Shlomi Fish > Math-Cephes-0.5303 > Math::Cephes

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NAME ^

Math::Cephes - perl interface to the cephes math library

SYNOPSIS ^

  use Math::Cephes qw(:all);

DESCRIPTION ^

  This module provides an interface to over 150 functions of the
  cephes math library of Stephen Moshier. No functions are exported
  by default, but rather must be imported explicitly, as in

     use Math::Cephes qw(sin cos);

  There are a number of export tags defined which allow
  importing groups of functions:
use Math::Cephes qw(:constants);
  imports the variables

  $PI      :   3.14159265358979323846      #  pi
  $PIO2    :   1.57079632679489661923      #  pi/2
  $PIO4    :   0.785398163397448309616     #  pi/4
  $SQRT2   :   1.41421356237309504880      #  sqrt(2)
  $SQRTH   :   0.707106781186547524401     #  sqrt(2)/2
  $LOG2E   :   1.4426950408889634073599    #  1/log(2)
  $SQ2OPI  :   0.79788456080286535587989   #  sqrt( 2/pi )
  $LOGE2   :   0.693147180559945309417     #  log(2)
  $LOGSQ2  :   0.346573590279972654709     #  log(2)/2
  $THPIO4  :   2.35619449019234492885      #  3*pi/4
  $TWOOPI  :   0.636619772367581343075535  #  2/pi

  As well, there are 4 machine-specific numbers available:

   $MACHEP : machine roundoff error
   $MAXLOG : maximum log on the machine
   $MINLOG : minimum log on the machine
   $MAXNUM : largest number represented
use Math::Cephes qw(:trigs);
  imports

 acos:  Inverse circular cosine
 asin:  Inverse circular sine
 atan:  Inverse circular tangent (arctangent)
 atan2:  Quadrant correct inverse circular tangent
 cos:  Circular cosine
 cosdg:  Circular cosine of angle in degrees
 cot:  Circular cotangent
 cotdg:  Circular cotangent of argument in degrees
 hypot: hypotenuse associated with the sides of a right triangle
 radian: Degrees, minutes, seconds to radians
 sin:  Circular sine
 sindg:  Circular sine of angle in degrees
 tan:  Circular tangent
 tandg:  Circular tangent of argument in degrees
 cosm1:  Relative error approximations for function arguments near unity
use Math::Cephes qw(:hypers);
  imports

 acosh:  Inverse hyperbolic cosine
 asinh:  Inverse hyperbolic sine
 atanh:  Inverse hyperbolic tangent
 cosh:  Hyperbolic cosine
 sinh:  Hyperbolic sine
 tanh:  Hyperbolic tangent
use Math::Cephes qw(:explog);
  imports

 exp:  Exponential function
 expxx: exp(x*x)
 exp10:  Base 10 exponential function (Common antilogarithm)
 exp2:  Base 2 exponential function
 log:  Natural logarithm
 log10:  Common logarithm
 log2:  Base 2 logarithm
 log1p,expm1:  Relative error approximations for function arguments near unity.
use Math::Cephes qw(:cmplx);
  imports

 new_cmplx: create a new complex number object
 cabs:  Complex absolute value
 cacos:  Complex circular arc cosine
 cacosh: Complex inverse hyperbolic cosine
 casin:  Complex circular arc sine
 casinh: Complex inverse hyperbolic sine
 catan:  Complex circular arc tangent
 catanh: Complex inverse hyperbolic tangent
 ccos:  Complex circular cosine
 ccosh: Complex hyperbolic cosine
 ccot:  Complex circular cotangent
 cexp:  Complex exponential function
 clog:  Complex natural logarithm
 cadd: add two complex numbers
 csub: subtract two complex numbers
 cmul: multiply two complex numbers
 cdiv: divide two complex numbers
 cmov: copy one complex number to another
 cneg: negate a complex number
 cpow: Complex power function
 csin:  Complex circular sine
 csinh: Complex hyperbolic sine
 csqrt:  Complex square root
 ctan:  Complex circular tangent
 ctanh: Complex hyperbolic tangent
use Math::Cephes qw(:utils);
  imports

 cbrt:  Cube root
 ceil:  ceil
 drand:  Pseudorandom number generator
 fabs:  Absolute value
 fac:  Factorial function
 floor:  floor
 frexp:  frexp
 ldexp:  multiplies x by 2**n.
 lrand:  Pseudorandom number generator
 lsqrt:  Integer square root
 pow:  Power function
 powi:  Real raised to integer power
 round:  Round double to nearest or even integer valued double
 sqrt:  Square root
use Math::Cephes qw(:bessels);
  imports

 i0:  Modified Bessel function of order zero
 i0e:  Modified Bessel function of order zero, exponentially scaled
 i1:  Modified Bessel function of order one
 i1e:  Modified Bessel function of order one, exponentially scaled
 iv:  Modified Bessel function of noninteger order
 j0:  Bessel function of order zero
 j1:  Bessel function of 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
 y0:  Bessel function of the second kind, order zero
 y1:  Bessel function of second kind of order one
 yn:  Bessel function of second kind of integer order
 yv:  Bessel function Yv with noninteger v
use Math::Cephes qw(:dists);
  imports

 bdtr:  Binomial distribution
 bdtrc:  Complemented binomial distribution
 bdtri:  Inverse binomial distribution
 btdtr:  Beta distribution
 chdtr:  Chi-square distribution
 chdtrc:  Complemented Chi-square distribution
 chdtri:  Inverse of complemented Chi-square distribution
 fdtr:  F distribution
 fdtrc:  Complemented F distribution
 fdtri:  Inverse of complemented F distribution
 gdtr:  Gamma distribution function
 gdtrc:  Complemented gamma distribution function
 nbdtr:  Negative binomial distribution
 nbdtrc:  Complemented negative binomial distribution
 nbdtri:  Functional inverse of negative binomial distribution
 ndtr:  Normal distribution function
 ndtri:  Inverse of Normal distribution function
 pdtr:  Poisson distribution
 pdtrc:  Complemented poisson distribution
 pdtri:  Inverse Poisson distribution
 stdtr:  Student's t distribution
 stdtri:  Functional inverse of Student's t distribution
use Math::Cephes qw(:gammas);
  imports

 fac:  Factorial function
 gamma:  Gamma function
 igam:  Incomplete gamma integral
 igamc:  Complemented incomplete gamma integral
 igami:  Inverse of complemented imcomplete gamma integral
 psi:  Psi (digamma) function
 rgamma:  Reciprocal gamma function
use Math::Cephes qw(:betas);
  imports

 beta:  Beta function
 incbet:  Incomplete beta integral
 incbi:  Inverse of imcomplete beta integral
 lbeta:  Natural logarithm of |beta|
use Math::Cephes qw(:elliptics);
  imports

 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
use Math::Cephes qw(:hypergeometrics);
  imports

 hyp2f0:  Gauss hypergeometric function   F
 hyp2f1:  Gauss hypergeometric function   F
 hyperg:  Confluent hypergeometric function
 onef2:  Hypergeometric function 1F2
 threef0:  Hypergeometric function 3F0
use Math::Cephes qw(:misc);
  imports

 airy:  Airy function
 bernum: Bernoulli numbers
 dawsn:  Dawson's Integral
 ei: Exponential integral
 erf:  Error function
 erfc:  Complementary error function
 expn:  Exponential integral En
 fresnl:  Fresnel integral
 plancki: Integral of Planck's black body radiation formula
 polylog: Polylogarithm function
 shichi:  Hyperbolic sine and cosine integrals
 sici:  Sine and cosine integrals
 simpson: Simpson's rule to find an integral
 spence:  Dilogarithm
 struve:  Struve function
 vecang: angle between two vectors
 zeta:  Riemann zeta function of two arguments
 zetac:  Riemann zeta function
use Math::Cephes qw(:fract);
  imports

 new_fract: create a new fraction object
 radd: add two fractions
 rmul: multiply two fractions
 rsub: subtracttwo fractions
 rdiv: divide two fractions
 euclid: finds the greatest common divisor

FUNCTIONS ^

  A description of the various functions available follows.
acosh: 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: Airy function
 SYNOPSIS:

 # double x, ai, aiprime, bi, biprime;
 # int airy();

 ($flag, $ai, $aiprime, $bi, $biprime) = airy( $x );

 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
radian: Degrees, minutes, seconds to radians
 SYNOPSIS:

 # double d, m, s, radian();

 $r = radian( $d, $m, $s );

 DESCRIPTION:

 Converts an angle of degrees, minutes, seconds to radians.
hypot: returns the hypotenuse associated with the sides of a right triangle
 SYNOPSIS:

 # double a, b, c, hypot();

 $c = hypot( $a, $b );

 DESCRIPTION:

 Calculates the hypotenuse associated with the sides of a
 right triangle, according to

        c = sqrt( a**2 + b**2)
asin: 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: 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: 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: 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: 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: 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
lbeta: Natural logarithm of |beta|
 SYNOPSIS:

 # double a, b;

 # double lbeta( a, b );

 $y = lbeta( $a, $b);
btdtr: 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: 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
chdtr: Chi-square distribution
 SYNOPSIS:

 # double v, x, y, chdtr();

 $y = chdtr( $v, $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 = chdtrc( v, x ) = igamc( v/2.0, x/2.0 ).

 The arguments must both be positive.

 ACCURACY:

 See igamc().

 ERROR MESSAGES:

   message         condition      value returned
 chdtrc domain  x < 0 or v < 1        0.0
chdtri: Inverse of complemented Chi-square distribution
 SYNOPSIS:

 # double df, x, y, chdtri();

 $x = chdtri( $df, $y );

 DESCRIPTION:

 Finds the Chi-square argument x such that the integral
 from x to infinity of the Chi-square density is equal
 to the given cumulative probability y.

 This is accomplished using the inverse gamma integral
 function and the relation

    x/2 = igami( df/2, y );

 ACCURACY:

 See igami.c.

 ERROR MESSAGES:

   message         condition      value returned
 chdtri domain   y < 0 or y > 1        0.0
                     v < 1
clog: Complex natural logarithm
 SYNOPSIS:

 # void clog();
 # cmplx z, w;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 clog($z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $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;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 cexp($z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $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;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 csin($z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $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;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 ccos($z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $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;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 ctan($z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $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;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 ccot($z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $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;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 casin($z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $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;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 cacos($z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $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;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 catan($z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $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;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 csinh($z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $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;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 casinh($z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
 print_new_cmplx($w);                 # prints $w as Re($w) + i Im($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;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 ccosh($z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $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;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 cacosh($z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $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;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 ctanh($z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $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;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 catanh($z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $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;

 $a = new_cmplx(5, 6);    # $z = 5 + 6 i
 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 cpow($a, $z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $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: Complex number arithmetic
 SYNOPSIS:

 # typedef struct {
 #     double r;     real part
 #     double i;     imaginary part
 #    }cmplx;

 # cmplx *a, *b, *c;

 $a = new_cmplx(3, 5);   # $a = 3 + 5 i
 $b = new_cmplx(2, 3);   # $b = 2 + 3 i
 $c = new_cmplx();

 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

 print $c->{r}, '  ', $c->{i};   # prints real and imaginary parts of $c

 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 a, cabs();
 # cmplx z;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $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;

 $z = new_cmplx(2, 3);    # $z = 2 + 3 i
 $w = new_cmplx();
 csqrt($z, $w );
 print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $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.
machconst: 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

 These lists are subject to change.
cosh: 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: 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: Pseudorandom number generator
 SYNOPSIS:

 # double y, drand();

 ($flag, $y) = drand( );

 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.
ellie: 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: 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: 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: Jacobian Elliptic Functions
 SYNOPSIS:

 # double u, m, sn, cn, dn, phi;
 # int ellpj();

 ($flag, $sn, $cn, $dn, $phi) = ellpj( $u, $m );

 DESCRIPTION:

 Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m),
 and dn(u|m) of parameter m between 0 and 1, and real
 argument u.

 These functions are periodic, with quarter-period on the
 real axis equal to the complete elliptic integral
 ellpk(1.0-m).

 Relation to incomplete elliptic integral:
 If u = ellik(phi,m), then sn(u|m) = sin(phi),
 and cn(u|m) = cos(phi).  Phi is called the amplitude of u.

 Computation is by means of the arithmetic-geometric mean
 algorithm, except when m is within 1e-9 of 0 or 1.  In the
 latter case with m close to 1, the approximation applies
 only for phi < pi/2.

 ACCURACY:

 Tested at random points with u between 0 and 10, m between
 0 and 1.

            Absolute error (* = relative error):
 arithmetic   function   # trials      peak         rms
    DEC       sn           1800       4.5e-16     8.7e-17
    IEEE      phi         10000       9.2e-16*    1.4e-16*
    IEEE      sn          50000       4.1e-15     4.6e-16
    IEEE      cn          40000       3.6e-15     4.4e-16
    IEEE      dn          10000       1.3e-12     1.8e-14

  Peak error observed in consistency check using addition
 theorem for sn(u+v) was 4e-16 (absolute).  Also tested by
 the above relation to the incomplete elliptic integral.
 Accuracy deteriorates when u is large.
ellpk: 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: Rational arithmetic routines
 SYNOPSIS:


 # typedef struct
 #     {
 #     double n;  numerator
 #     double d;  denominator
 #     }fract;

 $a = new_fract(3, 4);  # a = 3 / 4
 $b = new_fract(2, 3);  # b = 2 / 3
 $c = new_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
 print $c->{n}, ' ', $c->{d};  # prints numerator and denominator of $c

 ($gcd, $m_reduced, $n_reduced) = euclid($m, $n);
 # returns the greatest common divisor of $m and $n, as well as
 # the result of reducing $m and $n by $gcd

 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: 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
expxx: exp(x*x)
 #  double x, y, expxx();
 # int sign;

   $y = expxx( $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, exp(-x*x) is returned.
  If sign > 0, or omitted, exp(x*x) is returned.

 ACCURACY:

                       Relative error:
 arithmetic    domain     # trials      peak         rms
    IEEE      -26.6, 26.6    10^7       3.9e-16     8.9e-17
exp10: 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: 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.
ei: 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
expn: Exponential integral En
 SYNOPSIS:

 # int n;
 # double x, y, expn();

 $y = expn( $n, $x );

 DESCRIPTION:

 Evaluates the exponential integral

                 inf.
                   -
                  | |   -xt
                  |    e
      E (x)  =    |    ----  dt.
       n          |      n
                | |     t
                 -
                  1

 Both n and x must be nonnegative.

 The routine employs either a power series, a continued
 fraction, or an asymptotic formula depending on the
 relative values of n and x.

 ACCURACY:

                      Relative error:
 arithmetic   domain     # trials      peak         rms
    DEC       0, 30        5000       2.0e-16     4.6e-17
    IEEE      0, 30       10000       1.7e-15     3.6e-16
fabs: Absolute value
 SYNOPSIS:

 # double x, y;

 $y = fabs( $x );

 DESCRIPTION:

 Returns the absolute value of the argument.
fac: 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: 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
ceil: ceil
 ceil() returns the smallest integer greater than or equal
 to x.  It truncates toward plus infinity.

 SYNOPSIS:

 # double x, y, ceil();

 $y = ceil( $x );
floor: floor
 floor() returns the largest integer less than or equal to x.
 It truncates toward minus infinity.

 SYNOPSIS:

 # double x, y, floor();

 $y = floor( $x );
frexp: frexp
 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.

 SYNOPSIS:

 # double x, y, frexp();
 # int expnt;

 ($y, $expnt)  = frexp( $x );
ldexp: multiplies x by 2**n.
 SYNOPSIS:

 # double x, y, ldexp();
 # int n;

 $y = ldexp( $x, $n );
fresnl: Fresnel integral
 SYNOPSIS:

 # double x, S, C;
 # void fresnl();

 ($flag, $S, $C) = fresnl( $x );

 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: 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: 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: 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
hyp2f0: Gauss hypergeometric function 2F0
 SYNOPSIS:

 # double a, b, x, value, *err;
 # int type;    /* determines what converging factor to use */

 ($value, $err) =  hyp2f0( $a, $b, $x, $type )
hyp2f1: Gauss hypergeometric function 2F1
 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: Confluent hypergeometric function
 SYNOPSIS:

 # double a, b, x, y, hyperg();

 $y = hyperg( $a, $b, $x );

 DESCRIPTION:

 Computes the confluent hypergeometric function

                          1           2
                       a x    a(a+1) x
   F ( a,b;x )  =  1 + ---- + --------- + ...
  1 1                  b 1!   b(b+1) 2!

 Many higher transcendental functions are special cases of
 this power series.

 As is evident from the formula, b must not be a negative
 integer or zero unless a is an integer with 0 >= a > b.

 The routine attempts both a direct summation of the series
 and an asymptotic expansion.  In each case error due to
 roundoff, cancellation, and nonconvergence is estimated.
 The result with smaller estimated error is returned.

 ACCURACY:

 Tested at random points (a, b, x), all three variables
 ranging from 0 to 30.
                      Relative error:
 arithmetic   domain     # trials      peak         rms
    DEC       0,30         2000       1.2e-15     1.3e-16
    IEEE      0,30        30000       1.8e-14     1.1e-15

 Larger errors can be observed when b is near a negative
 integer or zero.  Certain combinations of arguments yield
 serious cancellation error in the power series summation
 and also are not in the region of near convergence of the
 asymptotic series.  An error message is printed if the
 self-estimated relative error is greater than 1.0e-12.
i0: 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: 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: 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: 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: 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: Incomplete beta integral
 SYNOPSIS:

 # double a, b, x, y, incbet();

 $y = incbet( $a, $b, $x );

 DESCRIPTION:

 Returns incomplete beta integral of the arguments, evaluated
 from zero to x.  The function is defined as

                  x
     -            -
    | (a+b)      | |  a-1     b-1
  -----------    |   t   (1-t)   dt.
   -     -     | |
  | (a) | (b)   -
                 0

 The domain of definition is 0 <= x <= 1.  In this
 implementation a and b are restricted to positive values.
 The integral from x to 1 may be obtained by the symmetry
 relation

    1 - incbet( a, b, x )  =  incbet( b, a, 1-x ).

 The integral is evaluated by a continued fraction expansion
 or, when b*x is small, by a power series.

 ACCURACY:

 Tested at uniformly distributed random points (a,b,x) with a and b
 in "domain" and x between 0 and 1.
                                        Relative error
 arithmetic   domain     # trials      peak         rms
    IEEE      0,5         10000       6.9e-15     4.5e-16
    IEEE      0,85       250000       2.2e-13     1.7e-14
    IEEE      0,1000      30000       5.3e-12     6.3e-13
    IEEE      0,10000    250000       9.3e-11     7.1e-12
    IEEE      0,100000    10000       8.7e-10     4.8e-11
 Outputs smaller than the IEEE gradual underflow threshold
 were excluded from these statistics.

 ERROR MESSAGES:
   message         condition      value returned
 incbet domain      x<0, x>1          0.0
 incbet underflow                     0.0
incbi: Inverse of imcomplete beta integral
 SYNOPSIS:

 # double a, b, x, y, incbi();

 $x = incbi( $a, $b, $y );

 DESCRIPTION:

 Given y, the function finds x such that

  incbet( a, b, x ) = y .

 The routine performs interval halving or Newton iterations to find the
 root of incbet(a,b,x) - y = 0.

 ACCURACY:

                      Relative error:
                x     a,b
 arithmetic   domain  domain  # trials    peak       rms
    IEEE      0,1    .5,10000   50000    5.8e-12   1.3e-13
    IEEE      0,1   .25,100    100000    1.8e-13   3.9e-15
    IEEE      0,1     0,5       50000    1.1e-12   5.5e-15
    VAX       0,1    .5,100     25000    3.5e-14   1.1e-15
 With a and b constrained to half-integer or integer values:
    IEEE      0,1    .5,10000   50000    5.8e-12   1.1e-13
    IEEE      0,1    .5,100    100000    1.7e-14   7.9e-16
 With a = .5, b constrained to half-integer or integer values:
    IEEE      0,1    .5,10000   10000    8.3e-11   1.0e-11
iv: 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: 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: 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: 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: 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: 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: 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: 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.

      k0e(x) = exp(x) * k0(x).

 ACCURACY:

                      Relative error:
 arithmetic   domain     # trials      peak         rms
    IEEE      0, 30       30000       1.4e-15     1.4e-16
 See k0().
k1: 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: 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: 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.
log: 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: 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: 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: Pseudorandom number generator
 SYNOPSIS:

 long y, lrand();

 $y = lrand( );

 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: 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.
mtherr: Library common error handling routine
 SYNOPSIS:

 char *fctnam;
 # int code;
 # int mtherr();

 mtherr( $fctnam, $code );

 DESCRIPTION:

 This routine may be called to report one of the following
 error conditions (in the include file mconf.h).

   Mnemonic        Value          Significance

    DOMAIN            1       argument domain error
    SING              2       function singularity
    OVERFLOW          3       overflow range error
    UNDERFLOW         4       underflow range error
    TLOSS             5       total loss of precision
    PLOSS             6       partial loss of precision
    EDOM             33       Unix domain error code
    ERANGE           34       Unix range error code

 The default version of the file prints the function name,
 passed to it by the pointer fctnam, followed by the
 error condition.  The display is directed to the standard
 output device.  The routine then returns to the calling
 program.  Users may wish to modify the program to abort by
 calling exit() under severe error conditions such as domain
 errors.

 Since all error conditions pass control to this function,
 the display may be easily changed, eliminated, or directed
 to an error logging device.

 SEE ALSO: mconf.h
nbdtr: Negative binomial distribution
 SYNOPSIS:

 # int k, n;
 # double p, y, nbdtr();

 $y = nbdtr( $k, $n, $p );

 DESCRIPTION:

 Returns the sum of the terms 0 through k of the negative
 binomial distribution:

   k
   --  ( n+j-1 )   n      j
   >   (       )  p  (1-p)
   --  (   j   )
  j=0

 In a sequence of Bernoulli trials, this is the probability
 that k or fewer failures precede the nth success.

 The terms are not computed individually; instead the incomplete
 beta integral is employed, according to the formula

 y = nbdtr( k, n, p ) = incbet( n, k+1, p ).

 The arguments must be positive, with p ranging from 0 to 1.

 ACCURACY:

 Tested at random points (a,b,p), with p between 0 and 1.

               a,b                     Relative error:
 arithmetic  domain     # trials      peak         rms
    IEEE     0,100       100000      1.7e-13     8.8e-15
 See also incbet.c.
nbdtrc: Complemented negative binomial distribution
 SYNOPSIS:

 # int k, n;
 # double p, y, nbdtrc();

 $y = nbdtrc( $k, $n, $p );

 DESCRIPTION:

 Returns the sum of the terms k+1 to infinity of the negative
 binomial distribution:

   inf
   --  ( n+j-1 )   n      j
   >   (       )  p  (1-p)
   --  (   j   )
  j=k+1

 The terms are not computed individually; instead the incomplete
 beta integral is employed, according to the formula

 y = nbdtrc( k, n, p ) = incbet( k+1, n, 1-p ).

 The arguments must be positive, with p ranging from 0 to 1.

 ACCURACY:

 Tested at random points (a,b,p), with p between 0 and 1.

               a,b                     Relative error:
 arithmetic  domain     # trials      peak         rms
    IEEE     0,100       100000      1.7e-13     8.8e-15
 See also incbet.c.
nbdtrc: Complemented negative binomial distribution
 SYNOPSIS:

 # int k, n;
 # double p, y, nbdtrc();

 $y = nbdtrc( $k, $n, $p );

 DESCRIPTION:

 Returns the sum of the terms k+1 to infinity of the negative
 binomial distribution:

   inf
   --  ( n+j-1 )   n      j
   >   (       )  p  (1-p)
   --  (   j   )
  j=k+1

 The terms are not computed individually; instead the incomplete
 beta integral is employed, according to the formula

 y = nbdtrc( k, n, p ) = incbet( k+1, n, 1-p ).

 The arguments must be positive, with p ranging from 0 to 1.

 ACCURACY:

 See incbet.c.
nbdtri: Functional inverse of negative binomial distribution
 SYNOPSIS:

 # int k, n;
 # double p, y, nbdtri();

 $p = nbdtri( $k, $n, $y );

 DESCRIPTION:

 Finds the argument p such that nbdtr(k,n,p) is equal to y.

 ACCURACY:

 Tested at random points (a,b,y), with y between 0 and 1.

               a,b                     Relative error:
 arithmetic  domain     # trials      peak         rms
    IEEE     0,100       100000      1.5e-14     8.5e-16
 See also incbi.c.
ndtr: 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

 where z = x/sqrt(2). Computation is via the functions
 erf and erfc.

 ACCURACY:

                      Relative error:
 arithmetic   domain     # trials      peak         rms
    DEC      -13,0         8000       2.1e-15     4.8e-16
    IEEE     -13,0        30000       3.4e-14     6.7e-15

 ERROR MESSAGES:

   message         condition         value returned
 erfc underflow    x > 37.519379347       0.0
erf: Error function
 SYNOPSIS:

 # double x, y, erf();

 $y = erf( $x );

 DESCRIPTION:

 The integral is

                           x
                            -
                 2         | |          2
   erf(x)  =  --------     |    exp( - t  ) dt.
              sqrt(pi)   | |
                          -
                           0

 The magnitude of x is limited to 9.231948545 for DEC
 arithmetic; 1 or -1 is returned outside this range.

 For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise
 erf(x) = 1 - erfc(x).

 ACCURACY:

                      Relative error:
 arithmetic   domain     # trials      peak         rms
    DEC       0,1         14000       4.7e-17     1.5e-17
    IEEE      0,1         30000       3.7e-16     1.0e-16
erfc: Complementary error function
 SYNOPSIS:

 # double x, y, erfc();

 $y = erfc( $x );

 DESCRIPTION:

  1 - erf(x) =

                           inf.
                             -
                  2         | |          2
   erfc(x)  =  --------     |    exp( - t  ) dt
               sqrt(pi)   | |
                           -
                            x

 For small x, erfc(x) = 1 - erf(x); otherwise rational
 approximations are computed.

 ACCURACY:

                      Relative error:
 arithmetic   domain     # trials      peak         rms
    DEC       0, 9.2319   12000       5.1e-16     1.2e-16
    IEEE      0,26.6417   30000       5.7e-14     1.5e-14

 ERROR MESSAGES:

   message         condition              value returned
 erfc underflow    x > 9.231948545 (DEC)       0.0
ndtri: 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: 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
pow: 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: 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: 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
rgamma: 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: 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: Hyperbolic sine and cosine integrals
 SYNOPSIS:

 # double x, Chi, Shi, shichi();

 ($flag, $Shi, $Chi) = shichi( $x );

 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: Sine and cosine integrals
 SYNOPSIS:

 # double x, Ci, Si, sici();

 ($flag, $Si, $Ci) = sici( $x );

 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
sin: 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: 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
sindg: 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: 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: 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: 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: 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: 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: 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: 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.

 ACCURACY:

 Not accurately characterized, but spot checked against tables.
plancki: 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 = 3.7417749e-16 and c2 = 0.01438769 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
       =  5.6705032e-8 T^4
 where sigma = 5.6705032e-8 W m^2 K^-4 is the Stefan-Boltzmann constant.


 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
polylog: polylogarithm function 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

  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
bernum: Bernoulli numbers
 SYNOPSIS:

    ($num, $den) = bernum( $n);
    ($num_array, $den_array) = bernum();

 DESCRIPTION:

 This calculates the Bernoulli numbers, up to 30th order.
 If called with an integer argument, the numerator and denominator
 of that Bernoulli number is returned; if called with no argument,
 two array references representing the numerator and denominators
 of the first 30 Bernoulli numbers are returned.
simpson: Simpson's rule to find an integral
 SYNOPSIS:

    $result = simpson(\&fun, $a, $b, $abs_err, $rel_err, $nmax);

    sub fun {
       my $x = shift;
       return cos($x)*exp($x);
    }

 DESCRIPTION:

 This evaluates the area under the graph of a function,
 represented in a subroutine, from $a to $b, using an 8-point
 Newton-Cotes formula. The routine divides up the interval into
 equal segments, evaluates the integral, then compares that
 to the result with double the number of segments. If the two
 results agree, to within an absolute error $abs_err or a
 relative error $rel_err, the result is returned; otherwise,
 the number of segments is doubled again, and the results
 compared. This continues until the desired accuracy is attained,
 or until the maximum number of iterations $nmax is reached.
vecang: angle between two vectors
 SYNOPSIS:

 # double p[3], q[3], vecang();

    $y = vecang( $p, $q );

 DESCRIPTION:

 For two vectors p, q, the angle A between them is given by

      p.q / (|p| |q|)  = cos A  .

 where "." represents inner product, "|x|" the length of vector x.
 If the angle is small, an expression in sin A is preferred.
 Set r = q - p.  Then

     p.q = p.p + p.r ,

     |p|^2 = p.p ,

     |q|^2 = p.p + 2 p.r + r.r ,

                  p.p^2 + 2 p.p p.r + p.r^2
     cos^2 A  =  ----------------------------
                    p.p (p.p + 2 p.r + r.r)

                  p.p + 2 p.r + p.r^2 / p.p
              =  --------------------------- ,
                     p.p + 2 p.r + r.r

     sin^2 A  =  1 - cos^2 A

                   r.r - p.r^2 / p.p
              =  --------------------
                  p.p + 2 p.r + r.r

              =   (r.r - p.r^2 / p.p) / q.q  .

 ACCURACY:

                      Relative error:
 arithmetic   domain     # trials      peak         rms
    IEEE      -1, 1        10^6       1.7e-16     4.2e-17
onef2: Hypergeometric function 1F2
 SYNOPSIS:

 # double a, b, c, x, value;

 # double *err;

 ($value, $err) = onef2( $a, $b, $c, $x)

 ACCURACY:

 Not accurately characterized, but spot checked against tables.
threef0: Hypergeometric function 3F0
 SYNOPSIS:

 # double a, b, c, x, value;

 # double *err;

 ($value, $err) = threef0( $a, $b, $c, $x )

 ACCURACY:

 Not accurately characterized, but spot checked against tables.
yv: Bessel function Yv with noninteger v
 SYNOPSIS:

 # double v, x;

 # double yv( v, x );

 $y = yv( $v, $x );

 ACCURACY:

 Not accurately characterized, but spot checked against tables.
tan: 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: 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: 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: 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: 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: Relative error approximations for function arguments near unity.
 SYNOPSIS:

# log1p(x) = log(1+x)

 $y = log1p( $x );

# expm1(x) = exp(x) - 1

 $y = expm1( $x );

# cosm1(x) = cos(x) - 1

 $y = cosm1( $x );
yn: 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: 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: 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

TODO ^

MAINTAINER ^

Shlomi Fish, http://www.shlomifish.org/, https://metacpan.org/author/SHLOMIF .

BUGS ^

Please report any on the rt.cpan.org interface: https://rt.cpan.org/Dist/Display.html?Queue=Math-Cephes

VERSION CONTROL ^

This distribution is maintained in this GitHub repository:

https://github.com/shlomif/Math-Cephes.

SEE ALSO ^

For interfaces to programs which can do symbolic manipulation, see PDL, Math::Pari, and Math::ematica. For a command line interface to the routines of Math::Cephes, see the included pmath script. For a different interface to the fraction and complex number routines, see Math::Cephes::Fraction and Math::Cephes::Complex. For an interface to some polynomial routines, see Math::Cephes::Polynomial, and for some matrix routines, see Math::Cephes::Matrix.

COPYRIGHT ^

The C code for the Cephes Math Library is Copyright 1984, 1987, 1989, 2002 by Stephen L. Moshier, and is available at http://www.netlib.org/cephes/. Direct inquiries to 30 Frost Street, Cambridge, MA 02140.

The file arrays.c included here to handle passing arrays into and out of C routines comes from the PGPLOT module of Karl Glazebrook <kgb@zzoepp.aao.gov.au>.

The perl interface is copyright 2000, 2002 by Randy Kobes. This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself.

Perl interface maintained by Shlomi Fish starting from 2012. All explicit or implicit copyrights on the changes are disclaimed by him.

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