.\" Automatically generated by Pod::Man 4.14 (Pod::Simple 3.40) .\" .\" Standard preamble: .\" ======================================================================== .de Sp \" Vertical space (when we can't use .PP) .if t .sp .5v .if n .sp .. .de Vb \" Begin verbatim text .ft CW .nf .ne \\$1 .. .de Ve \" End verbatim text .ft R .fi .. .\" Set up some character translations and predefined strings. \*(-- will .\" give an unbreakable dash, \*(PI will give pi, \*(L" will give a left .\" double quote, and \*(R" will give a right double quote. \*(C+ will .\" give a nicer C++. Capital omega is used to do unbreakable dashes and .\" therefore won't be available. \*(C` and \*(C' expand to `' in nroff, .\" nothing in troff, for use with C<>. .tr \(*W- .ds C+ C\v'-.1v'\h'-1p'\s-2+\h'-1p'+\s0\v'.1v'\h'-1p' .ie n \{\ . ds -- \(*W- . ds PI pi . if (\n(.H=4u)&(1m=24u) .ds -- \(*W\h'-12u'\(*W\h'-12u'-\" diablo 10 pitch . if (\n(.H=4u)&(1m=20u) .ds -- \(*W\h'-12u'\(*W\h'-8u'-\" diablo 12 pitch . ds L" "" . ds R" "" . ds C` "" . ds C' "" 'br\} .el\{\ . ds -- \|\(em\| . ds PI \(*p . ds L" `` . ds R" '' . ds C` . ds C' 'br\} .\" .\" Escape single quotes in literal strings from groff's Unicode transform. .ie \n(.g .ds Aq \(aq .el .ds Aq ' .\" .\" If the F register is >0, we'll generate index entries on stderr for .\" titles (.TH), headers (.SH), subsections (.SS), items (.Ip), and index .\" entries marked with X<> in POD. Of course, you'll have to process the .\" output yourself in some meaningful fashion. .\" .\" Avoid warning from groff about undefined register 'F'. .de IX .. .nr rF 0 .if \n(.g .if rF .nr rF 1 .if (\n(rF:(\n(.g==0)) \{\ . if \nF \{\ . de IX . tm Index:\\$1\t\\n%\t"\\$2" .. . if !\nF==2 \{\ . nr % 0 . nr F 2 . \} . \} .\} .rr rF .\" ======================================================================== .\" .IX Title "Math::Cephes 3pm" .TH Math::Cephes 3pm "2021-06-22" "perl v5.32.1" "User Contributed Perl Documentation" .\" For nroff, turn off justification. Always turn off hyphenation; it makes .\" way too many mistakes in technical documents. .if n .ad l .nh .SH "NAME" Math::Cephes \- perl interface to the cephes math library .SH "SYNOPSIS" .IX Header "SYNOPSIS" .Vb 1 \& use Math::Cephes qw(:all); .Ve .SH "DESCRIPTION" .IX Header "DESCRIPTION" .Vb 3 \& 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: .Ve .IP "use Math::Cephes qw(:constants);" 4 .IX Item "use Math::Cephes qw(:constants);" .Vb 1 \& 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 .Ve .IP "use Math::Cephes qw(:trigs);" 4 .IX Item "use Math::Cephes qw(:trigs);" .Vb 1 \& 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 .Ve .IP "use Math::Cephes qw(:hypers);" 4 .IX Item "use Math::Cephes qw(:hypers);" .Vb 1 \& imports \& \& acosh: Inverse hyperbolic cosine \& asinh: Inverse hyperbolic sine \& atanh: Inverse hyperbolic tangent \& cosh: Hyperbolic cosine \& sinh: Hyperbolic sine \& tanh: Hyperbolic tangent .Ve .IP "use Math::Cephes qw(:explog);" 4 .IX Item "use Math::Cephes qw(:explog);" .Vb 1 \& 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. .Ve .IP "use Math::Cephes qw(:cmplx);" 4 .IX Item "use Math::Cephes qw(:cmplx);" .Vb 1 \& 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 .Ve .IP "use Math::Cephes qw(:utils);" 4 .IX Item "use Math::Cephes qw(:utils);" .Vb 1 \& 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 .Ve .IP "use Math::Cephes qw(:bessels);" 4 .IX Item "use Math::Cephes qw(:bessels);" .Vb 1 \& 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 .Ve .IP "use Math::Cephes qw(:dists);" 4 .IX Item "use Math::Cephes qw(:dists);" .Vb 1 \& 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\*(Aqs t distribution \& stdtri: Functional inverse of Student\*(Aqs t distribution .Ve .IP "use Math::Cephes qw(:gammas);" 4 .IX Item "use Math::Cephes qw(:gammas);" .Vb 1 \& 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 .Ve .IP "use Math::Cephes qw(:betas);" 4 .IX Item "use Math::Cephes qw(:betas);" .Vb 1 \& imports \& \& beta: Beta function \& incbet: Incomplete beta integral \& incbi: Inverse of imcomplete beta integral \& lbeta: Natural logarithm of |beta| .Ve .IP "use Math::Cephes qw(:elliptics);" 4 .IX Item "use Math::Cephes qw(:elliptics);" .Vb 1 \& 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 .Ve .IP "use Math::Cephes qw(:hypergeometrics);" 4 .IX Item "use Math::Cephes qw(:hypergeometrics);" .Vb 1 \& imports \& \& hyp2f0: Gauss hypergeometric function F \& hyp2f1: Gauss hypergeometric function F \& hyperg: Confluent hypergeometric function \& onef2: Hypergeometric function 1F2 \& threef0: Hypergeometric function 3F0 .Ve .IP "use Math::Cephes qw(:misc);" 4 .IX Item "use Math::Cephes qw(:misc);" .Vb 1 \& imports \& \& airy: Airy function \& bernum: Bernoulli numbers \& dawsn: Dawson\*(Aqs Integral \& ei: Exponential integral \& erf: Error function \& erfc: Complementary error function \& expn: Exponential integral En \& fresnl: Fresnel integral \& plancki: Integral of Planck\*(Aqs black body radiation formula \& polylog: Polylogarithm function \& shichi: Hyperbolic sine and cosine integrals \& sici: Sine and cosine integrals \& simpson: Simpson\*(Aqs 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 .Ve .IP "use Math::Cephes qw(:fract);" 4 .IX Item "use Math::Cephes qw(:fract);" .Vb 1 \& 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 .Ve .SH "FUNCTIONS" .IX Header "FUNCTIONS" .Vb 1 \& A description of the various functions available follows. .Ve .IP "\fIacosh\fR: Inverse hyperbolic cosine" 4 .IX Item "acosh: Inverse hyperbolic cosine" .Vb 1 \& SYNOPSIS: \& \& # double x, y, acosh(); \& \& $y = acosh( $x ); \& \& DESCRIPTION: \& \& Returns inverse hyperbolic cosine of argument. \& \& If 1 <= x < 1.5, a rational approximation \& \& sqrt(z) * P(z)/Q(z) \& \& where z = x\-1, is used. Otherwise, \& \& acosh(x) = log( x + sqrt( (x\-1)(x+1) ). \& \& ACCURACY: \& Relative error: \& arithmetic domain # trials peak rms \& DEC 1,3 30000 4.2e\-17 1.1e\-17 \& IEEE 1,3 30000 4.6e\-16 8.7e\-17 \& \& ERROR MESSAGES: \& \& message condition value returned \& acosh domain |x| < 1 NAN .Ve .IP "\fIairy\fR: Airy function" 4 .IX Item "airy: Airy function" .Vb 1 \& SYNOPSIS: \& \& # double x, ai, aiprime, bi, biprime; \& # int airy(); \& \& ($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\*(Aq(x), Bi\*(Aq(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\*(Aq 10000 4.6e\-15 7.6e\-16 \& IEEE 0, 10 Ai\*(Aq 10000 1.8e\-14* 1.5e\-15* \& IEEE \-10, 10 Bi 30000 4.2e\-15 5.3e\-16 \& IEEE \-10, 10 Bi\*(Aq 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\*(Aq 5000 4.7e\-16 7.8e\-17 \& DEC 0, 10 Ai\*(Aq 12000 1.8e\-15* 1.5e\-16* \& DEC \-10, 10 Bi 10000 5.5e\-16 6.8e\-17 \& DEC \-10, 10 Bi\*(Aq 7000 5.3e\-16 8.7e\-17 .Ve .IP "\fIradian\fR: Degrees, minutes, seconds to radians" 4 .IX Item "radian: Degrees, minutes, seconds to radians" .Vb 1 \& SYNOPSIS: \& \& # double d, m, s, radian(); \& \& $r = radian( $d, $m, $s ); \& \& DESCRIPTION: \& \& Converts an angle of degrees, minutes, seconds to radians. .Ve .IP "\fIhypot\fR: returns the hypotenuse associated with the sides of a right triangle" 4 .IX Item "hypot: returns the hypotenuse associated with the sides of a right triangle" .Vb 1 \& 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) .Ve .IP "\fIasin\fR: Inverse circular sine" 4 .IX Item "asin: Inverse circular sine" .Vb 1 \& 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 .Ve .IP "\fIacos\fR: Inverse circular cosine" 4 .IX Item "acos: Inverse circular cosine" .Vb 1 \& 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 .Ve .IP "\fIasinh\fR: Inverse hyperbolic sine" 4 .IX Item "asinh: Inverse hyperbolic sine" .Vb 1 \& 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 .Ve .IP "\fIatan\fR: Inverse circular tangent (arctangent)" 4 .IX Item "atan: Inverse circular tangent (arctangent)" .Vb 1 \& 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 .Ve .IP "\fIatan2\fR: Quadrant correct inverse circular tangent" 4 .IX Item "atan2: Quadrant correct inverse circular tangent" .Vb 1 \& 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. .Ve .IP "\fIatanh\fR: Inverse hyperbolic tangent" 4 .IX Item "atanh: Inverse hyperbolic tangent" .Vb 1 \& 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 .Ve .IP "\fIbdtr\fR: Binomial distribution" 4 .IX Item "bdtr: Binomial distribution" .Vb 1 \& 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 .Ve .IP "\fIbdtrc\fR: Complemented binomial distribution" 4 .IX Item "bdtrc: Complemented binomial distribution" .Vb 1 \& 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 1 .Ve .IP "\fIbeta\fR: Beta function" 4 .IX Item "beta: Beta function" .Vb 1 \& 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 .Ve .IP "\fIlbeta\fR: Natural logarithm of |beta|" 4 .IX Item "lbeta: Natural logarithm of |beta|" .Vb 1 \& SYNOPSIS: \& \& # double a, b; \& \& # double lbeta( a, b ); \& \& $y = lbeta( $a, $b); .Ve .IP "\fIbtdtr\fR: Beta distribution" 4 .IX Item "btdtr: Beta distribution" .Vb 1 \& 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. .Ve .IP "\fIcbrt\fR: Cube root" 4 .IX Item "cbrt: Cube root" .Vb 1 \& 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\*(Aqs \& 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 .Ve .IP "\fIchdtr\fR: Chi-square distribution" 4 .IX Item "chdtr: Chi-square distribution" .Vb 1 \& 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 .Ve .IP "\fIchdtrc\fR: Complemented Chi-square distribution" 4 .IX Item "chdtrc: Complemented Chi-square distribution" .Vb 1 \& 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 .Ve .IP "\fIchdtri\fR: Inverse of complemented Chi-square distribution" 4 .IX Item "chdtri: Inverse of complemented Chi-square distribution" .Vb 1 \& 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 .Ve .IP "\fIclog\fR: Complex natural logarithm" 4 .IX Item "clog: Complex natural logarithm" .Vb 1 \& SYNOPSIS: \& \& # void clog(); \& # cmplx z, w; \& \& $z = new_cmplx(2, 3); # $z = 2 + 3 i \& $w = new_cmplx(); \& clog($z, $w ); \& print $w\->{r}, \*(Aq \*(Aq, $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. .Ve .IP "\fIcexp\fR: Complex exponential function" 4 .IX Item "cexp: Complex exponential function" .Vb 1 \& SYNOPSIS: \& \& # void cexp(); \& # cmplx z, w; \& \& $z = new_cmplx(2, 3); # $z = 2 + 3 i \& $w = new_cmplx(); \& cexp($z, $w ); \& print $w\->{r}, \*(Aq \*(Aq, $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 .Ve .IP "\fIcsin\fR: Complex circular sine" 4 .IX Item "csin: Complex circular sine" .Vb 1 \& SYNOPSIS: \& \& # void csin(); \& # cmplx z, w; \& \& $z = new_cmplx(2, 3); # $z = 2 + 3 i \& $w = new_cmplx(); \& csin($z, $w ); \& print $w\->{r}, \*(Aq \*(Aq, $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. .Ve .IP "\fIccos\fR: Complex circular cosine" 4 .IX Item "ccos: Complex circular cosine" .Vb 1 \& SYNOPSIS: \& \& # void ccos(); \& # cmplx z, w; \& \& $z = new_cmplx(2, 3); # $z = 2 + 3 i \& $w = new_cmplx(); \& ccos($z, $w ); \& print $w\->{r}, \*(Aq \*(Aq, $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 .Ve .IP "\fIctan\fR: Complex circular tangent" 4 .IX Item "ctan: Complex circular tangent" .Vb 1 \& SYNOPSIS: \& \& # void ctan(); \& # cmplx z, w; \& \& $z = new_cmplx(2, 3); # $z = 2 + 3 i \& $w = new_cmplx(); \& ctan($z, $w ); \& print $w\->{r}, \*(Aq \*(Aq, $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. .Ve .IP "\fIccot\fR: Complex circular cotangent" 4 .IX Item "ccot: Complex circular cotangent" .Vb 1 \& SYNOPSIS: \& \& # void ccot(); \& # cmplx z, w; \& \& $z = new_cmplx(2, 3); # $z = 2 + 3 i \& $w = new_cmplx(); \& ccot($z, $w ); \& print $w\->{r}, \*(Aq \*(Aq, $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. .Ve .IP "\fIcasin\fR: Complex circular arc sine" 4 .IX Item "casin: Complex circular arc sine" .Vb 1 \& SYNOPSIS: \& \& # void casin(); \& # cmplx z, w; \& \& $z = new_cmplx(2, 3); # $z = 2 + 3 i \& $w = new_cmplx(); \& casin($z, $w ); \& print $w\->{r}, \*(Aq \*(Aq, $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. .Ve .IP "\fIcacos\fR: Complex circular arc cosine" 4 .IX Item "cacos: Complex circular arc cosine" .Vb 1 \& SYNOPSIS: \& \& # void cacos(); \& # cmplx z, w; \& \& $z = new_cmplx(2, 3); # $z = 2 + 3 i \& $w = new_cmplx(); \& cacos($z, $w ); \& print $w\->{r}, \*(Aq \*(Aq, $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 .Ve .IP "\fIcatan\fR: Complex circular arc tangent" 4 .IX Item "catan: Complex circular arc tangent" .Vb 1 \& SYNOPSIS: \& \& # void catan(); \& # cmplx z, w; \& \& $z = new_cmplx(2, 3); # $z = 2 + 3 i \& $w = new_cmplx(); \& catan($z, $w ); \& print $w\->{r}, \*(Aq \*(Aq, $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(). .Ve .IP "\fIcsinh\fR: Complex hyperbolic sine" 4 .IX Item "csinh: Complex hyperbolic sine" .Vb 1 \& SYNOPSIS: \& \& # void csinh(); \& # cmplx z, w; \& \& $z = new_cmplx(2, 3); # $z = 2 + 3 i \& $w = new_cmplx(); \& csinh($z, $w ); \& print $w\->{r}, \*(Aq \*(Aq, $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 .Ve .IP "\fIcasinh\fR: Complex inverse hyperbolic sine" 4 .IX Item "casinh: Complex inverse hyperbolic sine" .Vb 1 \& SYNOPSIS: \& \& # void casinh(); \& # cmplx z, w; \& \& $z = new_cmplx(2, 3); # $z = 2 + 3 i \& $w = new_cmplx(); \& casinh($z, $w ); \& print $w\->{r}, \*(Aq \*(Aq, $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 .Ve .IP "\fIccosh\fR: Complex hyperbolic cosine" 4 .IX Item "ccosh: Complex hyperbolic cosine" .Vb 1 \& SYNOPSIS: \& \& # void ccosh(); \& # cmplx z, w; \& \& $z = new_cmplx(2, 3); # $z = 2 + 3 i \& $w = new_cmplx(); \& ccosh($z, $w ); \& print $w\->{r}, \*(Aq \*(Aq, $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 .Ve .IP "\fIcacosh\fR: Complex inverse hyperbolic cosine" 4 .IX Item "cacosh: Complex inverse hyperbolic cosine" .Vb 1 \& SYNOPSIS: \& \& # void cacosh(); \& # cmplx z, w; \& \& $z = new_cmplx(2, 3); # $z = 2 + 3 i \& $w = new_cmplx(); \& cacosh($z, $w ); \& print $w\->{r}, \*(Aq \*(Aq, $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 .Ve .IP "\fIctanh\fR: Complex hyperbolic tangent" 4 .IX Item "ctanh: Complex hyperbolic tangent" .Vb 1 \& SYNOPSIS: \& \& # void ctanh(); \& # cmplx z, w; \& \& $z = new_cmplx(2, 3); # $z = 2 + 3 i \& $w = new_cmplx(); \& ctanh($z, $w ); \& print $w\->{r}, \*(Aq \*(Aq, $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 .Ve .IP "\fIcatanh\fR: Complex inverse hyperbolic tangent" 4 .IX Item "catanh: Complex inverse hyperbolic tangent" .Vb 1 \& SYNOPSIS: \& \& # void catanh(); \& # cmplx z, w; \& \& $z = new_cmplx(2, 3); # $z = 2 + 3 i \& $w = new_cmplx(); \& catanh($z, $w ); \& print $w\->{r}, \*(Aq \*(Aq, $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 .Ve .IP "\fIcpow\fR: Complex power function" 4 .IX Item "cpow: Complex power function" .Vb 1 \& 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}, \*(Aq \*(Aq, $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 .Ve .IP "\fIcmplx\fR: Complex number arithmetic" 4 .IX Item "cmplx: Complex number arithmetic" .Vb 1 \& 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}, \*(Aq \*(Aq, $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 .Ve .IP "\fIcabs\fR: Complex absolute value" 4 .IX Item "cabs: Complex absolute value" .Vb 1 \& 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 .Ve .IP "\fIcsqrt\fR: Complex square root" 4 .IX Item "csqrt: Complex square root" .Vb 1 \& SYNOPSIS: \& \& # void csqrt(); \& # cmplx z, w; \& \& $z = new_cmplx(2, 3); # $z = 2 + 3 i \& $w = new_cmplx(); \& csqrt($z, $w ); \& print $w\->{r}, \*(Aq \*(Aq, $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. .Ve .IP "\fImachconst\fR: Globally declared constants" 4 .IX Item "machconst: Globally declared constants" .Vb 1 \& 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. .Ve .IP "\fIcosh\fR: Hyperbolic cosine" 4 .IX Item "cosh: Hyperbolic cosine" .Vb 1 \& 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 .Ve .IP "\fIdawsn\fR: Dawson's Integral" 4 .IX Item "dawsn: Dawson's Integral" .Vb 1 \& SYNOPSIS: \& \& # double x, y, dawsn(); \& \& $y = dawsn( $x ); \& \& DESCRIPTION: \& \& Approximates the integral \& \& x \& \- \& 2 | | 2 \& dawsn(x) = exp( \-x ) | exp( t ) dt \& | | \& \- \& 0 \& \& Three different rational approximations are employed, for \& the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up. \& \& ACCURACY: \& \& Relative error: \& arithmetic domain # trials peak rms \& IEEE 0,10 10000 6.9e\-16 1.0e\-16 \& DEC 0,10 6000 7.4e\-17 1.4e\-17 .Ve .IP "\fIdrand\fR: Pseudorandom number generator" 4 .IX Item "drand: Pseudorandom number generator" .Vb 1 \& SYNOPSIS: \& \& # double y, drand(); \& \& ($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. .Ve .IP "\fIellie\fR: Incomplete elliptic integral of the second kind" 4 .IX Item "ellie: Incomplete elliptic integral of the second kind" .Vb 1 \& SYNOPSIS: \& \& # double phi, m, y, ellie(); \& \& $y = ellie( $phi, $m ); \& \& DESCRIPTION: \& \& Approximates the integral \& \& phi \& \- \& | | \& | 2 \& E(phi_\em) = | 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 .Ve .IP "\fIellik\fR: Incomplete elliptic integral of the first kind" 4 .IX Item "ellik: Incomplete elliptic integral of the first kind" .Vb 1 \& SYNOPSIS: \& \& # double phi, m, y, ellik(); \& \& $y = ellik( $phi, $m ); \& \& DESCRIPTION: \& \& Approximates the integral \& \& phi \& \- \& | | \& | dt \& F(phi_\em) = | \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& | 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 .Ve .IP "\fIellpe\fR: Complete elliptic integral of the second kind" 4 .IX Item "ellpe: Complete elliptic integral of the second kind" .Vb 1 \& SYNOPSIS: \& \& # double m1, y, ellpe(); \& \& $y = ellpe( $m1 ); \& \& DESCRIPTION: \& \& Approximates the integral \& \& pi/2 \& \- \& | | 2 \& E(m) = | sqrt( 1 \- m sin t ) dt \& | | \& \- \& 0 \& \& Where m = 1 \- m1, using the approximation \& \& P(x) \- x log x Q(x). \& \& Though there are no singularities, the argument m1 is used \& rather than m for compatibility with ellpk(). \& \& E(1) = 1; E(0) = pi/2. \& \& ACCURACY: \& \& Relative error: \& arithmetic domain # trials peak rms \& DEC 0, 1 13000 3.1e\-17 9.4e\-18 \& IEEE 0, 1 10000 2.1e\-16 7.3e\-17 \& \& ERROR MESSAGES: \& \& message condition value returned \& ellpe domain x<0, x>1 0.0 .Ve .IP "\fIellpj\fR: Jacobian Elliptic Functions" 4 .IX Item "ellpj: Jacobian Elliptic Functions" .Vb 1 \& SYNOPSIS: \& \& # double u, m, sn, cn, dn, phi; \& # int ellpj(); \& \& ($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. .Ve .IP "\fIellpk\fR: Complete elliptic integral of the first kind" 4 .IX Item "ellpk: Complete elliptic integral of the first kind" .Vb 1 \& 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 .Ve .IP "\fIeuclid\fR: Rational arithmetic routines" 4 .IX Item "euclid: Rational arithmetic routines" .Vb 1 \& 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}, \*(Aq \*(Aq, $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. .Ve .IP "\fIexp\fR: Exponential function" 4 .IX Item "exp: Exponential function" .Vb 1 \& 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\*(Aq 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 .Ve .IP "\fIexpxx\fR: exp(x*x)" 4 .IX Item "expxx: exp(x*x)" .Vb 2 \& # 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 .Ve .IP "\fIexp10\fR: Base 10 exponential function (Common antilogarithm)" 4 .IX Item "exp10: Base 10 exponential function (Common antilogarithm)" .Vb 1 \& 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\*(Aq 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. .Ve .IP "\fIexp2\fR: Base 2 exponential function" 4 .IX Item "exp2: Base 2 exponential function" .Vb 1 \& 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\*(Aq 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. .Ve .IP "\fIei\fR: Exponential integral" 4 .IX Item "ei: Exponential integral" .Vb 1 \& 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 .Ve .IP "\fIexpn\fR: Exponential integral En" 4 .IX Item "expn: Exponential integral En" .Vb 1 \& 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 .Ve .IP "\fIfabs\fR: Absolute value" 4 .IX Item "fabs: Absolute value" .Vb 1 \& SYNOPSIS: \& \& # double x, y; \& \& $y = fabs( $x ); \& \& DESCRIPTION: \& \& Returns the absolute value of the argument. .Ve .IP "\fIfac\fR: Factorial function" 4 .IX Item "fac: Factorial function" .Vb 1 \& 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 .Ve .IP "\fIfdtr\fR: F distribution" 4 .IX Item "fdtr: F distribution" .Vb 1 \& 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\*(Aqs 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 .Ve .IP "\fIfdtrc\fR: Complemented F distribution" 4 .IX Item "fdtrc: Complemented F distribution" .Vb 1 \& 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\*(Aqs 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 .Ve .IP "\fIfdtri\fR: Inverse of complemented F distribution" 4 .IX Item "fdtri: Inverse of complemented F distribution" .Vb 1 \& 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 .Ve .IP "\fIceil\fR: ceil" 4 .IX Item "ceil: ceil" .Vb 2 \& ceil() returns the smallest integer greater than or equal \& to x. It truncates toward plus infinity. \& \& SYNOPSIS: \& \& # double x, y, ceil(); \& \& $y = ceil( $x ); .Ve .IP "\fIfloor\fR: floor" 4 .IX Item "floor: floor" .Vb 2 \& floor() returns the largest integer less than or equal to x. \& It truncates toward minus infinity. \& \& SYNOPSIS: \& \& # double x, y, floor(); \& \& $y = floor( $x ); .Ve .IP "\fIfrexp\fR: frexp" 4 .IX Item "frexp: frexp" .Vb 3 \& frexp() extracts the exponent from x. It returns an integer \& power of two to expnt and the significand between 0.5 and 1 \& to y. Thus x = y * 2**expn. \& \& SYNOPSIS: \& \& # double x, y, frexp(); \& # int expnt; \& \& ($y, $expnt) = frexp( $x ); .Ve .IP "\fIldexp\fR: multiplies x by 2**n." 4 .IX Item "ldexp: multiplies x by 2**n." .Vb 1 \& SYNOPSIS: \& \& # double x, y, ldexp(); \& # int n; \& \& $y = ldexp( $x, $n ); .Ve .IP "\fIfresnl\fR: Fresnel integral" 4 .IX Item "fresnl: Fresnel integral" .Vb 1 \& SYNOPSIS: \& \& # double x, S, C; \& # void fresnl(); \& \& ($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 .Ve .IP "\fIgamma\fR: Gamma function" 4 .IX Item "gamma: Gamma function" .Vb 1 \& 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\*(Aqs \& 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. .Ve .IP "\fIlgam\fR: Natural logarithm of gamma function" 4 .IX Item "lgam: Natural logarithm of gamma function" .Vb 1 \& 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\*(Aqs 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 .Ve .IP "\fIgdtr\fR: Gamma distribution function" 4 .IX Item "gdtr: Gamma distribution function" .Vb 1 \& 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 .Ve .IP "\fIgdtrc\fR: Complemented gamma distribution function" 4 .IX Item "gdtrc: Complemented gamma distribution function" .Vb 1 \& SYNOPSIS: \& \& # double a, b, x, y, gdtrc(); \& \& $y = gdtrc( $a, $b, $x ); \& \& DESCRIPTION: \& \& Returns the integral from x to infinity of the gamma \& probability density function: \& \& inf. \& b \- \& a | | b\-1 \-at \& y = \-\-\-\-\- | t e dt \& \- | | \& | (b) \- \& x \& \& The incomplete gamma integral is used, according to the \& relation \& \& y = igamc( b, ax ). \& \& ACCURACY: \& \& See igamc(). \& \& ERROR MESSAGES: \& \& message condition value returned \& gdtrc domain x < 0 0.0 .Ve .IP "\fIhyp2f0\fR: Gauss hypergeometric function 2F0" 4 .IX Item "hyp2f0: Gauss hypergeometric function 2F0" .Vb 1 \& SYNOPSIS: \& \& # double a, b, x, value, *err; \& # int type; /* determines what converging factor to use */ \& \& ($value, $err) = hyp2f0( $a, $b, $x, $type ) .Ve .IP "\fIhyp2f1\fR: Gauss hypergeometric function 2F1" 4 .IX Item "hyp2f1: Gauss hypergeometric function 2F1" .Vb 1 \& SYNOPSIS: \& \& # double a, b, c, x, y, hyp2f1(); \& \& $y = hyp2f1( $a, $b, $c, $x ); \& \& DESCRIPTION: \& \& hyp2f1( a, b, c, x ) = F ( a, b; c; x ) \& 2 1 \& \& inf. \& \- a(a+1)...(a+k) b(b+1)...(b+k) k+1 \& = 1 + > \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- x . \& \- c(c+1)...(c+k) (k+1)! \& k = 0 \& \& Cases addressed are \& Tests and escapes for negative integer a, b, or c \& Linear transformation if c \- a or c \- b negative integer \& Special case c = a or c = b \& Linear transformation for x near +1 \& Transformation for x < \-0.5 \& Psi function expansion if x > 0.5 and c \- a \- b integer \& Conditionally, a recurrence on c to make c\-a\-b > 0 \& \& |x| > 1 is rejected. \& \& The parameters a, b, c are considered to be integer \& valued if they are within 1.0e\-14 of the nearest integer \& (1.0e\-13 for IEEE arithmetic). \& \& ACCURACY: \& \& Relative error (\-1 < x < 1): \& arithmetic domain # trials peak rms \& IEEE \-1,7 230000 1.2e\-11 5.2e\-14 \& \& Several special cases also tested with a, b, c in \& the range \-7 to 7. \& \& ERROR MESSAGES: \& \& A "partial loss of precision" message is printed if \& the internally estimated relative error exceeds 1^\-12. \& A "singularity" message is printed on overflow or \& in cases not addressed (such as x < \-1). .Ve .IP "\fIhyperg\fR: Confluent hypergeometric function" 4 .IX Item "hyperg: Confluent hypergeometric function" .Vb 1 \& 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. .Ve .IP "\fIi0\fR: Modified Bessel function of order zero" 4 .IX Item "i0: Modified Bessel function of order zero" .Vb 1 \& 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 .Ve .IP "\fIi0e\fR: Modified Bessel function of order zero, exponentially scaled" 4 .IX Item "i0e: Modified Bessel function of order zero, exponentially scaled" .Vb 1 \& 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(). .Ve .IP "\fIi1\fR: Modified Bessel function of order one" 4 .IX Item "i1: Modified Bessel function of order one" .Vb 1 \& 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 .Ve .IP "\fIi1e\fR: Modified Bessel function of order one, exponentially scaled" 4 .IX Item "i1e: Modified Bessel function of order one, exponentially scaled" .Vb 1 \& 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(). .Ve .IP "\fIigam\fR: Incomplete gamma integral" 4 .IX Item "igam: Incomplete gamma integral" .Vb 1 \& 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 .Ve .IP "\fIigamc\fR: Complemented incomplete gamma integral" 4 .IX Item "igamc: Complemented incomplete gamma integral" .Vb 1 \& 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 .Ve .IP "\fIigami\fR: Inverse of complemented imcomplete gamma integral" 4 .IX Item "igami: Inverse of complemented imcomplete gamma integral" .Vb 1 \& 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 .Ve .IP "\fIincbet\fR: Incomplete beta integral" 4 .IX Item "incbet: Incomplete beta integral" .Vb 1 \& 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 .Ve .IP "\fIincbi\fR: Inverse of imcomplete beta integral" 4 .IX Item "incbi: Inverse of imcomplete beta integral" .Vb 1 \& 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 .Ve .IP "\fIiv\fR: Modified Bessel function of noninteger order" 4 .IX Item "iv: Modified Bessel function of noninteger order" .Vb 1 \& 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. .Ve .IP "\fIj0\fR: Bessel function of order zero" 4 .IX Item "j0: Bessel function of order zero" .Vb 1 \& 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\*(Aqs 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 .Ve .IP "\fIy0\fR: Bessel function of the second kind, order zero" 4 .IX Item "y0: Bessel function of the second kind, order zero" .Vb 1 \& 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 .Ve .IP "\fIj1\fR: Bessel function of order one" 4 .IX Item "j1: Bessel function of order one" .Vb 1 \& 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 .Ve .IP "\fIy1\fR: Bessel function of second kind of order one" 4 .IX Item "y1: Bessel function of second kind of order one" .Vb 1 \& 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). .Ve .IP "\fIjn\fR: Bessel function of integer order" 4 .IX Item "jn: Bessel function of integer order" .Vb 1 \& 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. .Ve .IP "\fIjv\fR: Bessel function of noninteger order" 4 .IX Item "jv: Bessel function of noninteger order" .Vb 1 \& 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* .Ve .IP "\fIk0\fR: Modified Bessel function, third kind, order zero" 4 .IX Item "k0: Modified Bessel function, third kind, order zero" .Vb 1 \& 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 .Ve .IP "\fIk0e\fR: Modified Bessel function, third kind, order zero, exponentially scaled" 4 .IX Item "k0e: Modified Bessel function, third kind, order zero, exponentially scaled" .Vb 1 \& 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(). .Ve .IP "\fIk1\fR: Modified Bessel function, third kind, order one" 4 .IX Item "k1: Modified Bessel function, third kind, order one" .Vb 1 \& 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 .Ve .IP "\fIk1e\fR: Modified Bessel function, third kind, order one, exponentially scaled" 4 .IX Item "k1e: Modified Bessel function, third kind, order one, exponentially scaled" .Vb 1 \& 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(). .Ve .IP "\fIkn\fR: Modified Bessel function, third kind, integer order" 4 .IX Item "kn: Modified Bessel function, third kind, integer order" .Vb 1 \& 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. .Ve .IP "\fIlog\fR: Natural logarithm" 4 .IX Item "log: Natural logarithm" .Vb 1 \& 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 .Ve .IP "\fIlog10\fR: Common logarithm" 4 .IX Item "log10: Common logarithm" .Vb 1 \& 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 .Ve .IP "\fIlog2\fR: Base 2 logarithm" 4 .IX Item "log2: Base 2 logarithm" .Vb 1 \& 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 .Ve .IP "\fIlrand\fR: Pseudorandom number generator" 4 .IX Item "lrand: Pseudorandom number generator" .Vb 1 \& 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. .Ve .IP "\fIlsqrt\fR: Integer square root" 4 .IX Item "lsqrt: Integer square root" .Vb 1 \& 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. .Ve .IP "\fImtherr\fR: Library common error handling routine" 4 .IX Item "mtherr: Library common error handling routine" .Vb 1 \& 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 .Ve .IP "\fInbdtr\fR: Negative binomial distribution" 4 .IX Item "nbdtr: Negative binomial distribution" .Vb 1 \& 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. .Ve .IP "\fInbdtrc\fR: Complemented negative binomial distribution" 4 .IX Item "nbdtrc: Complemented negative binomial distribution" .Vb 1 \& 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. .Ve .IP "\fInbdtrc\fR: Complemented negative binomial distribution" 4 .IX Item "nbdtrc: Complemented negative binomial distribution" .Vb 1 \& 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. .Ve .IP "\fInbdtri\fR: Functional inverse of negative binomial distribution" 4 .IX Item "nbdtri: Functional inverse of negative binomial distribution" .Vb 1 \& 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. .Ve .IP "\fIndtr\fR: Normal distribution function" 4 .IX Item "ndtr: Normal distribution function" .Vb 1 \& 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 .Ve .IP "\fIerf\fR: Error function" 4 .IX Item "erf: Error function" .Vb 1 \& 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 .Ve .IP "\fIerfc\fR: Complementary error function" 4 .IX Item "erfc: Complementary error function" .Vb 1 \& 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 .Ve .IP "\fIndtri\fR: Inverse of Normal distribution function" 4 .IX Item "ndtri: Inverse of Normal distribution function" .Vb 1 \& 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 .Ve .IP "\fIpdtr\fR: Poisson distribution" 4 .IX Item "pdtr: Poisson distribution" .Vb 1 \& 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(). .Ve .IP "\fIpdtrc\fR: Complemented poisson distribution" 4 .IX Item "pdtrc: Complemented poisson distribution" .Vb 1 \& 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. .Ve .IP "\fIpdtri\fR: Inverse Poisson distribution" 4 .IX Item "pdtri: Inverse Poisson distribution" .Vb 1 \& 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 .Ve .IP "\fIpow\fR: Power function" 4 .IX Item "pow: Power function" .Vb 1 \& 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 .Ve .IP "\fIpowi\fR: Real raised to integer power" 4 .IX Item "powi: Real raised to integer power" .Vb 1 \& 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. .Ve .IP "\fIpsi\fR: Psi (digamma) function" 4 .IX Item "psi: Psi (digamma) function" .Vb 1 \& 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 .Ve .IP "\fIrgamma\fR: Reciprocal gamma function" 4 .IX Item "rgamma: Reciprocal gamma function" .Vb 1 \& 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. .Ve .IP "\fIround\fR: Round double to nearest or even integer valued double" 4 .IX Item "round: Round double to nearest or even integer valued double" .Vb 1 \& SYNOPSIS: \& \& # double x, y, round(); \& \& $y = round( $x ); \& \& DESCRIPTION: \& \& Returns the nearest integer to x as a double precision \& floating point result. If x ends in 0.5 exactly, the \& nearest even integer is chosen. \& \& ACCURACY: \& \& If x is greater than 1/(2*MACHEP), its closest machine \& representation is already an integer, so rounding does \& not change it. .Ve .IP "\fIshichi\fR: Hyperbolic sine and cosine integrals" 4 .IX Item "shichi: Hyperbolic sine and cosine integrals" .Vb 1 \& SYNOPSIS: \& \& # double x, Chi, Shi, shichi(); \& \& ($flag, $Shi, $Chi) = shichi( $x ); \& \& DESCRIPTION: \& \& Approximates the integrals \& \& x \& \- \& | | cosh t \- 1 \& Chi(x) = eul + ln x + | \-\-\-\-\-\-\-\-\-\-\- dt, \& | | t \& \- \& 0 \& \& x \& \- \& | | sinh t \& Shi(x) = | \-\-\-\-\-\- dt \& | | t \& \- \& 0 \& \& where eul = 0.57721566490153286061 is Euler\*(Aqs constant. \& The integrals are evaluated by power series for x < 8 \& and by Chebyshev expansions for x between 8 and 88. \& For large x, both functions approach exp(x)/2x. \& Arguments greater than 88 in magnitude return MAXNUM. \& \& ACCURACY: \& \& Test interval 0 to 88. \& Relative error: \& arithmetic function # trials peak rms \& DEC Shi 3000 9.1e\-17 \& IEEE Shi 30000 6.9e\-16 1.6e\-16 \& Absolute error, except relative when |Chi| > 1: \& DEC Chi 2500 9.3e\-17 \& IEEE Chi 30000 8.4e\-16 1.4e\-16 .Ve .IP "\fIsici\fR: Sine and cosine integrals" 4 .IX Item "sici: Sine and cosine integrals" .Vb 1 \& SYNOPSIS: \& \& # double x, Ci, Si, sici(); \& \& ($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\*(Aqs 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 .Ve .IP "\fIsin\fR: Circular sine" 4 .IX Item "sin: Circular sine" .Vb 1 \& 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. .Ve .IP "\fIcos\fR: Circular cosine" 4 .IX Item "cos: Circular cosine" .Vb 1 \& 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 .Ve .IP "\fIsindg\fR: Circular sine of angle in degrees" 4 .IX Item "sindg: Circular sine of angle in degrees" .Vb 1 \& 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) .Ve .IP "\fIcosdg\fR: Circular cosine of angle in degrees" 4 .IX Item "cosdg: Circular cosine of angle in degrees" .Vb 1 \& 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(). .Ve .IP "\fIsinh\fR: Hyperbolic sine" 4 .IX Item "sinh: Hyperbolic sine" .Vb 1 \& 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 .Ve .IP "\fIspence\fR: Dilogarithm" 4 .IX Item "spence: Dilogarithm" .Vb 1 \& 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 .Ve .IP "\fIsqrt\fR: Square root" 4 .IX Item "sqrt: Square root" .Vb 1 \& 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\*(Aqs 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 .Ve .IP "\fIstdtr\fR: Student's t distribution" 4 .IX Item "stdtr: Student's t distribution" .Vb 1 \& 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 .Ve .IP "\fIstdtri\fR: Functional inverse of Student's t distribution" 4 .IX Item "stdtri: Functional inverse of Student's t distribution" .Vb 1 \& 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 .Ve .IP "\fIstruve\fR: Struve function" 4 .IX Item "struve: Struve function" .Vb 1 \& 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. .Ve .IP "\fIplancki\fR: Integral of Planck's black body radiation formula" 4 .IX Item "plancki: Integral of Planck's black body radiation formula" .Vb 1 \& SYNOPSIS: \& \& # double lambda, T, y, plancki() \& \& $y = plancki( $lambda, $T ); \& \& DESCRIPTION: \& \& Evaluates the definite integral, from wavelength 0 to lambda, \& of Planck\*(Aqs 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 .Ve .IP "\fIpolylog\fR: polylogarithm function \s-1SYNOPSIS:\s0" 4 .IX Item "polylog: polylogarithm function SYNOPSIS:" .Vb 2 \& # double x, y, polylog(); \& # int n; \& \& $y = polylog( $n, $x ); \& \& The polylogarithm of order n is defined by the series \& \& inf k \& \- x \& Li (x) = > \-\-\- . \& n \- n \& k=1 k \& \& For x = 1, \& \& inf \& \- 1 \& Li (1) = > \-\-\- = Riemann zeta function (n) . \& n \- n \& k=1 k \& \& When n = 2, the function is the dilogarithm, related to Spence\*(Aqs integral: \& \& x 1\-x \& \- \- \& | | \-ln(1\-t) | | ln t \& Li (x) = | \-\-\-\-\-\-\-\- dt = | \-\-\-\-\-\- dt = spence(1\-x) . \& 2 | | t | | 1 \- t \& \- \- \& 0 1 \& \& ACCURACY: \& \& Relative error: \& arithmetic domain n # trials peak rms \& IEEE 0, 1 2 50000 6.2e\-16 8.0e\-17 \& IEEE 0, 1 3 100000 2.5e\-16 6.6e\-17 \& IEEE 0, 1 4 30000 1.7e\-16 4.9e\-17 \& IEEE 0, 1 5 30000 5.1e\-16 7.8e\-17 .Ve .IP "\fIbernum\fR: Bernoulli numbers" 4 .IX Item "bernum: Bernoulli numbers" .Vb 1 \& SYNOPSIS: \& \& ($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. .Ve .IP "\fIsimpson\fR: Simpson's rule to find an integral" 4 .IX Item "simpson: Simpson's rule to find an integral" .Vb 1 \& SYNOPSIS: \& \& $result = simpson(\e&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. .Ve .IP "\fIvecang\fR: angle between two vectors" 4 .IX Item "vecang: angle between two vectors" .Vb 1 \& SYNOPSIS: \& \& # double p[3], q[3], vecang(); \& \& $y = vecang( $p, $q ); \& \& DESCRIPTION: \& \& For two vectors p, q, the angle A between them is given by \& \& p.q / (|p| |q|) = cos A . \& \& where "." represents inner product, "|x|" the length of vector x. \& If the angle is small, an expression in sin A is preferred. \& Set r = q \- p. Then \& \& p.q = p.p + p.r , \& \& |p|^2 = p.p , \& \& |q|^2 = p.p + 2 p.r + r.r , \& \& p.p^2 + 2 p.p p.r + p.r^2 \& cos^2 A = \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& p.p (p.p + 2 p.r + r.r) \& \& p.p + 2 p.r + p.r^2 / p.p \& = \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- , \& p.p + 2 p.r + r.r \& \& sin^2 A = 1 \- cos^2 A \& \& r.r \- p.r^2 / p.p \& = \-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\- \& p.p + 2 p.r + r.r \& \& = (r.r \- p.r^2 / p.p) / q.q . \& \& ACCURACY: \& \& Relative error: \& arithmetic domain # trials peak rms \& IEEE \-1, 1 10^6 1.7e\-16 4.2e\-17 .Ve .IP "\fIonef2\fR: Hypergeometric function 1F2" 4 .IX Item "onef2: Hypergeometric function 1F2" .Vb 1 \& SYNOPSIS: \& \& # double a, b, c, x, value; \& \& # double *err; \& \& ($value, $err) = onef2( $a, $b, $c, $x) \& \& ACCURACY: \& \& Not accurately characterized, but spot checked against tables. .Ve .IP "\fIthreef0\fR: Hypergeometric function 3F0" 4 .IX Item "threef0: Hypergeometric function 3F0" .Vb 1 \& SYNOPSIS: \& \& # double a, b, c, x, value; \& \& # double *err; \& \& ($value, $err) = threef0( $a, $b, $c, $x ) \& \& ACCURACY: \& \& Not accurately characterized, but spot checked against tables. .Ve .IP "\fIyv\fR: Bessel function Yv with noninteger v" 4 .IX Item "yv: Bessel function Yv with noninteger v" .Vb 1 \& SYNOPSIS: \& \& # double v, x; \& \& # double yv( v, x ); \& \& $y = yv( $v, $x ); \& \& ACCURACY: \& \& Not accurately characterized, but spot checked against tables. .Ve .IP "\fItan\fR: Circular tangent" 4 .IX Item "tan: Circular tangent" .Vb 1 \& 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 .Ve .IP "\fIcot\fR: Circular cotangent" 4 .IX Item "cot: Circular cotangent" .Vb 1 \& 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 .Ve .IP "\fItandg\fR: Circular tangent of argument in degrees" 4 .IX Item "tandg: Circular tangent of argument in degrees" .Vb 1 \& 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 .Ve .IP "\fIcotdg\fR: Circular cotangent of argument in degrees" 4 .IX Item "cotdg: Circular cotangent of argument in degrees" .Vb 1 \& 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 .Ve .IP "\fItanh\fR: Hyperbolic tangent" 4 .IX Item "tanh: Hyperbolic tangent" .Vb 1 \& 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 .Ve .IP "\fIunity\fR: Relative error approximations for function arguments near unity." 4 .IX Item "unity: Relative error approximations for function arguments near unity." .Vb 1 \& SYNOPSIS: .Ve .Sp # log1p(x) = log(1+x) .Sp .Vb 1 \& $y = log1p( $x ); .Ve .Sp # expm1(x) = exp(x) \- 1 .Sp .Vb 1 \& $y = expm1( $x ); .Ve .Sp # cosm1(x) = cos(x) \- 1 .Sp .Vb 1 \& $y = cosm1( $x ); .Ve .IP "\fIyn\fR: Bessel function of second kind of integer order" 4 .IX Item "yn: Bessel function of second kind of integer order" .Vb 1 \& 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. .Ve .IP "\fIzeta\fR: Riemann zeta function of two arguments" 4 .IX Item "zeta: Riemann zeta function of two arguments" .Vb 1 \& 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. .Ve .IP "\fIzetac\fR: Riemann zeta function" 4 .IX Item "zetac: Riemann zeta function" .Vb 1 \& 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 .Ve .SH "TODO" .IX Header "TODO" .IP "\(bu" 4 Include more operating systems when generating mconf.h. .SH "MAINTAINER" .IX Header "MAINTAINER" Shlomi Fish, , . .SH "BUGS" .IX Header "BUGS" Please report any on the rt.cpan.org interface: .SH "VERSION CONTROL" .IX Header "VERSION CONTROL" This distribution is maintained in this GitHub repository: .PP . .SH "SEE ALSO" .IX Header "SEE ALSO" For interfaces to programs which can do symbolic manipulation, see \s-1PDL\s0, Math::Pari, and Math::ematica. For a command line interface to the routines of \fIMath::Cephes\fR, see the included \f(CW\*(C`pmath\*(C'\fR 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. .SH "COPYRIGHT" .IX Header "COPYRIGHT" The C code for the Cephes Math Library is Copyright 1984, 1987, 1989, 2002 by Stephen L. Moshier, and is available at . Direct inquiries to 30 Frost Street, Cambridge, \s-1MA 02140.\s0 .PP The file arrays.c included here to handle passing arrays into and out of C routines comes from the \s-1PGPLOT\s0 module of Karl Glazebrook . .PP 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. .PP Perl interface maintained by Shlomi Fish starting from 2012. All explicit or implicit copyrights on the changes are disclaimed by him.