							yn.c
     *
     *	Bessel function of second kind of integer order
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, yn();
     * int n;
     *
     * y = yn( n, x );
     *
     *
     *
     * DESCRIPTION:
     *
     * Returns Bessel function of order n, where n is a
     * (possibly negative) integer.
     *
     * The function is evaluated by forward recurrence on
     * n, starting with values computed by the routines
     * y0() and y1().
     *
     * If n = 0 or 1 the routine for y0 or y1 is called
     * directly.
     *
     *
     *
     * ACCURACY:
     *
     *
     *                      Absolute error, except relative
     *                      when y > 1:
     * arithmetic   domain     # trials      peak         rms
     *    DEC       0, 30        2200       2.9e-16     5.3e-17
     *    IEEE      0, 30       30000       3.4e-15     4.3e-16
     *
     *
     * ERROR MESSAGES:
     *
     *   message         condition      value returned
     * yn singularity   x = 0              MAXNUM
     * yn overflow                         MAXNUM
     *
     * Spot checked against tables for x, n between 0 and 100.
     *
     */
    
     
    /*							zeta.c
     *
     *	Riemann zeta function of two arguments
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, q, y, zeta();
     *
     * y = zeta( x, q );
     *
     *
     *
     * DESCRIPTION:
     *
     *
     *
     *                 inf.
     *                  -        -x
     *   zeta(x,q)  =   >   (k+q)  
     *                  -
     *                 k=0
     *
     * where x > 1 and q is not a negative integer or zero.
     * The Euler-Maclaurin summation formula is used to obtain
     * the expansion
     *
     *                n         
     *                -       -x
     * zeta(x,q)  =   >  (k+q)  
     *                -         
     *               k=1        
     *
     *           1-x                 inf.  B   x(x+1)...(x+2j)
     *      (n+q)           1         -     2j
     *  +  ---------  -  -------  +   >    --------------------
     *        x-1              x      -                   x+2j+1
     *                   2(n+q)      j=1       (2j)! (n+q)
     *
     * where the B2j are Bernoulli numbers.  Note that (see zetac.c)
     * zeta(x,1) = zetac(x) + 1.
     *
     *
     *
     * ACCURACY:
     *
     *
     *
     * REFERENCE:
     *
     * Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals,
     * Series, and Products, p. 1073; Academic Press, 1980.
     *
     */
    
     
    /*							zetac.c
     *
     *	Riemann zeta function
     *
     *
     *
     * SYNOPSIS:
     *
     * double x, y, zetac();
     *
     * y = zetac( x );
     *
     *
     *
     * DESCRIPTION:
     *
     *
     *
     *                inf.
     *                 -    -x
     *   zetac(x)  =   >   k   ,   x > 1,
     *                 -
     *                k=2
     *
     * is related to the Riemann zeta function by
     *
     *	Riemann zeta(x) = zetac(x) + 1.
     *
     * Extension of the function definition for x < 1 is implemented.
     * Zero is returned for x > log2(MAXNUM).
     *
     * An overflow error may occur for large negative x, due to the
     * gamma function in the reflection formula.
     *
     * ACCURACY:
     *
     * Tabulated values have full machine accuracy.
     *
     *                      Relative error:
     * arithmetic   domain     # trials      peak         rms
     *    IEEE      1,50        10000       9.8e-16	    1.3e-16
     *    DEC       1,50         2000       1.1e-16     1.9e-17
     *
     *
     */
    

      To Cephes home page www.moshier.net:

      Get double.zip:
    Last update: 5 October 2014