Provided by: libmath-cephes-perl_0.5305-5build1_amd64 
NAME
Math::Cephes - perl interface to the cephes math library
SYNOPSIS
use Math::Cephes qw(:all);
DESCRIPTION
This module provides an interface to over 150 functions of the
cephes math library of Stephen Moshier. No functions are exported
by default, but rather must be imported explicitly, as in
use Math::Cephes qw(sin cos);
There are a number of export tags defined which allow
importing groups of functions:
use Math::Cephes qw(:constants);
imports the variables
$PI : 3.14159265358979323846 # pi
$PIO2 : 1.57079632679489661923 # pi/2
$PIO4 : 0.785398163397448309616 # pi/4
$SQRT2 : 1.41421356237309504880 # sqrt(2)
$SQRTH : 0.707106781186547524401 # sqrt(2)/2
$LOG2E : 1.4426950408889634073599 # 1/log(2)
$SQ2OPI : 0.79788456080286535587989 # sqrt( 2/pi )
$LOGE2 : 0.693147180559945309417 # log(2)
$LOGSQ2 : 0.346573590279972654709 # log(2)/2
$THPIO4 : 2.35619449019234492885 # 3*pi/4
$TWOOPI : 0.636619772367581343075535 # 2/pi
As well, there are 4 machine-specific numbers available:
$MACHEP : machine roundoff error
$MAXLOG : maximum log on the machine
$MINLOG : minimum log on the machine
$MAXNUM : largest number represented
use Math::Cephes qw(:trigs);
imports
acos: Inverse circular cosine
asin: Inverse circular sine
atan: Inverse circular tangent (arctangent)
atan2: Quadrant correct inverse circular tangent
cos: Circular cosine
cosdg: Circular cosine of angle in degrees
cot: Circular cotangent
cotdg: Circular cotangent of argument in degrees
hypot: hypotenuse associated with the sides of a right triangle
radian: Degrees, minutes, seconds to radians
sin: Circular sine
sindg: Circular sine of angle in degrees
tan: Circular tangent
tandg: Circular tangent of argument in degrees
cosm1: Relative error approximations for function arguments near unity
use Math::Cephes qw(:hypers);
imports
acosh: Inverse hyperbolic cosine
asinh: Inverse hyperbolic sine
atanh: Inverse hyperbolic tangent
cosh: Hyperbolic cosine
sinh: Hyperbolic sine
tanh: Hyperbolic tangent
use Math::Cephes qw(:explog);
imports
exp: Exponential function
expxx: exp(x*x)
exp10: Base 10 exponential function (Common antilogarithm)
exp2: Base 2 exponential function
log: Natural logarithm
log10: Common logarithm
log2: Base 2 logarithm
log1p,expm1: Relative error approximations for function arguments near unity.
use Math::Cephes qw(:cmplx);
imports
new_cmplx: create a new complex number object
cabs: Complex absolute value
cacos: Complex circular arc cosine
cacosh: Complex inverse hyperbolic cosine
casin: Complex circular arc sine
casinh: Complex inverse hyperbolic sine
catan: Complex circular arc tangent
catanh: Complex inverse hyperbolic tangent
ccos: Complex circular cosine
ccosh: Complex hyperbolic cosine
ccot: Complex circular cotangent
cexp: Complex exponential function
clog: Complex natural logarithm
cadd: add two complex numbers
csub: subtract two complex numbers
cmul: multiply two complex numbers
cdiv: divide two complex numbers
cmov: copy one complex number to another
cneg: negate a complex number
cpow: Complex power function
csin: Complex circular sine
csinh: Complex hyperbolic sine
csqrt: Complex square root
ctan: Complex circular tangent
ctanh: Complex hyperbolic tangent
use Math::Cephes qw(:utils);
imports
cbrt: Cube root
ceil: ceil
drand: Pseudorandom number generator
fabs: Absolute value
fac: Factorial function
floor: floor
frexp: frexp
ldexp: multiplies x by 2**n.
lrand: Pseudorandom number generator
lsqrt: Integer square root
pow: Power function
powi: Real raised to integer power
round: Round double to nearest or even integer valued double
sqrt: Square root
use Math::Cephes qw(:bessels);
imports
i0: Modified Bessel function of order zero
i0e: Modified Bessel function of order zero, exponentially scaled
i1: Modified Bessel function of order one
i1e: Modified Bessel function of order one, exponentially scaled
iv: Modified Bessel function of noninteger order
j0: Bessel function of order zero
j1: Bessel function of order one
jn: Bessel function of integer order
jv: Bessel function of noninteger order
k0: Modified Bessel function, third kind, order zero
k0e: Modified Bessel function, third kind, order zero, exponentially scaled
k1: Modified Bessel function, third kind, order one
k1e: Modified Bessel function, third kind, order one, exponentially scaled
kn: Modified Bessel function, third kind, integer order
y0: Bessel function of the second kind, order zero
y1: Bessel function of second kind of order one
yn: Bessel function of second kind of integer order
yv: Bessel function Yv with noninteger v
use Math::Cephes qw(:dists);
imports
bdtr: Binomial distribution
bdtrc: Complemented binomial distribution
bdtri: Inverse binomial distribution
btdtr: Beta distribution
chdtr: Chi-square distribution
chdtrc: Complemented Chi-square distribution
chdtri: Inverse of complemented Chi-square distribution
fdtr: F distribution
fdtrc: Complemented F distribution
fdtri: Inverse of complemented F distribution
gdtr: Gamma distribution function
gdtrc: Complemented gamma distribution function
nbdtr: Negative binomial distribution
nbdtrc: Complemented negative binomial distribution
nbdtri: Functional inverse of negative binomial distribution
ndtr: Normal distribution function
ndtri: Inverse of Normal distribution function
pdtr: Poisson distribution
pdtrc: Complemented poisson distribution
pdtri: Inverse Poisson distribution
stdtr: Student's t distribution
stdtri: Functional inverse of Student's t distribution
use Math::Cephes qw(:gammas);
imports
fac: Factorial function
gamma: Gamma function
igam: Incomplete gamma integral
igamc: Complemented incomplete gamma integral
igami: Inverse of complemented imcomplete gamma integral
psi: Psi (digamma) function
rgamma: Reciprocal gamma function
use Math::Cephes qw(:betas);
imports
beta: Beta function
incbet: Incomplete beta integral
incbi: Inverse of imcomplete beta integral
lbeta: Natural logarithm of |beta|
use Math::Cephes qw(:elliptics);
imports
ellie: Incomplete elliptic integral of the second kind
ellik: Incomplete elliptic integral of the first kind
ellpe: Complete elliptic integral of the second kind
ellpj: Jacobian Elliptic Functions
ellpk: Complete elliptic integral of the first kind
use Math::Cephes qw(:hypergeometrics);
imports
hyp2f0: Gauss hypergeometric function F
hyp2f1: Gauss hypergeometric function F
hyperg: Confluent hypergeometric function
onef2: Hypergeometric function 1F2
threef0: Hypergeometric function 3F0
use Math::Cephes qw(:misc);
imports
airy: Airy function
bernum: Bernoulli numbers
dawsn: Dawson's Integral
ei: Exponential integral
erf: Error function
erfc: Complementary error function
expn: Exponential integral En
fresnl: Fresnel integral
plancki: Integral of Planck's black body radiation formula
polylog: Polylogarithm function
shichi: Hyperbolic sine and cosine integrals
sici: Sine and cosine integrals
simpson: Simpson's rule to find an integral
spence: Dilogarithm
struve: Struve function
vecang: angle between two vectors
zeta: Riemann zeta function of two arguments
zetac: Riemann zeta function
use Math::Cephes qw(:fract);
imports
new_fract: create a new fraction object
radd: add two fractions
rmul: multiply two fractions
rsub: subtracttwo fractions
rdiv: divide two fractions
euclid: finds the greatest common divisor
FUNCTIONS
A description of the various functions available follows.
acosh: Inverse hyperbolic cosine
SYNOPSIS:
# double x, y, acosh();
$y = acosh( $x );
DESCRIPTION:
Returns inverse hyperbolic cosine of argument.
If 1 <= x < 1.5, a rational approximation
sqrt(z) * P(z)/Q(z)
where z = x-1, is used. Otherwise,
acosh(x) = log( x + sqrt( (x-1)(x+1) ).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC 1,3 30000 4.2e-17 1.1e-17
IEEE 1,3 30000 4.6e-16 8.7e-17
ERROR MESSAGES:
message condition value returned
acosh domain |x| < 1 NAN
airy: Airy function
SYNOPSIS:
# double x, ai, aiprime, bi, biprime;
# int airy();
($flag, $ai, $aiprime, $bi, $biprime) = airy( $x );
DESCRIPTION:
Solution of the differential equation
y"(x) = xy.
The function returns the two independent solutions Ai, Bi
and their first derivatives Ai'(x), Bi'(x).
Evaluation is by power series summation for small x,
by rational minimax approximations for large x.
ACCURACY:
Error criterion is absolute when function <= 1, relative
when function > 1, except * denotes relative error criterion.
For large negative x, the absolute error increases as x^1.5.
For large positive x, the relative error increases as x^1.5.
Arithmetic domain function # trials peak rms
IEEE -10, 0 Ai 10000 1.6e-15 2.7e-16
IEEE 0, 10 Ai 10000 2.3e-14* 1.8e-15*
IEEE -10, 0 Ai' 10000 4.6e-15 7.6e-16
IEEE 0, 10 Ai' 10000 1.8e-14* 1.5e-15*
IEEE -10, 10 Bi 30000 4.2e-15 5.3e-16
IEEE -10, 10 Bi' 30000 4.9e-15 7.3e-16
DEC -10, 0 Ai 5000 1.7e-16 2.8e-17
DEC 0, 10 Ai 5000 2.1e-15* 1.7e-16*
DEC -10, 0 Ai' 5000 4.7e-16 7.8e-17
DEC 0, 10 Ai' 12000 1.8e-15* 1.5e-16*
DEC -10, 10 Bi 10000 5.5e-16 6.8e-17
DEC -10, 10 Bi' 7000 5.3e-16 8.7e-17
radian: Degrees, minutes, seconds to radians
SYNOPSIS:
# double d, m, s, radian();
$r = radian( $d, $m, $s );
DESCRIPTION:
Converts an angle of degrees, minutes, seconds to radians.
hypot: returns the hypotenuse associated with the sides of a right triangle
SYNOPSIS:
# double a, b, c, hypot();
$c = hypot( $a, $b );
DESCRIPTION:
Calculates the hypotenuse associated with the sides of a
right triangle, according to
c = sqrt( a**2 + b**2)
asin: Inverse circular sine
SYNOPSIS:
# double x, y, asin();
$y = asin( $x );
DESCRIPTION:
Returns radian angle between -pi/2 and +pi/2 whose sine is x.
A rational function of the form x + x**3 P(x**2)/Q(x**2)
is used for |x| in the interval [0, 0.5]. If |x| > 0.5 it is
transformed by the identity
asin(x) = pi/2 - 2 asin( sqrt( (1-x)/2 ) ).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -1, 1 40000 2.6e-17 7.1e-18
IEEE -1, 1 10^6 1.9e-16 5.4e-17
ERROR MESSAGES:
message condition value returned
asin domain |x| > 1 NAN
acos: Inverse circular cosine
SYNOPSIS:
# double x, y, acos();
$y = acos( $x );
DESCRIPTION:
Returns radian angle between 0 and pi whose cosine
is x.
Analytically, acos(x) = pi/2 - asin(x). However if |x| is
near 1, there is cancellation error in subtracting asin(x)
from pi/2. Hence if x < -0.5,
acos(x) = pi - 2.0 * asin( sqrt((1+x)/2) );
or if x > +0.5,
acos(x) = 2.0 * asin( sqrt((1-x)/2) ).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -1, 1 50000 3.3e-17 8.2e-18
IEEE -1, 1 10^6 2.2e-16 6.5e-17
ERROR MESSAGES:
message condition value returned
asin domain |x| > 1 NAN
asinh: Inverse hyperbolic sine
SYNOPSIS:
# double x, y, asinh();
$y = asinh( $x );
DESCRIPTION:
Returns inverse hyperbolic sine of argument.
If |x| < 0.5, the function is approximated by a rational
form x + x**3 P(x)/Q(x). Otherwise,
asinh(x) = log( x + sqrt(1 + x*x) ).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -3,3 75000 4.6e-17 1.1e-17
IEEE -1,1 30000 3.7e-16 7.8e-17
IEEE 1,3 30000 2.5e-16 6.7e-17
atan: Inverse circular tangent (arctangent)
SYNOPSIS:
# double x, y, atan();
$y = atan( $x );
DESCRIPTION:
Returns radian angle between -pi/2 and +pi/2 whose tangent
is x.
Range reduction is from three intervals into the interval
from zero to 0.66. The approximant uses a rational
function of degree 4/5 of the form x + x**3 P(x)/Q(x).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -10, 10 50000 2.4e-17 8.3e-18
IEEE -10, 10 10^6 1.8e-16 5.0e-17
atan2: Quadrant correct inverse circular tangent
SYNOPSIS:
# double x, y, z, atan2();
$z = atan2( $y, $x );
DESCRIPTION:
Returns radian angle whose tangent is y/x.
Define compile time symbol ANSIC = 1 for ANSI standard,
range -PI < z <= +PI, args (y,x); else ANSIC = 0 for range
0 to 2PI, args (x,y).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE -10, 10 10^6 2.5e-16 6.9e-17
See atan.c.
atanh: Inverse hyperbolic tangent
SYNOPSIS:
# double x, y, atanh();
$y = atanh( $x );
DESCRIPTION:
Returns inverse hyperbolic tangent of argument in the range
MINLOG to MAXLOG.
If |x| < 0.5, the rational form x + x**3 P(x)/Q(x) is
employed. Otherwise,
atanh(x) = 0.5 * log( (1+x)/(1-x) ).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -1,1 50000 2.4e-17 6.4e-18
IEEE -1,1 30000 1.9e-16 5.2e-17
bdtr: Binomial distribution
SYNOPSIS:
# int k, n;
# double p, y, bdtr();
$y = bdtr( $k, $n, $p );
DESCRIPTION:
Returns the sum of the terms 0 through k of the Binomial
probability density:
k
-- ( n ) j n-j
> ( ) p (1-p)
-- ( j )
j=0
The terms are not summed directly; instead the incomplete
beta integral is employed, according to the formula
y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ).
The arguments must be positive, with p ranging from 0 to 1.
ACCURACY:
Tested at random points (a,b,p), with p between 0 and 1.
a,b Relative error:
arithmetic domain # trials peak rms
For p between 0.001 and 1:
IEEE 0,100 100000 4.3e-15 2.6e-16
See also incbet.c.
ERROR MESSAGES:
message condition value returned
bdtr domain k < 0 0.0
n < k
x < 0, x > 1
bdtrc: Complemented binomial distribution
SYNOPSIS:
# int k, n;
# double p, y, bdtrc();
$y = bdtrc( $k, $n, $p );
DESCRIPTION:
Returns the sum of the terms k+1 through n of the Binomial
probability density:
n
-- ( n ) j n-j
> ( ) p (1-p)
-- ( j )
j=k+1
The terms are not summed directly; instead the incomplete
beta integral is employed, according to the formula
y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ).
The arguments must be positive, with p ranging from 0 to 1.
ACCURACY:
Tested at random points (a,b,p).
a,b Relative error:
arithmetic domain # trials peak rms
For p between 0.001 and 1:
IEEE 0,100 100000 6.7e-15 8.2e-16
For p between 0 and .001:
IEEE 0,100 100000 1.5e-13 2.7e-15
ERROR MESSAGES:
message condition value returned
bdtrc domain x<0, x>1, n<k 0.0
bdtri: Inverse binomial distribution
SYNOPSIS:
# int k, n;
# double p, y, bdtri();
$p = bdtr( $k, $n, $y );
DESCRIPTION:
Finds the event probability p such that the sum of the
terms 0 through k of the Binomial probability density
is equal to the given cumulative probability y.
This is accomplished using the inverse beta integral
function and the relation
1 - p = incbi( n-k, k+1, y ).
ACCURACY:
Tested at random points (a,b,p).
a,b Relative error:
arithmetic domain # trials peak rms
For p between 0.001 and 1:
IEEE 0,100 100000 2.3e-14 6.4e-16
IEEE 0,10000 100000 6.6e-12 1.2e-13
For p between 10^-6 and 0.001:
IEEE 0,100 100000 2.0e-12 1.3e-14
IEEE 0,10000 100000 1.5e-12 3.2e-14
See also incbi.c.
ERROR MESSAGES:
message condition value returned
bdtri domain k < 0, n <= k 0.0
x < 0, x > 1
beta: Beta function
SYNOPSIS:
# double a, b, y, beta();
$y = beta( $a, $b );
DESCRIPTION:
- -
| (a) | (b)
beta( a, b ) = -----------.
-
| (a+b)
For large arguments the logarithm of the function is
evaluated using lgam(), then exponentiated.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC 0,30 1700 7.7e-15 1.5e-15
IEEE 0,30 30000 8.1e-14 1.1e-14
ERROR MESSAGES:
message condition value returned
beta overflow log(beta) > MAXLOG 0.0
a or b <0 integer 0.0
lbeta: Natural logarithm of |beta|
SYNOPSIS:
# double a, b;
# double lbeta( a, b );
$y = lbeta( $a, $b);
btdtr: Beta distribution
SYNOPSIS:
# double a, b, x, y, btdtr();
$y = btdtr( $a, $b, $x );
DESCRIPTION:
Returns the area from zero to x under the beta density
function:
x
- -
| (a+b) | | a-1 b-1
P(x) = ---------- | t (1-t) dt
- - | |
| (a) | (b) -
0
This function is identical to the incomplete beta
integral function incbet(a, b, x).
The complemented function is
1 - P(1-x) = incbet( b, a, x );
ACCURACY:
See incbet.c.
cbrt: Cube root
SYNOPSIS:
# double x, y, cbrt();
$y = cbrt( $x );
DESCRIPTION:
Returns the cube root of the argument, which may be negative.
Range reduction involves determining the power of 2 of
the argument. A polynomial of degree 2 applied to the
mantissa, and multiplication by the cube root of 1, 2, or 4
approximates the root to within about 0.1%. Then Newton's
iteration is used three times to converge to an accurate
result.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -10,10 200000 1.8e-17 6.2e-18
IEEE 0,1e308 30000 1.5e-16 5.0e-17
chdtr: Chi-square distribution
SYNOPSIS:
# double v, x, y, chdtr();
$y = chdtr( $v, $x );
DESCRIPTION:
Returns the area under the left hand tail (from 0 to x)
of the Chi square probability density function with
v degrees of freedom.
inf.
-
1 | | v/2-1 -t/2
P( x | v ) = ----------- | t e dt
v/2 - | |
2 | (v/2) -
x
where x is the Chi-square variable.
The incomplete gamma integral is used, according to the
formula
y = chdtr( v, x ) = igam( v/2.0, x/2.0 ).
The arguments must both be positive.
ACCURACY:
See igam().
ERROR MESSAGES:
message condition value returned
chdtr domain x < 0 or v < 1 0.0
chdtrc: Complemented Chi-square distribution
SYNOPSIS:
# double v, x, y, chdtrc();
$y = chdtrc( $v, $x );
DESCRIPTION:
Returns the area under the right hand tail (from x to
infinity) of the Chi square probability density function
with v degrees of freedom:
inf.
-
1 | | v/2-1 -t/2
P( x | v ) = ----------- | t e dt
v/2 - | |
2 | (v/2) -
x
where x is the Chi-square variable.
The incomplete gamma integral is used, according to the
formula
y = chdtrc( v, x ) = igamc( v/2.0, x/2.0 ).
The arguments must both be positive.
ACCURACY:
See igamc().
ERROR MESSAGES:
message condition value returned
chdtrc domain x < 0 or v < 1 0.0
chdtri: Inverse of complemented Chi-square distribution
SYNOPSIS:
# double df, x, y, chdtri();
$x = chdtri( $df, $y );
DESCRIPTION:
Finds the Chi-square argument x such that the integral
from x to infinity of the Chi-square density is equal
to the given cumulative probability y.
This is accomplished using the inverse gamma integral
function and the relation
x/2 = igami( df/2, y );
ACCURACY:
See igami.c.
ERROR MESSAGES:
message condition value returned
chdtri domain y < 0 or y > 1 0.0
v < 1
clog: Complex natural logarithm
SYNOPSIS:
# void clog();
# cmplx z, w;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
clog($z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
DESCRIPTION:
Returns complex logarithm to the base e (2.718...) of
the complex argument x.
If z = x + iy, r = sqrt( x**2 + y**2 ),
then
w = log(r) + i arctan(y/x).
The arctangent ranges from -PI to +PI.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -10,+10 7000 8.5e-17 1.9e-17
IEEE -10,+10 30000 5.0e-15 1.1e-16
Larger relative error can be observed for z near 1 +i0.
In IEEE arithmetic the peak absolute error is 5.2e-16, rms
absolute error 1.0e-16.
cexp: Complex exponential function
SYNOPSIS:
# void cexp();
# cmplx z, w;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
cexp($z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
DESCRIPTION:
Returns the exponential of the complex argument z
into the complex result w.
If
z = x + iy,
r = exp(x),
then
w = r cos y + i r sin y.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -10,+10 8700 3.7e-17 1.1e-17
IEEE -10,+10 30000 3.0e-16 8.7e-17
csin: Complex circular sine
SYNOPSIS:
# void csin();
# cmplx z, w;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
csin($z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
DESCRIPTION:
If
z = x + iy,
then
w = sin x cosh y + i cos x sinh y.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -10,+10 8400 5.3e-17 1.3e-17
IEEE -10,+10 30000 3.8e-16 1.0e-16
Also tested by csin(casin(z)) = z.
ccos: Complex circular cosine
SYNOPSIS:
# void ccos();
# cmplx z, w;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
ccos($z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
DESCRIPTION:
If
z = x + iy,
then
w = cos x cosh y - i sin x sinh y.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -10,+10 8400 4.5e-17 1.3e-17
IEEE -10,+10 30000 3.8e-16 1.0e-16
ctan: Complex circular tangent
SYNOPSIS:
# void ctan();
# cmplx z, w;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
ctan($z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
DESCRIPTION:
If
z = x + iy,
then
sin 2x + i sinh 2y
w = --------------------.
cos 2x + cosh 2y
On the real axis the denominator is zero at odd multiples
of PI/2. The denominator is evaluated by its Taylor
series near these points.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -10,+10 5200 7.1e-17 1.6e-17
IEEE -10,+10 30000 7.2e-16 1.2e-16
Also tested by ctan * ccot = 1 and catan(ctan(z)) = z.
ccot: Complex circular cotangent
SYNOPSIS:
# void ccot();
# cmplx z, w;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
ccot($z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
DESCRIPTION:
If
z = x + iy,
then
sin 2x - i sinh 2y
w = --------------------.
cosh 2y - cos 2x
On the real axis, the denominator has zeros at even
multiples of PI/2. Near these points it is evaluated
by a Taylor series.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -10,+10 3000 6.5e-17 1.6e-17
IEEE -10,+10 30000 9.2e-16 1.2e-16
Also tested by ctan * ccot = 1 + i0.
casin: Complex circular arc sine
SYNOPSIS:
# void casin();
# cmplx z, w;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
casin($z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
DESCRIPTION:
Inverse complex sine:
2
w = -i clog( iz + csqrt( 1 - z ) ).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -10,+10 10100 2.1e-15 3.4e-16
IEEE -10,+10 30000 2.2e-14 2.7e-15
Larger relative error can be observed for z near zero.
Also tested by csin(casin(z)) = z.
cacos: Complex circular arc cosine
SYNOPSIS:
# void cacos();
# cmplx z, w;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
cacos($z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
DESCRIPTION:
w = arccos z = PI/2 - arcsin z.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -10,+10 5200 1.6e-15 2.8e-16
IEEE -10,+10 30000 1.8e-14 2.2e-15
catan: Complex circular arc tangent
SYNOPSIS:
# void catan();
# cmplx z, w;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
catan($z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
DESCRIPTION:
If
z = x + iy,
then
1 ( 2x )
Re w = - arctan(-----------) + k PI
2 ( 2 2)
(1 - x - y )
( 2 2)
1 (x + (y+1) )
Im w = - log(------------)
4 ( 2 2)
(x + (y-1) )
Where k is an arbitrary integer.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -10,+10 5900 1.3e-16 7.8e-18
IEEE -10,+10 30000 2.3e-15 8.5e-17
The check catan( ctan(z) ) = z, with |x| and |y| < PI/2,
had peak relative error 1.5e-16, rms relative error
2.9e-17. See also clog().
csinh: Complex hyperbolic sine
SYNOPSIS:
# void csinh();
# cmplx z, w;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
csinh($z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
DESCRIPTION:
csinh z = (cexp(z) - cexp(-z))/2
= sinh x * cos y + i cosh x * sin y .
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE -10,+10 30000 3.1e-16 8.2e-17
casinh: Complex inverse hyperbolic sine
SYNOPSIS:
# void casinh();
# cmplx z, w;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
casinh($z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
print_new_cmplx($w); # prints $w as Re($w) + i Im($w)
DESCRIPTION:
casinh z = -i casin iz .
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE -10,+10 30000 1.8e-14 2.6e-15
ccosh: Complex hyperbolic cosine
SYNOPSIS:
# void ccosh();
# cmplx z, w;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
ccosh($z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
DESCRIPTION:
ccosh(z) = cosh x cos y + i sinh x sin y .
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE -10,+10 30000 2.9e-16 8.1e-17
cacosh: Complex inverse hyperbolic cosine
SYNOPSIS:
# void cacosh();
# cmplx z, w;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
cacosh($z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
DESCRIPTION:
acosh z = i acos z .
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE -10,+10 30000 1.6e-14 2.1e-15
ctanh: Complex hyperbolic tangent
SYNOPSIS:
# void ctanh();
# cmplx z, w;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
ctanh($z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
DESCRIPTION:
tanh z = (sinh 2x + i sin 2y) / (cosh 2x + cos 2y) .
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE -10,+10 30000 1.7e-14 2.4e-16
catanh: Complex inverse hyperbolic tangent
SYNOPSIS:
# void catanh();
# cmplx z, w;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
catanh($z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
DESCRIPTION:
Inverse tanh, equal to -i catan (iz);
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE -10,+10 30000 2.3e-16 6.2e-17
cpow: Complex power function
SYNOPSIS:
# void cpow();
# cmplx a, z, w;
$a = new_cmplx(5, 6); # $z = 5 + 6 i
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
cpow($a, $z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
DESCRIPTION:
Raises complex A to the complex Zth power.
Definition is per AMS55 # 4.2.8,
analytically equivalent to cpow(a,z) = cexp(z clog(a)).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE -10,+10 30000 9.4e-15 1.5e-15
cmplx: Complex number arithmetic
SYNOPSIS:
# typedef struct {
# double r; real part
# double i; imaginary part
# }cmplx;
# cmplx *a, *b, *c;
$a = new_cmplx(3, 5); # $a = 3 + 5 i
$b = new_cmplx(2, 3); # $b = 2 + 3 i
$c = new_cmplx();
cadd( $a, $b, $c ); # c = b + a
csub( $a, $b, $c ); # c = b - a
cmul( $a, $b, $c ); # c = b * a
cdiv( $a, $b, $c ); # c = b / a
cneg( $c ); # c = -c
cmov( $b, $c ); # c = b
print $c->{r}, ' ', $c->{i}; # prints real and imaginary parts of $c
DESCRIPTION:
Addition:
c.r = b.r + a.r
c.i = b.i + a.i
Subtraction:
c.r = b.r - a.r
c.i = b.i - a.i
Multiplication:
c.r = b.r * a.r - b.i * a.i
c.i = b.r * a.i + b.i * a.r
Division:
d = a.r * a.r + a.i * a.i
c.r = (b.r * a.r + b.i * a.i)/d
c.i = (b.i * a.r - b.r * a.i)/d
ACCURACY:
In DEC arithmetic, the test (1/z) * z = 1 had peak relative
error 3.1e-17, rms 1.2e-17. The test (y/z) * (z/y) = 1 had
peak relative error 8.3e-17, rms 2.1e-17.
Tests in the rectangle {-10,+10}:
Relative error:
arithmetic function # trials peak rms
DEC cadd 10000 1.4e-17 3.4e-18
IEEE cadd 100000 1.1e-16 2.7e-17
DEC csub 10000 1.4e-17 4.5e-18
IEEE csub 100000 1.1e-16 3.4e-17
DEC cmul 3000 2.3e-17 8.7e-18
IEEE cmul 100000 2.1e-16 6.9e-17
DEC cdiv 18000 4.9e-17 1.3e-17
IEEE cdiv 100000 3.7e-16 1.1e-16
cabs: Complex absolute value
SYNOPSIS:
# double a, cabs();
# cmplx z;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$a = cabs( $z );
DESCRIPTION:
If z = x + iy
then
a = sqrt( x**2 + y**2 ).
Overflow and underflow are avoided by testing the magnitudes
of x and y before squaring. If either is outside half of
the floating point full scale range, both are rescaled.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -30,+30 30000 3.2e-17 9.2e-18
IEEE -10,+10 100000 2.7e-16 6.9e-17
csqrt: Complex square root
SYNOPSIS:
# void csqrt();
# cmplx z, w;
$z = new_cmplx(2, 3); # $z = 2 + 3 i
$w = new_cmplx();
csqrt($z, $w );
print $w->{r}, ' ', $w->{i}; # prints real and imaginary parts of $w
DESCRIPTION:
If z = x + iy, r = |z|, then
1/2
Im w = [ (r - x)/2 ] ,
Re w = y / 2 Im w.
Note that -w is also a square root of z. The root chosen
is always in the upper half plane.
Because of the potential for cancellation error in r - x,
the result is sharpened by doing a Heron iteration
(see sqrt.c) in complex arithmetic.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -10,+10 25000 3.2e-17 9.6e-18
IEEE -10,+10 100000 3.2e-16 7.7e-17
2
Also tested by csqrt( z ) = z, and tested by arguments
close to the real axis.
machconst: Globally declared constants
SYNOPSIS:
extern double nameofconstant;
DESCRIPTION:
This file contains a number of mathematical constants and
also some needed size parameters of the computer arithmetic.
The values are supplied as arrays of hexadecimal integers
for IEEE arithmetic; arrays of octal constants for DEC
arithmetic; and in a normal decimal scientific notation for
other machines. The particular notation used is determined
by a symbol (DEC, IBMPC, or UNK) defined in the include file
mconf.h.
The default size parameters are as follows.
For DEC and UNK modes:
MACHEP = 1.38777878078144567553E-17 2**-56
MAXLOG = 8.8029691931113054295988E1 log(2**127)
MINLOG = -8.872283911167299960540E1 log(2**-128)
MAXNUM = 1.701411834604692317316873e38 2**127
For IEEE arithmetic (IBMPC):
MACHEP = 1.11022302462515654042E-16 2**-53
MAXLOG = 7.09782712893383996843E2 log(2**1024)
MINLOG = -7.08396418532264106224E2 log(2**-1022)
MAXNUM = 1.7976931348623158E308 2**1024
These lists are subject to change.
cosh: Hyperbolic cosine
SYNOPSIS:
# double x, y, cosh();
$y = cosh( $x );
DESCRIPTION:
Returns hyperbolic cosine of argument in the range MINLOG to
MAXLOG.
cosh(x) = ( exp(x) + exp(-x) )/2.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC +- 88 50000 4.0e-17 7.7e-18
IEEE +-MAXLOG 30000 2.6e-16 5.7e-17
ERROR MESSAGES:
message condition value returned
cosh overflow |x| > MAXLOG MAXNUM
dawsn: Dawson's Integral
SYNOPSIS:
# double x, y, dawsn();
$y = dawsn( $x );
DESCRIPTION:
Approximates the integral
x
-
2 | | 2
dawsn(x) = exp( -x ) | exp( t ) dt
| |
-
0
Three different rational approximations are employed, for
the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,10 10000 6.9e-16 1.0e-16
DEC 0,10 6000 7.4e-17 1.4e-17
drand: Pseudorandom number generator
SYNOPSIS:
# double y, drand();
($flag, $y) = drand( );
DESCRIPTION:
Yields a random number 1.0 <= y < 2.0.
The three-generator congruential algorithm by Brian
Wichmann and David Hill (BYTE magazine, March, 1987,
pp 127-8) is used. The period, given by them, is
6953607871644.
Versions invoked by the different arithmetic compile
time options DEC, IBMPC, and MIEEE, produce
approximately the same sequences, differing only in the
least significant bits of the numbers. The UNK option
implements the algorithm as recommended in the BYTE
article. It may be used on all computers. However,
the low order bits of a double precision number may
not be adequately random, and may vary due to arithmetic
implementation details on different computers.
The other compile options generate an additional random
integer that overwrites the low order bits of the double
precision number. This reduces the period by a factor of
two but tends to overcome the problems mentioned.
ellie: Incomplete elliptic integral of the second kind
SYNOPSIS:
# double phi, m, y, ellie();
$y = ellie( $phi, $m );
DESCRIPTION:
Approximates the integral
phi
-
| |
| 2
E(phi_\m) = | sqrt( 1 - m sin t ) dt
|
| |
-
0
of amplitude phi and modulus m, using the arithmetic -
geometric mean algorithm.
ACCURACY:
Tested at random arguments with phi in [-10, 10] and m in
[0, 1].
Relative error:
arithmetic domain # trials peak rms
DEC 0,2 2000 1.9e-16 3.4e-17
IEEE -10,10 150000 3.3e-15 1.4e-16
ellik: Incomplete elliptic integral of the first kind
SYNOPSIS:
# double phi, m, y, ellik();
$y = ellik( $phi, $m );
DESCRIPTION:
Approximates the integral
phi
-
| |
| dt
F(phi_\m) = | ------------------
| 2
| | sqrt( 1 - m sin t )
-
0
of amplitude phi and modulus m, using the arithmetic -
geometric mean algorithm.
ACCURACY:
Tested at random points with m in [0, 1] and phi as indicated.
Relative error:
arithmetic domain # trials peak rms
IEEE -10,10 200000 7.4e-16 1.0e-16
ellpe: Complete elliptic integral of the second kind
SYNOPSIS:
# double m1, y, ellpe();
$y = ellpe( $m1 );
DESCRIPTION:
Approximates the integral
pi/2
-
| | 2
E(m) = | sqrt( 1 - m sin t ) dt
| |
-
0
Where m = 1 - m1, using the approximation
P(x) - x log x Q(x).
Though there are no singularities, the argument m1 is used
rather than m for compatibility with ellpk().
E(1) = 1; E(0) = pi/2.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC 0, 1 13000 3.1e-17 9.4e-18
IEEE 0, 1 10000 2.1e-16 7.3e-17
ERROR MESSAGES:
message condition value returned
ellpe domain x<0, x>1 0.0
ellpj: Jacobian Elliptic Functions
SYNOPSIS:
# double u, m, sn, cn, dn, phi;
# int ellpj();
($flag, $sn, $cn, $dn, $phi) = ellpj( $u, $m );
DESCRIPTION:
Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m),
and dn(u|m) of parameter m between 0 and 1, and real
argument u.
These functions are periodic, with quarter-period on the
real axis equal to the complete elliptic integral
ellpk(1.0-m).
Relation to incomplete elliptic integral:
If u = ellik(phi,m), then sn(u|m) = sin(phi),
and cn(u|m) = cos(phi). Phi is called the amplitude of u.
Computation is by means of the arithmetic-geometric mean
algorithm, except when m is within 1e-9 of 0 or 1. In the
latter case with m close to 1, the approximation applies
only for phi < pi/2.
ACCURACY:
Tested at random points with u between 0 and 10, m between
0 and 1.
Absolute error (* = relative error):
arithmetic function # trials peak rms
DEC sn 1800 4.5e-16 8.7e-17
IEEE phi 10000 9.2e-16* 1.4e-16*
IEEE sn 50000 4.1e-15 4.6e-16
IEEE cn 40000 3.6e-15 4.4e-16
IEEE dn 10000 1.3e-12 1.8e-14
Peak error observed in consistency check using addition
theorem for sn(u+v) was 4e-16 (absolute). Also tested by
the above relation to the incomplete elliptic integral.
Accuracy deteriorates when u is large.
ellpk: Complete elliptic integral of the first kind
SYNOPSIS:
# double m1, y, ellpk();
$y = ellpk( $m1 );
DESCRIPTION:
Approximates the integral
pi/2
-
| |
| dt
K(m) = | ------------------
| 2
| | sqrt( 1 - m sin t )
-
0
where m = 1 - m1, using the approximation
P(x) - log x Q(x).
The argument m1 is used rather than m so that the logarithmic
singularity at m = 1 will be shifted to the origin; this
preserves maximum accuracy.
K(0) = pi/2.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC 0,1 16000 3.5e-17 1.1e-17
IEEE 0,1 30000 2.5e-16 6.8e-17
ERROR MESSAGES:
message condition value returned
ellpk domain x<0, x>1 0.0
euclid: Rational arithmetic routines
SYNOPSIS:
# typedef struct
# {
# double n; numerator
# double d; denominator
# }fract;
$a = new_fract(3, 4); # a = 3 / 4
$b = new_fract(2, 3); # b = 2 / 3
$c = new_fract();
radd( $a, $b, $c ); # c = b + a
rsub( $a, $b, $c ); # c = b - a
rmul( $a, $b, $c ); # c = b * a
rdiv( $a, $b, $c ); # c = b / a
print $c->{n}, ' ', $c->{d}; # prints numerator and denominator of $c
($gcd, $m_reduced, $n_reduced) = euclid($m, $n);
# returns the greatest common divisor of $m and $n, as well as
# the result of reducing $m and $n by $gcd
Arguments of the routines are pointers to the structures.
The double precision numbers are assumed, without checking,
to be integer valued. Overflow conditions are reported.
exp: Exponential function
SYNOPSIS:
# double x, y, exp();
$y = exp( $x );
DESCRIPTION:
Returns e (2.71828...) raised to the x power.
Range reduction is accomplished by separating the argument
into an integer k and fraction f such that
x k f
e = 2 e.
A Pade' form 1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
of degree 2/3 is used to approximate exp(f) in the basic
interval [-0.5, 0.5].
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC +- 88 50000 2.8e-17 7.0e-18
IEEE +- 708 40000 2.0e-16 5.6e-17
Error amplification in the exponential function can be
a serious matter. The error propagation involves
exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ),
which shows that a 1 lsb error in representing X produces
a relative error of X times 1 lsb in the function.
While the routine gives an accurate result for arguments
that are exactly represented by a double precision
computer number, the result contains amplified roundoff
error for large arguments not exactly represented.
ERROR MESSAGES:
message condition value returned
exp underflow x < MINLOG 0.0
exp overflow x > MAXLOG INFINITY
expxx: exp(x*x)
# double x, y, expxx();
# int sign;
$y = expxx( $x, $sign );
DESCRIPTION:
Computes y = exp(x*x) while suppressing error amplification
that would ordinarily arise from the inexactness of the
exponential argument x*x.
If sign < 0, exp(-x*x) is returned.
If sign > 0, or omitted, exp(x*x) is returned.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE -26.6, 26.6 10^7 3.9e-16 8.9e-17
exp10: Base 10 exponential function (Common antilogarithm)
SYNOPSIS:
# double x, y, exp10();
$y = exp10( $x );
DESCRIPTION:
Returns 10 raised to the x power.
Range reduction is accomplished by expressing the argument
as 10**x = 2**n 10**f, with |f| < 0.5 log10(2).
The Pade' form
1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
is used to approximate 10**f.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE -307,+307 30000 2.2e-16 5.5e-17
Test result from an earlier version (2.1):
DEC -38,+38 70000 3.1e-17 7.0e-18
ERROR MESSAGES:
message condition value returned
exp10 underflow x < -MAXL10 0.0
exp10 overflow x > MAXL10 MAXNUM
DEC arithmetic: MAXL10 = 38.230809449325611792.
IEEE arithmetic: MAXL10 = 308.2547155599167.
exp2: Base 2 exponential function
SYNOPSIS:
# double x, y, exp2();
$y = exp2( $x );
DESCRIPTION:
Returns 2 raised to the x power.
Range reduction is accomplished by separating the argument
into an integer k and fraction f such that
x k f
2 = 2 2.
A Pade' form
1 + 2x P(x**2) / (Q(x**2) - x P(x**2) )
approximates 2**x in the basic range [-0.5, 0.5].
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE -1022,+1024 30000 1.8e-16 5.4e-17
See exp.c for comments on error amplification.
ERROR MESSAGES:
message condition value returned
exp underflow x < -MAXL2 0.0
exp overflow x > MAXL2 MAXNUM
For DEC arithmetic, MAXL2 = 127.
For IEEE arithmetic, MAXL2 = 1024.
ei: Exponential integral
SYNOPSIS:
#double x, y, ei();
$y = ei( $x );
DESCRIPTION:
x
- t
| | e
Ei(x) = -|- --- dt .
| | t
-
-inf
Not defined for x <= 0.
See also expn.c.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,100 50000 8.6e-16 1.3e-16
expn: Exponential integral En
SYNOPSIS:
# int n;
# double x, y, expn();
$y = expn( $n, $x );
DESCRIPTION:
Evaluates the exponential integral
inf.
-
| | -xt
| e
E (x) = | ---- dt.
n | n
| | t
-
1
Both n and x must be nonnegative.
The routine employs either a power series, a continued
fraction, or an asymptotic formula depending on the
relative values of n and x.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC 0, 30 5000 2.0e-16 4.6e-17
IEEE 0, 30 10000 1.7e-15 3.6e-16
fabs: Absolute value
SYNOPSIS:
# double x, y;
$y = fabs( $x );
DESCRIPTION:
Returns the absolute value of the argument.
fac: Factorial function
SYNOPSIS:
# double y, fac();
# int i;
$y = fac( $i );
DESCRIPTION:
Returns factorial of i = 1 * 2 * 3 * ... * i.
fac(0) = 1.0.
Due to machine arithmetic bounds the largest value of
i accepted is 33 in DEC arithmetic or 170 in IEEE
arithmetic. Greater values, or negative ones,
produce an error message and return MAXNUM.
ACCURACY:
For i < 34 the values are simply tabulated, and have
full machine accuracy. If i > 55, fac(i) = gamma(i+1);
see gamma.c.
Relative error:
arithmetic domain peak
IEEE 0, 170 1.4e-15
DEC 0, 33 1.4e-17
fdtr: F distribution
SYNOPSIS:
# int df1, df2;
# double x, y, fdtr();
$y = fdtr( $df1, $df2, $x );
DESCRIPTION:
Returns the area from zero to x under the F density
function (also known as Snedcor's density or the
variance ratio density). This is the density
of x = (u1/df1)/(u2/df2), where u1 and u2 are random
variables having Chi square distributions with df1
and df2 degrees of freedom, respectively.
The incomplete beta integral is used, according to the
formula
P(x) = incbet( df1/2, df2/2, df1*x/(df2 + df1*x) ).
The arguments a and b are greater than zero, and x is
nonnegative.
ACCURACY:
Tested at random points (a,b,x).
x a,b Relative error:
arithmetic domain domain # trials peak rms
IEEE 0,1 0,100 100000 9.8e-15 1.7e-15
IEEE 1,5 0,100 100000 6.5e-15 3.5e-16
IEEE 0,1 1,10000 100000 2.2e-11 3.3e-12
IEEE 1,5 1,10000 100000 1.1e-11 1.7e-13
See also incbet.c.
ERROR MESSAGES:
message condition value returned
fdtr domain a<0, b<0, x<0 0.0
fdtrc: Complemented F distribution
SYNOPSIS:
# int df1, df2;
# double x, y, fdtrc();
$y = fdtrc( $df1, $df2, $x );
DESCRIPTION:
Returns the area from x to infinity under the F density
function (also known as Snedcor's density or the
variance ratio density).
inf.
-
1 | | a-1 b-1
1-P(x) = ------ | t (1-t) dt
B(a,b) | |
-
x
The incomplete beta integral is used, according to the
formula
P(x) = incbet( df2/2, df1/2, df2/(df2 + df1*x) ).
ACCURACY:
Tested at random points (a,b,x) in the indicated intervals.
x a,b Relative error:
arithmetic domain domain # trials peak rms
IEEE 0,1 1,100 100000 3.7e-14 5.9e-16
IEEE 1,5 1,100 100000 8.0e-15 1.6e-15
IEEE 0,1 1,10000 100000 1.8e-11 3.5e-13
IEEE 1,5 1,10000 100000 2.0e-11 3.0e-12
See also incbet.c.
ERROR MESSAGES:
message condition value returned
fdtrc domain a<0, b<0, x<0 0.0
fdtri: Inverse of complemented F distribution
SYNOPSIS:
# int df1, df2;
# double x, p, fdtri();
$x = fdtri( $df1, $df2, $p );
DESCRIPTION:
Finds the F density argument x such that the integral
from x to infinity of the F density is equal to the
given probability p.
This is accomplished using the inverse beta integral
function and the relations
z = incbi( df2/2, df1/2, p )
x = df2 (1-z) / (df1 z).
Note: the following relations hold for the inverse of
the uncomplemented F distribution:
z = incbi( df1/2, df2/2, p )
x = df2 z / (df1 (1-z)).
ACCURACY:
Tested at random points (a,b,p).
a,b Relative error:
arithmetic domain # trials peak rms
For p between .001 and 1:
IEEE 1,100 100000 8.3e-15 4.7e-16
IEEE 1,10000 100000 2.1e-11 1.4e-13
For p between 10^-6 and 10^-3:
IEEE 1,100 50000 1.3e-12 8.4e-15
IEEE 1,10000 50000 3.0e-12 4.8e-14
See also fdtrc.c.
ERROR MESSAGES:
message condition value returned
fdtri domain p <= 0 or p > 1 0.0
v < 1
ceil: ceil
ceil() returns the smallest integer greater than or equal
to x. It truncates toward plus infinity.
SYNOPSIS:
# double x, y, ceil();
$y = ceil( $x );
floor: floor
floor() returns the largest integer less than or equal to x.
It truncates toward minus infinity.
SYNOPSIS:
# double x, y, floor();
$y = floor( $x );
frexp: frexp
frexp() extracts the exponent from x. It returns an integer
power of two to expnt and the significand between 0.5 and 1
to y. Thus x = y * 2**expn.
SYNOPSIS:
# double x, y, frexp();
# int expnt;
($y, $expnt) = frexp( $x );
ldexp: multiplies x by 2**n.
SYNOPSIS:
# double x, y, ldexp();
# int n;
$y = ldexp( $x, $n );
fresnl: Fresnel integral
SYNOPSIS:
# double x, S, C;
# void fresnl();
($flag, $S, $C) = fresnl( $x );
DESCRIPTION:
Evaluates the Fresnel integrals
x
-
| |
C(x) = | cos(pi/2 t**2) dt,
| |
-
0
x
-
| |
S(x) = | sin(pi/2 t**2) dt.
| |
-
0
The integrals are evaluated by a power series for x < 1.
For x >= 1 auxiliary functions f(x) and g(x) are employed
such that
C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 )
S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 )
ACCURACY:
Relative error.
Arithmetic function domain # trials peak rms
IEEE S(x) 0, 10 10000 2.0e-15 3.2e-16
IEEE C(x) 0, 10 10000 1.8e-15 3.3e-16
DEC S(x) 0, 10 6000 2.2e-16 3.9e-17
DEC C(x) 0, 10 5000 2.3e-16 3.9e-17
gamma: Gamma function
SYNOPSIS:
# double x, y, gamma();
# extern int sgngam;
$y = gamma( $x );
DESCRIPTION:
Returns gamma function of the argument. The result is
correctly signed, and the sign (+1 or -1) is also
returned in a global (extern) variable named sgngam.
This variable is also filled in by the logarithmic gamma
function lgam().
Arguments |x| <= 34 are reduced by recurrence and the function
approximated by a rational function of degree 6/7 in the
interval (2,3). Large arguments are handled by Stirling's
formula. Large negative arguments are made positive using
a reflection formula.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -34, 34 10000 1.3e-16 2.5e-17
IEEE -170,-33 20000 2.3e-15 3.3e-16
IEEE -33, 33 20000 9.4e-16 2.2e-16
IEEE 33, 171.6 20000 2.3e-15 3.2e-16
Error for arguments outside the test range will be larger
owing to error amplification by the exponential function.
lgam: Natural logarithm of gamma function
SYNOPSIS:
# double x, y, lgam();
# extern int sgngam;
$y = lgam( $x );
DESCRIPTION:
Returns the base e (2.718...) logarithm of the absolute
value of the gamma function of the argument.
The sign (+1 or -1) of the gamma function is returned in a
global (extern) variable named sgngam.
For arguments greater than 13, the logarithm of the gamma
function is approximated by the logarithmic version of
Stirling's formula using a polynomial approximation of
degree 4. Arguments between -33 and +33 are reduced by
recurrence to the interval [2,3] of a rational approximation.
The cosecant reflection formula is employed for arguments
less than -33.
Arguments greater than MAXLGM return MAXNUM and an error
message. MAXLGM = 2.035093e36 for DEC
arithmetic or 2.556348e305 for IEEE arithmetic.
ACCURACY:
arithmetic domain # trials peak rms
DEC 0, 3 7000 5.2e-17 1.3e-17
DEC 2.718, 2.035e36 5000 3.9e-17 9.9e-18
IEEE 0, 3 28000 5.4e-16 1.1e-16
IEEE 2.718, 2.556e305 40000 3.5e-16 8.3e-17
The error criterion was relative when the function magnitude
was greater than one but absolute when it was less than one.
The following test used the relative error criterion, though
at certain points the relative error could be much higher than
indicated.
IEEE -200, -4 10000 4.8e-16 1.3e-16
gdtr: Gamma distribution function
SYNOPSIS:
# double a, b, x, y, gdtr();
$y = gdtr( $a, $b, $x );
DESCRIPTION:
Returns the integral from zero to x of the gamma probability
density function:
x
b -
a | | b-1 -at
y = ----- | t e dt
- | |
| (b) -
0
The incomplete gamma integral is used, according to the
relation
y = igam( b, ax ).
ACCURACY:
See igam().
ERROR MESSAGES:
message condition value returned
gdtr domain x < 0 0.0
gdtrc: Complemented gamma distribution function
SYNOPSIS:
# double a, b, x, y, gdtrc();
$y = gdtrc( $a, $b, $x );
DESCRIPTION:
Returns the integral from x to infinity of the gamma
probability density function:
inf.
b -
a | | b-1 -at
y = ----- | t e dt
- | |
| (b) -
x
The incomplete gamma integral is used, according to the
relation
y = igamc( b, ax ).
ACCURACY:
See igamc().
ERROR MESSAGES:
message condition value returned
gdtrc domain x < 0 0.0
hyp2f0: Gauss hypergeometric function 2F0
SYNOPSIS:
# double a, b, x, value, *err;
# int type; /* determines what converging factor to use */
($value, $err) = hyp2f0( $a, $b, $x, $type )
hyp2f1: Gauss hypergeometric function 2F1
SYNOPSIS:
# double a, b, c, x, y, hyp2f1();
$y = hyp2f1( $a, $b, $c, $x );
DESCRIPTION:
hyp2f1( a, b, c, x ) = F ( a, b; c; x )
2 1
inf.
- a(a+1)...(a+k) b(b+1)...(b+k) k+1
= 1 + > ----------------------------- x .
- c(c+1)...(c+k) (k+1)!
k = 0
Cases addressed are
Tests and escapes for negative integer a, b, or c
Linear transformation if c - a or c - b negative integer
Special case c = a or c = b
Linear transformation for x near +1
Transformation for x < -0.5
Psi function expansion if x > 0.5 and c - a - b integer
Conditionally, a recurrence on c to make c-a-b > 0
|x| > 1 is rejected.
The parameters a, b, c are considered to be integer
valued if they are within 1.0e-14 of the nearest integer
(1.0e-13 for IEEE arithmetic).
ACCURACY:
Relative error (-1 < x < 1):
arithmetic domain # trials peak rms
IEEE -1,7 230000 1.2e-11 5.2e-14
Several special cases also tested with a, b, c in
the range -7 to 7.
ERROR MESSAGES:
A "partial loss of precision" message is printed if
the internally estimated relative error exceeds 1^-12.
A "singularity" message is printed on overflow or
in cases not addressed (such as x < -1).
hyperg: Confluent hypergeometric function
SYNOPSIS:
# double a, b, x, y, hyperg();
$y = hyperg( $a, $b, $x );
DESCRIPTION:
Computes the confluent hypergeometric function
1 2
a x a(a+1) x
F ( a,b;x ) = 1 + ---- + --------- + ...
1 1 b 1! b(b+1) 2!
Many higher transcendental functions are special cases of
this power series.
As is evident from the formula, b must not be a negative
integer or zero unless a is an integer with 0 >= a > b.
The routine attempts both a direct summation of the series
and an asymptotic expansion. In each case error due to
roundoff, cancellation, and nonconvergence is estimated.
The result with smaller estimated error is returned.
ACCURACY:
Tested at random points (a, b, x), all three variables
ranging from 0 to 30.
Relative error:
arithmetic domain # trials peak rms
DEC 0,30 2000 1.2e-15 1.3e-16
IEEE 0,30 30000 1.8e-14 1.1e-15
Larger errors can be observed when b is near a negative
integer or zero. Certain combinations of arguments yield
serious cancellation error in the power series summation
and also are not in the region of near convergence of the
asymptotic series. An error message is printed if the
self-estimated relative error is greater than 1.0e-12.
i0: Modified Bessel function of order zero
SYNOPSIS:
# double x, y, i0();
$y = i0( $x );
DESCRIPTION:
Returns modified Bessel function of order zero of the
argument.
The function is defined as i0(x) = j0( ix ).
The range is partitioned into the two intervals [0,8] and
(8, infinity). Chebyshev polynomial expansions are employed
in each interval.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC 0,30 6000 8.2e-17 1.9e-17
IEEE 0,30 30000 5.8e-16 1.4e-16
i0e: Modified Bessel function of order zero, exponentially scaled
SYNOPSIS:
# double x, y, i0e();
$y = i0e( $x );
DESCRIPTION:
Returns exponentially scaled modified Bessel function
of order zero of the argument.
The function is defined as i0e(x) = exp(-|x|) j0( ix ).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,30 30000 5.4e-16 1.2e-16
See i0().
i1: Modified Bessel function of order one
SYNOPSIS:
# double x, y, i1();
$y = i1( $x );
DESCRIPTION:
Returns modified Bessel function of order one of the
argument.
The function is defined as i1(x) = -i j1( ix ).
The range is partitioned into the two intervals [0,8] and
(8, infinity). Chebyshev polynomial expansions are employed
in each interval.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC 0, 30 3400 1.2e-16 2.3e-17
IEEE 0, 30 30000 1.9e-15 2.1e-16
i1e: Modified Bessel function of order one, exponentially scaled
SYNOPSIS:
# double x, y, i1e();
$y = i1e( $x );
DESCRIPTION:
Returns exponentially scaled modified Bessel function
of order one of the argument.
The function is defined as i1(x) = -i exp(-|x|) j1( ix ).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0, 30 30000 2.0e-15 2.0e-16
See i1().
igam: Incomplete gamma integral
SYNOPSIS:
# double a, x, y, igam();
$y = igam( $a, $x );
DESCRIPTION:
The function is defined by
x
-
1 | | -t a-1
igam(a,x) = ----- | e t dt.
- | |
| (a) -
0
In this implementation both arguments must be positive.
The integral is evaluated by either a power series or
continued fraction expansion, depending on the relative
values of a and x.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,30 200000 3.6e-14 2.9e-15
IEEE 0,100 300000 9.9e-14 1.5e-14
igamc: Complemented incomplete gamma integral
SYNOPSIS:
# double a, x, y, igamc();
$y = igamc( $a, $x );
DESCRIPTION:
The function is defined by
igamc(a,x) = 1 - igam(a,x)
inf.
-
1 | | -t a-1
= ----- | e t dt.
- | |
| (a) -
x
In this implementation both arguments must be positive.
The integral is evaluated by either a power series or
continued fraction expansion, depending on the relative
values of a and x.
ACCURACY:
Tested at random a, x.
a x Relative error:
arithmetic domain domain # trials peak rms
IEEE 0.5,100 0,100 200000 1.9e-14 1.7e-15
IEEE 0.01,0.5 0,100 200000 1.4e-13 1.6e-15
igami: Inverse of complemented imcomplete gamma integral
SYNOPSIS:
# double a, x, p, igami();
$x = igami( $a, $p );
DESCRIPTION:
Given p, the function finds x such that
igamc( a, x ) = p.
It is valid in the right-hand tail of the distribution, p < 0.5.
Starting with the approximate value
3
x = a t
where
t = 1 - d - ndtri(p) sqrt(d)
and
d = 1/9a,
the routine performs up to 10 Newton iterations to find the
root of igamc(a,x) - p = 0.
ACCURACY:
Tested at random a, p in the intervals indicated.
a p Relative error:
arithmetic domain domain # trials peak rms
IEEE 0.5,100 0,0.5 100000 1.0e-14 1.7e-15
IEEE 0.01,0.5 0,0.5 100000 9.0e-14 3.4e-15
IEEE 0.5,10000 0,0.5 20000 2.3e-13 3.8e-14
incbet: Incomplete beta integral
SYNOPSIS:
# double a, b, x, y, incbet();
$y = incbet( $a, $b, $x );
DESCRIPTION:
Returns incomplete beta integral of the arguments, evaluated
from zero to x. The function is defined as
x
- -
| (a+b) | | a-1 b-1
----------- | t (1-t) dt.
- - | |
| (a) | (b) -
0
The domain of definition is 0 <= x <= 1. In this
implementation a and b are restricted to positive values.
The integral from x to 1 may be obtained by the symmetry
relation
1 - incbet( a, b, x ) = incbet( b, a, 1-x ).
The integral is evaluated by a continued fraction expansion
or, when b*x is small, by a power series.
ACCURACY:
Tested at uniformly distributed random points (a,b,x) with a and b
in "domain" and x between 0 and 1.
Relative error
arithmetic domain # trials peak rms
IEEE 0,5 10000 6.9e-15 4.5e-16
IEEE 0,85 250000 2.2e-13 1.7e-14
IEEE 0,1000 30000 5.3e-12 6.3e-13
IEEE 0,10000 250000 9.3e-11 7.1e-12
IEEE 0,100000 10000 8.7e-10 4.8e-11
Outputs smaller than the IEEE gradual underflow threshold
were excluded from these statistics.
ERROR MESSAGES:
message condition value returned
incbet domain x<0, x>1 0.0
incbet underflow 0.0
incbi: Inverse of imcomplete beta integral
SYNOPSIS:
# double a, b, x, y, incbi();
$x = incbi( $a, $b, $y );
DESCRIPTION:
Given y, the function finds x such that
incbet( a, b, x ) = y .
The routine performs interval halving or Newton iterations to find the
root of incbet(a,b,x) - y = 0.
ACCURACY:
Relative error:
x a,b
arithmetic domain domain # trials peak rms
IEEE 0,1 .5,10000 50000 5.8e-12 1.3e-13
IEEE 0,1 .25,100 100000 1.8e-13 3.9e-15
IEEE 0,1 0,5 50000 1.1e-12 5.5e-15
VAX 0,1 .5,100 25000 3.5e-14 1.1e-15
With a and b constrained to half-integer or integer values:
IEEE 0,1 .5,10000 50000 5.8e-12 1.1e-13
IEEE 0,1 .5,100 100000 1.7e-14 7.9e-16
With a = .5, b constrained to half-integer or integer values:
IEEE 0,1 .5,10000 10000 8.3e-11 1.0e-11
iv: Modified Bessel function of noninteger order
SYNOPSIS:
# double v, x, y, iv();
$y = iv( $v, $x );
DESCRIPTION:
Returns modified Bessel function of order v of the
argument. If x is negative, v must be integer valued.
The function is defined as Iv(x) = Jv( ix ). It is
here computed in terms of the confluent hypergeometric
function, according to the formula
v -x
Iv(x) = (x/2) e hyperg( v+0.5, 2v+1, 2x ) / gamma(v+1)
If v is a negative integer, then v is replaced by -v.
ACCURACY:
Tested at random points (v, x), with v between 0 and
30, x between 0 and 28.
Relative error:
arithmetic domain # trials peak rms
DEC 0,30 2000 3.1e-15 5.4e-16
IEEE 0,30 10000 1.7e-14 2.7e-15
Accuracy is diminished if v is near a negative integer.
See also hyperg.c.
j0: Bessel function of order zero
SYNOPSIS:
# double x, y, j0();
$y = j0( $x );
DESCRIPTION:
Returns Bessel function of order zero of the argument.
The domain is divided into the intervals [0, 5] and
(5, infinity). In the first interval the following rational
approximation is used:
2 2
(w - r ) (w - r ) P (w) / Q (w)
1 2 3 8
2
where w = x and the two r's are zeros of the function.
In the second interval, the Hankel asymptotic expansion
is employed with two rational functions of degree 6/6
and 7/7.
ACCURACY:
Absolute error:
arithmetic domain # trials peak rms
DEC 0, 30 10000 4.4e-17 6.3e-18
IEEE 0, 30 60000 4.2e-16 1.1e-16
y0: Bessel function of the second kind, order zero
SYNOPSIS:
# double x, y, y0();
$y = y0( $x );
DESCRIPTION:
Returns Bessel function of the second kind, of order
zero, of the argument.
The domain is divided into the intervals [0, 5] and
(5, infinity). In the first interval a rational approximation
R(x) is employed to compute
y0(x) = R(x) + 2 * log(x) * j0(x) / PI.
Thus a call to j0() is required.
In the second interval, the Hankel asymptotic expansion
is employed with two rational functions of degree 6/6
and 7/7.
ACCURACY:
Absolute error, when y0(x) < 1; else relative error:
arithmetic domain # trials peak rms
DEC 0, 30 9400 7.0e-17 7.9e-18
IEEE 0, 30 30000 1.3e-15 1.6e-16
j1: Bessel function of order one
SYNOPSIS:
# double x, y, j1();
$y = j1( $x );
DESCRIPTION:
Returns Bessel function of order one of the argument.
The domain is divided into the intervals [0, 8] and
(8, infinity). In the first interval a 24 term Chebyshev
expansion is used. In the second, the asymptotic
trigonometric representation is employed using two
rational functions of degree 5/5.
ACCURACY:
Absolute error:
arithmetic domain # trials peak rms
DEC 0, 30 10000 4.0e-17 1.1e-17
IEEE 0, 30 30000 2.6e-16 1.1e-16
y1: Bessel function of second kind of order one
SYNOPSIS:
# double x, y, y1();
$y = y1( $x );
DESCRIPTION:
Returns Bessel function of the second kind of order one
of the argument.
The domain is divided into the intervals [0, 8] and
(8, infinity). In the first interval a 25 term Chebyshev
expansion is used, and a call to j1() is required.
In the second, the asymptotic trigonometric representation
is employed using two rational functions of degree 5/5.
ACCURACY:
Absolute error:
arithmetic domain # trials peak rms
DEC 0, 30 10000 8.6e-17 1.3e-17
IEEE 0, 30 30000 1.0e-15 1.3e-16
(error criterion relative when |y1| > 1).
jn: Bessel function of integer order
SYNOPSIS:
# int n;
# double x, y, jn();
$y = jn( $n, $x );
DESCRIPTION:
Returns Bessel function of order n, where n is a
(possibly negative) integer.
The ratio of jn(x) to j0(x) is computed by backward
recurrence. First the ratio jn/jn-1 is found by a
continued fraction expansion. Then the recurrence
relating successive orders is applied until j0 or j1 is
reached.
If n = 0 or 1 the routine for j0 or j1 is called
directly.
ACCURACY:
Absolute error:
arithmetic range # trials peak rms
DEC 0, 30 5500 6.9e-17 9.3e-18
IEEE 0, 30 5000 4.4e-16 7.9e-17
Not suitable for large n or x. Use jv() instead.
jv: Bessel function of noninteger order
SYNOPSIS:
# double v, x, y, jv();
$y = jv( $v, $x );
DESCRIPTION:
Returns Bessel function of order v of the argument,
where v is real. Negative x is allowed if v is an integer.
Several expansions are included: the ascending power
series, the Hankel expansion, and two transitional
expansions for large v. If v is not too large, it
is reduced by recurrence to a region of best accuracy.
The transitional expansions give 12D accuracy for v > 500.
ACCURACY:
Results for integer v are indicated by *, where x and v
both vary from -125 to +125. Otherwise,
x ranges from 0 to 125, v ranges as indicated by "domain."
Error criterion is absolute, except relative when |jv()| > 1.
arithmetic v domain x domain # trials peak rms
IEEE 0,125 0,125 100000 4.6e-15 2.2e-16
IEEE -125,0 0,125 40000 5.4e-11 3.7e-13
IEEE 0,500 0,500 20000 4.4e-15 4.0e-16
Integer v:
IEEE -125,125 -125,125 50000 3.5e-15* 1.9e-16*
k0: Modified Bessel function, third kind, order zero
SYNOPSIS:
# double x, y, k0();
$y = k0( $x );
DESCRIPTION:
Returns modified Bessel function of the third kind
of order zero of the argument.
The range is partitioned into the two intervals [0,8] and
(8, infinity). Chebyshev polynomial expansions are employed
in each interval.
ACCURACY:
Tested at 2000 random points between 0 and 8. Peak absolute
error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15.
Relative error:
arithmetic domain # trials peak rms
DEC 0, 30 3100 1.3e-16 2.1e-17
IEEE 0, 30 30000 1.2e-15 1.6e-16
ERROR MESSAGES:
message condition value returned
K0 domain x <= 0 MAXNUM
k0e: Modified Bessel function, third kind, order zero, exponentially scaled
SYNOPSIS:
# double x, y, k0e();
$y = k0e( $x );
DESCRIPTION:
Returns exponentially scaled modified Bessel function
of the third kind of order zero of the argument.
k0e(x) = exp(x) * k0(x).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0, 30 30000 1.4e-15 1.4e-16
See k0().
k1: Modified Bessel function, third kind, order one
SYNOPSIS:
# double x, y, k1();
$y = k1( $x );
DESCRIPTION:
Computes the modified Bessel function of the third kind
of order one of the argument.
The range is partitioned into the two intervals [0,2] and
(2, infinity). Chebyshev polynomial expansions are employed
in each interval.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC 0, 30 3300 8.9e-17 2.2e-17
IEEE 0, 30 30000 1.2e-15 1.6e-16
ERROR MESSAGES:
message condition value returned
k1 domain x <= 0 MAXNUM
k1e: Modified Bessel function, third kind, order one, exponentially scaled
SYNOPSIS:
# double x, y, k1e();
$y = k1e( $x );
DESCRIPTION:
Returns exponentially scaled modified Bessel function
of the third kind of order one of the argument:
k1e(x) = exp(x) * k1(x).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0, 30 30000 7.8e-16 1.2e-16
See k1().
kn: Modified Bessel function, third kind, integer order
SYNOPSIS:
# double x, y, kn();
# int n;
$y = kn( $n, $x );
DESCRIPTION:
Returns modified Bessel function of the third kind
of order n of the argument.
The range is partitioned into the two intervals [0,9.55] and
(9.55, infinity). An ascending power series is used in the
low range, and an asymptotic expansion in the high range.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC 0,30 3000 1.3e-9 5.8e-11
IEEE 0,30 90000 1.8e-8 3.0e-10
Error is high only near the crossover point x = 9.55
between the two expansions used.
log: Natural logarithm
SYNOPSIS:
# double x, y, log();
$y = log( $x );
DESCRIPTION:
Returns the base e (2.718...) logarithm of x.
The argument is separated into its exponent and fractional
parts. If the exponent is between -1 and +1, the logarithm
of the fraction is approximated by
log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
Otherwise, setting z = 2(x-1)/x+1),
log(x) = z + z**3 P(z)/Q(z).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0.5, 2.0 150000 1.44e-16 5.06e-17
IEEE +-MAXNUM 30000 1.20e-16 4.78e-17
DEC 0, 10 170000 1.8e-17 6.3e-18
In the tests over the interval [+-MAXNUM], the logarithms
of the random arguments were uniformly distributed over
[0, MAXLOG].
ERROR MESSAGES:
log singularity: x = 0; returns -INFINITY
log domain: x < 0; returns NAN
log10: Common logarithm
SYNOPSIS:
# double x, y, log10();
$y = log10( $x );
DESCRIPTION:
Returns logarithm to the base 10 of x.
The argument is separated into its exponent and fractional
parts. The logarithm of the fraction is approximated by
log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0.5, 2.0 30000 1.5e-16 5.0e-17
IEEE 0, MAXNUM 30000 1.4e-16 4.8e-17
DEC 1, MAXNUM 50000 2.5e-17 6.0e-18
In the tests over the interval [1, MAXNUM], the logarithms
of the random arguments were uniformly distributed over
[0, MAXLOG].
ERROR MESSAGES:
log10 singularity: x = 0; returns -INFINITY
log10 domain: x < 0; returns NAN
log2: Base 2 logarithm
SYNOPSIS:
# double x, y, log2();
$y = log2( $x );
DESCRIPTION:
Returns the base 2 logarithm of x.
The argument is separated into its exponent and fractional
parts. If the exponent is between -1 and +1, the base e
logarithm of the fraction is approximated by
log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
Otherwise, setting z = 2(x-1)/x+1),
log(x) = z + z**3 P(z)/Q(z).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0.5, 2.0 30000 2.0e-16 5.5e-17
IEEE exp(+-700) 40000 1.3e-16 4.6e-17
In the tests over the interval [exp(+-700)], the logarithms
of the random arguments were uniformly distributed.
ERROR MESSAGES:
log2 singularity: x = 0; returns -INFINITY
log2 domain: x < 0; returns NAN
lrand: Pseudorandom number generator
SYNOPSIS:
long y, lrand();
$y = lrand( );
DESCRIPTION:
Yields a long integer random number.
The three-generator congruential algorithm by Brian
Wichmann and David Hill (BYTE magazine, March, 1987,
pp 127-8) is used. The period, given by them, is
6953607871644.
lsqrt: Integer square root
SYNOPSIS:
long x, y;
long lsqrt();
$y = lsqrt( $x );
DESCRIPTION:
Returns a long integer square root of the long integer
argument. The computation is by binary long division.
The largest possible result is lsqrt(2,147,483,647)
= 46341.
If x < 0, the square root of |x| is returned, and an
error message is printed.
ACCURACY:
An extra, roundoff, bit is computed; hence the result
is the nearest integer to the actual square root.
NOTE: only DEC arithmetic is currently supported.
mtherr: Library common error handling routine
SYNOPSIS:
char *fctnam;
# int code;
# int mtherr();
mtherr( $fctnam, $code );
DESCRIPTION:
This routine may be called to report one of the following
error conditions (in the include file mconf.h).
Mnemonic Value Significance
DOMAIN 1 argument domain error
SING 2 function singularity
OVERFLOW 3 overflow range error
UNDERFLOW 4 underflow range error
TLOSS 5 total loss of precision
PLOSS 6 partial loss of precision
EDOM 33 Unix domain error code
ERANGE 34 Unix range error code
The default version of the file prints the function name,
passed to it by the pointer fctnam, followed by the
error condition. The display is directed to the standard
output device. The routine then returns to the calling
program. Users may wish to modify the program to abort by
calling exit() under severe error conditions such as domain
errors.
Since all error conditions pass control to this function,
the display may be easily changed, eliminated, or directed
to an error logging device.
SEE ALSO: mconf.h
nbdtr: Negative binomial distribution
SYNOPSIS:
# int k, n;
# double p, y, nbdtr();
$y = nbdtr( $k, $n, $p );
DESCRIPTION:
Returns the sum of the terms 0 through k of the negative
binomial distribution:
k
-- ( n+j-1 ) n j
> ( ) p (1-p)
-- ( j )
j=0
In a sequence of Bernoulli trials, this is the probability
that k or fewer failures precede the nth success.
The terms are not computed individually; instead the incomplete
beta integral is employed, according to the formula
y = nbdtr( k, n, p ) = incbet( n, k+1, p ).
The arguments must be positive, with p ranging from 0 to 1.
ACCURACY:
Tested at random points (a,b,p), with p between 0 and 1.
a,b Relative error:
arithmetic domain # trials peak rms
IEEE 0,100 100000 1.7e-13 8.8e-15
See also incbet.c.
nbdtrc: Complemented negative binomial distribution
SYNOPSIS:
# int k, n;
# double p, y, nbdtrc();
$y = nbdtrc( $k, $n, $p );
DESCRIPTION:
Returns the sum of the terms k+1 to infinity of the negative
binomial distribution:
inf
-- ( n+j-1 ) n j
> ( ) p (1-p)
-- ( j )
j=k+1
The terms are not computed individually; instead the incomplete
beta integral is employed, according to the formula
y = nbdtrc( k, n, p ) = incbet( k+1, n, 1-p ).
The arguments must be positive, with p ranging from 0 to 1.
ACCURACY:
Tested at random points (a,b,p), with p between 0 and 1.
a,b Relative error:
arithmetic domain # trials peak rms
IEEE 0,100 100000 1.7e-13 8.8e-15
See also incbet.c.
nbdtrc: Complemented negative binomial distribution
SYNOPSIS:
# int k, n;
# double p, y, nbdtrc();
$y = nbdtrc( $k, $n, $p );
DESCRIPTION:
Returns the sum of the terms k+1 to infinity of the negative
binomial distribution:
inf
-- ( n+j-1 ) n j
> ( ) p (1-p)
-- ( j )
j=k+1
The terms are not computed individually; instead the incomplete
beta integral is employed, according to the formula
y = nbdtrc( k, n, p ) = incbet( k+1, n, 1-p ).
The arguments must be positive, with p ranging from 0 to 1.
ACCURACY:
See incbet.c.
nbdtri: Functional inverse of negative binomial distribution
SYNOPSIS:
# int k, n;
# double p, y, nbdtri();
$p = nbdtri( $k, $n, $y );
DESCRIPTION:
Finds the argument p such that nbdtr(k,n,p) is equal to y.
ACCURACY:
Tested at random points (a,b,y), with y between 0 and 1.
a,b Relative error:
arithmetic domain # trials peak rms
IEEE 0,100 100000 1.5e-14 8.5e-16
See also incbi.c.
ndtr: Normal distribution function
SYNOPSIS:
# double x, y, ndtr();
$y = ndtr( $x );
DESCRIPTION:
Returns the area under the Gaussian probability density
function, integrated from minus infinity to x:
x
-
1 | | 2
ndtr(x) = --------- | exp( - t /2 ) dt
sqrt(2pi) | |
-
-inf.
= ( 1 + erf(z) ) / 2
where z = x/sqrt(2). Computation is via the functions
erf and erfc.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -13,0 8000 2.1e-15 4.8e-16
IEEE -13,0 30000 3.4e-14 6.7e-15
ERROR MESSAGES:
message condition value returned
erfc underflow x > 37.519379347 0.0
erf: Error function
SYNOPSIS:
# double x, y, erf();
$y = erf( $x );
DESCRIPTION:
The integral is
x
-
2 | | 2
erf(x) = -------- | exp( - t ) dt.
sqrt(pi) | |
-
0
The magnitude of x is limited to 9.231948545 for DEC
arithmetic; 1 or -1 is returned outside this range.
For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise
erf(x) = 1 - erfc(x).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC 0,1 14000 4.7e-17 1.5e-17
IEEE 0,1 30000 3.7e-16 1.0e-16
erfc: Complementary error function
SYNOPSIS:
# double x, y, erfc();
$y = erfc( $x );
DESCRIPTION:
1 - erf(x) =
inf.
-
2 | | 2
erfc(x) = -------- | exp( - t ) dt
sqrt(pi) | |
-
x
For small x, erfc(x) = 1 - erf(x); otherwise rational
approximations are computed.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC 0, 9.2319 12000 5.1e-16 1.2e-16
IEEE 0,26.6417 30000 5.7e-14 1.5e-14
ERROR MESSAGES:
message condition value returned
erfc underflow x > 9.231948545 (DEC) 0.0
ndtri: Inverse of Normal distribution function
SYNOPSIS:
# double x, y, ndtri();
$x = ndtri( $y );
DESCRIPTION:
Returns the argument, x, for which the area under the
Gaussian probability density function (integrated from
minus infinity to x) is equal to y.
For small arguments 0 < y < exp(-2), the program computes
z = sqrt( -2.0 * log(y) ); then the approximation is
x = z - log(z)/z - (1/z) P(1/z) / Q(1/z).
There are two rational functions P/Q, one for 0 < y < exp(-32)
and the other for y up to exp(-2). For larger arguments,
w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC 0.125, 1 5500 9.5e-17 2.1e-17
DEC 6e-39, 0.135 3500 5.7e-17 1.3e-17
IEEE 0.125, 1 20000 7.2e-16 1.3e-16
IEEE 3e-308, 0.135 50000 4.6e-16 9.8e-17
ERROR MESSAGES:
message condition value returned
ndtri domain x <= 0 -MAXNUM
ndtri domain x >= 1 MAXNUM
pdtr: Poisson distribution
SYNOPSIS:
# int k;
# double m, y, pdtr();
$y = pdtr( $k, $m );
DESCRIPTION:
Returns the sum of the first k terms of the Poisson
distribution:
k j
-- -m m
> e --
-- j!
j=0
The terms are not summed directly; instead the incomplete
gamma integral is employed, according to the relation
y = pdtr( k, m ) = igamc( k+1, m ).
The arguments must both be positive.
ACCURACY:
See igamc().
pdtrc: Complemented poisson distribution
SYNOPSIS:
# int k;
# double m, y, pdtrc();
$y = pdtrc( $k, $m );
DESCRIPTION:
Returns the sum of the terms k+1 to infinity of the Poisson
distribution:
inf. j
-- -m m
> e --
-- j!
j=k+1
The terms are not summed directly; instead the incomplete
gamma integral is employed, according to the formula
y = pdtrc( k, m ) = igam( k+1, m ).
The arguments must both be positive.
ACCURACY:
See igam.c.
pdtri: Inverse Poisson distribution
SYNOPSIS:
# int k;
# double m, y, pdtr();
$m = pdtri( $k, $y );
DESCRIPTION:
Finds the Poisson variable x such that the integral
from 0 to x of the Poisson density is equal to the
given probability y.
This is accomplished using the inverse gamma integral
function and the relation
m = igami( k+1, y ).
ACCURACY:
See igami.c.
ERROR MESSAGES:
message condition value returned
pdtri domain y < 0 or y >= 1 0.0
k < 0
pow: Power function
SYNOPSIS:
# double x, y, z, pow();
$z = pow( $x, $y );
DESCRIPTION:
Computes x raised to the yth power. Analytically,
x**y = exp( y log(x) ).
Following Cody and Waite, this program uses a lookup table
of 2**-i/16 and pseudo extended precision arithmetic to
obtain an extra three bits of accuracy in both the logarithm
and the exponential.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE -26,26 30000 4.2e-16 7.7e-17
DEC -26,26 60000 4.8e-17 9.1e-18
1/26 < x < 26, with log(x) uniformly distributed.
-26 < y < 26, y uniformly distributed.
IEEE 0,8700 30000 1.5e-14 2.1e-15
0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
ERROR MESSAGES:
message condition value returned
pow overflow x**y > MAXNUM INFINITY
pow underflow x**y < 1/MAXNUM 0.0
pow domain x<0 and y noninteger 0.0
powi: Real raised to integer power
SYNOPSIS:
# double x, y, powi();
# int n;
$y = powi( $x, $n );
DESCRIPTION:
Returns argument x raised to the nth power.
The routine efficiently decomposes n as a sum of powers of
two. The desired power is a product of two-to-the-kth
powers of x. Thus to compute the 32767 power of x requires
28 multiplications instead of 32767 multiplications.
ACCURACY:
Relative error:
arithmetic x domain n domain # trials peak rms
DEC .04,26 -26,26 100000 2.7e-16 4.3e-17
IEEE .04,26 -26,26 50000 2.0e-15 3.8e-16
IEEE 1,2 -1022,1023 50000 8.6e-14 1.6e-14
Returns MAXNUM on overflow, zero on underflow.
psi: Psi (digamma) function
SYNOPSIS:
# double x, y, psi();
$y = psi( $x );
DESCRIPTION:
d -
psi(x) = -- ln | (x)
dx
is the logarithmic derivative of the gamma function.
For integer x,
n-1
-
psi(n) = -EUL + > 1/k.
-
k=1
This formula is used for 0 < n <= 10. If x is negative, it
is transformed to a positive argument by the reflection
formula psi(1-x) = psi(x) + pi cot(pi x).
For general positive x, the argument is made greater than 10
using the recurrence psi(x+1) = psi(x) + 1/x.
Then the following asymptotic expansion is applied:
inf. B
- 2k
psi(x) = log(x) - 1/2x - > -------
- 2k
k=1 2k x
where the B2k are Bernoulli numbers.
ACCURACY:
Relative error (except absolute when |psi| < 1):
arithmetic domain # trials peak rms
DEC 0,30 2500 1.7e-16 2.0e-17
IEEE 0,30 30000 1.3e-15 1.4e-16
IEEE -30,0 40000 1.5e-15 2.2e-16
ERROR MESSAGES:
message condition value returned
psi singularity x integer <=0 MAXNUM
rgamma: Reciprocal gamma function
SYNOPSIS:
# double x, y, rgamma();
$y = rgamma( $x );
DESCRIPTION:
Returns one divided by the gamma function of the argument.
The function is approximated by a Chebyshev expansion in
the interval [0,1]. Range reduction is by recurrence
for arguments between -34.034 and +34.84425627277176174.
1/MAXNUM is returned for positive arguments outside this
range. For arguments less than -34.034 the cosecant
reflection formula is applied; lograrithms are employed
to avoid unnecessary overflow.
The reciprocal gamma function has no singularities,
but overflow and underflow may occur for large arguments.
These conditions return either MAXNUM or 1/MAXNUM with
appropriate sign.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -30,+30 4000 1.2e-16 1.8e-17
IEEE -30,+30 30000 1.1e-15 2.0e-16
For arguments less than -34.034 the peak error is on the
order of 5e-15 (DEC), excepting overflow or underflow.
round: Round double to nearest or even integer valued double
SYNOPSIS:
# double x, y, round();
$y = round( $x );
DESCRIPTION:
Returns the nearest integer to x as a double precision
floating point result. If x ends in 0.5 exactly, the
nearest even integer is chosen.
ACCURACY:
If x is greater than 1/(2*MACHEP), its closest machine
representation is already an integer, so rounding does
not change it.
shichi: Hyperbolic sine and cosine integrals
SYNOPSIS:
# double x, Chi, Shi, shichi();
($flag, $Shi, $Chi) = shichi( $x );
DESCRIPTION:
Approximates the integrals
x
-
| | cosh t - 1
Chi(x) = eul + ln x + | ----------- dt,
| | t
-
0
x
-
| | sinh t
Shi(x) = | ------ dt
| | t
-
0
where eul = 0.57721566490153286061 is Euler's constant.
The integrals are evaluated by power series for x < 8
and by Chebyshev expansions for x between 8 and 88.
For large x, both functions approach exp(x)/2x.
Arguments greater than 88 in magnitude return MAXNUM.
ACCURACY:
Test interval 0 to 88.
Relative error:
arithmetic function # trials peak rms
DEC Shi 3000 9.1e-17
IEEE Shi 30000 6.9e-16 1.6e-16
Absolute error, except relative when |Chi| > 1:
DEC Chi 2500 9.3e-17
IEEE Chi 30000 8.4e-16 1.4e-16
sici: Sine and cosine integrals
SYNOPSIS:
# double x, Ci, Si, sici();
($flag, $Si, $Ci) = sici( $x );
DESCRIPTION:
Evaluates the integrals
x
-
| cos t - 1
Ci(x) = eul + ln x + | --------- dt,
| t
-
0
x
-
| sin t
Si(x) = | ----- dt
| t
-
0
where eul = 0.57721566490153286061 is Euler's constant.
The integrals are approximated by rational functions.
For x > 8 auxiliary functions f(x) and g(x) are employed
such that
Ci(x) = f(x) sin(x) - g(x) cos(x)
Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x)
ACCURACY:
Test interval = [0,50].
Absolute error, except relative when > 1:
arithmetic function # trials peak rms
IEEE Si 30000 4.4e-16 7.3e-17
IEEE Ci 30000 6.9e-16 5.1e-17
DEC Si 5000 4.4e-17 9.0e-18
DEC Ci 5300 7.9e-17 5.2e-18
sin: Circular sine
SYNOPSIS:
# double x, y, sin();
$y = sin( $x );
DESCRIPTION:
Range reduction is into intervals of pi/4. The reduction
error is nearly eliminated by contriving an extended precision
modular arithmetic.
Two polynomial approximating functions are employed.
Between 0 and pi/4 the sine is approximated by
x + x**3 P(x**2).
Between pi/4 and pi/2 the cosine is represented as
1 - x**2 Q(x**2).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC 0, 10 150000 3.0e-17 7.8e-18
IEEE -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17
ERROR MESSAGES:
message condition value returned
sin total loss x > 1.073741824e9 0.0
Partial loss of accuracy begins to occur at x = 2**30
= 1.074e9. The loss is not gradual, but jumps suddenly to
about 1 part in 10e7. Results may be meaningless for
x > 2**49 = 5.6e14. The routine as implemented flags a
TLOSS error for x > 2**30 and returns 0.0.
cos: Circular cosine
SYNOPSIS:
# double x, y, cos();
$y = cos( $x );
DESCRIPTION:
Range reduction is into intervals of pi/4. The reduction
error is nearly eliminated by contriving an extended precision
modular arithmetic.
Two polynomial approximating functions are employed.
Between 0 and pi/4 the cosine is approximated by
1 - x**2 Q(x**2).
Between pi/4 and pi/2 the sine is represented as
x + x**3 P(x**2).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE -1.07e9,+1.07e9 130000 2.1e-16 5.4e-17
DEC 0,+1.07e9 17000 3.0e-17 7.2e-18
sindg: Circular sine of angle in degrees
SYNOPSIS:
# double x, y, sindg();
$y = sindg( $x );
DESCRIPTION:
Range reduction is into intervals of 45 degrees.
Two polynomial approximating functions are employed.
Between 0 and pi/4 the sine is approximated by
x + x**3 P(x**2).
Between pi/4 and pi/2 the cosine is represented as
1 - x**2 P(x**2).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC +-1000 3100 3.3e-17 9.0e-18
IEEE +-1000 30000 2.3e-16 5.6e-17
ERROR MESSAGES:
message condition value returned
sindg total loss x > 8.0e14 (DEC) 0.0
x > 1.0e14 (IEEE)
cosdg: Circular cosine of angle in degrees
SYNOPSIS:
# double x, y, cosdg();
$y = cosdg( $x );
DESCRIPTION:
Range reduction is into intervals of 45 degrees.
Two polynomial approximating functions are employed.
Between 0 and pi/4 the cosine is approximated by
1 - x**2 P(x**2).
Between pi/4 and pi/2 the sine is represented as
x + x**3 P(x**2).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC +-1000 3400 3.5e-17 9.1e-18
IEEE +-1000 30000 2.1e-16 5.7e-17
See also sin().
sinh: Hyperbolic sine
SYNOPSIS:
# double x, y, sinh();
$y = sinh( $x );
DESCRIPTION:
Returns hyperbolic sine of argument in the range MINLOG to
MAXLOG.
The range is partitioned into two segments. If |x| <= 1, a
rational function of the form x + x**3 P(x)/Q(x) is employed.
Otherwise the calculation is sinh(x) = ( exp(x) - exp(-x) )/2.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC +- 88 50000 4.0e-17 7.7e-18
IEEE +-MAXLOG 30000 2.6e-16 5.7e-17
spence: Dilogarithm
SYNOPSIS:
# double x, y, spence();
$y = spence( $x );
DESCRIPTION:
Computes the integral
x
-
| | log t
spence(x) = - | ----- dt
| | t - 1
-
1
for x >= 0. A rational approximation gives the integral in
the interval (0.5, 1.5). Transformation formulas for 1/x
and 1-x are employed outside the basic expansion range.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE 0,4 30000 3.9e-15 5.4e-16
DEC 0,4 3000 2.5e-16 4.5e-17
sqrt: Square root
SYNOPSIS:
# double x, y, sqrt();
$y = sqrt( $x );
DESCRIPTION:
Returns the square root of x.
Range reduction involves isolating the power of two of the
argument and using a polynomial approximation to obtain
a rough value for the square root. Then Heron's iteration
is used three times to converge to an accurate value.
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC 0, 10 60000 2.1e-17 7.9e-18
IEEE 0,1.7e308 30000 1.7e-16 6.3e-17
ERROR MESSAGES:
message condition value returned
sqrt domain x < 0 0.0
stdtr: Student's t distribution
SYNOPSIS:
# double t, stdtr();
short k;
$y = stdtr( $k, $t );
DESCRIPTION:
Computes the integral from minus infinity to t of the Student
t distribution with integer k > 0 degrees of freedom:
t
-
| |
- | 2 -(k+1)/2
| ( (k+1)/2 ) | ( x )
---------------------- | ( 1 + --- ) dx
- | ( k )
sqrt( k pi ) | ( k/2 ) |
| |
-
-inf.
Relation to incomplete beta integral:
1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z )
where
z = k/(k + t**2).
For t < -2, this is the method of computation. For higher t,
a direct method is derived from integration by parts.
Since the function is symmetric about t=0, the area under the
right tail of the density is found by calling the function
with -t instead of t.
ACCURACY:
Tested at random 1 <= k <= 25. The "domain" refers to t.
Relative error:
arithmetic domain # trials peak rms
IEEE -100,-2 50000 5.9e-15 1.4e-15
IEEE -2,100 500000 2.7e-15 4.9e-17
stdtri: Functional inverse of Student's t distribution
SYNOPSIS:
# double p, t, stdtri();
# int k;
$t = stdtri( $k, $p );
DESCRIPTION:
Given probability p, finds the argument t such that stdtr(k,t)
is equal to p.
ACCURACY:
Tested at random 1 <= k <= 100. The "domain" refers to p:
Relative error:
arithmetic domain # trials peak rms
IEEE .001,.999 25000 5.7e-15 8.0e-16
IEEE 10^-6,.001 25000 2.0e-12 2.9e-14
struve: Struve function
SYNOPSIS:
# double v, x, y, struve();
$y = struve( $v, $x );
DESCRIPTION:
Computes the Struve function Hv(x) of order v, argument x.
Negative x is rejected unless v is an integer.
ACCURACY:
Not accurately characterized, but spot checked against tables.
plancki: Integral of Planck's black body radiation formula
SYNOPSIS:
# double lambda, T, y, plancki()
$y = plancki( $lambda, $T );
DESCRIPTION:
Evaluates the definite integral, from wavelength 0 to lambda,
of Planck's radiation formula
-5
c1 lambda
E = ------------------
c2/(lambda T)
e - 1
Physical constants c1 = 3.7417749e-16 and c2 = 0.01438769 are built in
to the function program. They are scaled to provide a result
in watts per square meter. Argument T represents temperature in degrees
Kelvin; lambda is wavelength in meters.
The integral is expressed in closed form, in terms of polylogarithms
(see polylog.c).
The total area under the curve is
(-1/8) (42 zeta(4) - 12 pi^2 zeta(2) + pi^4 ) c1 (T/c2)^4
= (pi^4 / 15) c1 (T/c2)^4
= 5.6705032e-8 T^4
where sigma = 5.6705032e-8 W m^2 K^-4 is the Stefan-Boltzmann constant.
ACCURACY:
The left tail of the function experiences some relative error
amplification in computing the dominant term exp(-c2/(lambda T)).
For the right-hand tail see planckc, below.
Relative error.
The domain refers to lambda T / c2.
arithmetic domain # trials peak rms
IEEE 0.1, 10 50000 7.1e-15 5.4e-16
polylog: polylogarithm function SYNOPSIS:
# double x, y, polylog();
# int n;
$y = polylog( $n, $x );
The polylogarithm of order n is defined by the series
inf k
- x
Li (x) = > --- .
n - n
k=1 k
For x = 1,
inf
- 1
Li (1) = > --- = Riemann zeta function (n) .
n - n
k=1 k
When n = 2, the function is the dilogarithm, related to Spence's integral:
x 1-x
- -
| | -ln(1-t) | | ln t
Li (x) = | -------- dt = | ------ dt = spence(1-x) .
2 | | t | | 1 - t
- -
0 1
ACCURACY:
Relative error:
arithmetic domain n # trials peak rms
IEEE 0, 1 2 50000 6.2e-16 8.0e-17
IEEE 0, 1 3 100000 2.5e-16 6.6e-17
IEEE 0, 1 4 30000 1.7e-16 4.9e-17
IEEE 0, 1 5 30000 5.1e-16 7.8e-17
bernum: Bernoulli numbers
SYNOPSIS:
($num, $den) = bernum( $n);
($num_array, $den_array) = bernum();
DESCRIPTION:
This calculates the Bernoulli numbers, up to 30th order.
If called with an integer argument, the numerator and denominator
of that Bernoulli number is returned; if called with no argument,
two array references representing the numerator and denominators
of the first 30 Bernoulli numbers are returned.
simpson: Simpson's rule to find an integral
SYNOPSIS:
$result = simpson(\&fun, $a, $b, $abs_err, $rel_err, $nmax);
sub fun {
my $x = shift;
return cos($x)*exp($x);
}
DESCRIPTION:
This evaluates the area under the graph of a function,
represented in a subroutine, from $a to $b, using an 8-point
Newton-Cotes formula. The routine divides up the interval into
equal segments, evaluates the integral, then compares that
to the result with double the number of segments. If the two
results agree, to within an absolute error $abs_err or a
relative error $rel_err, the result is returned; otherwise,
the number of segments is doubled again, and the results
compared. This continues until the desired accuracy is attained,
or until the maximum number of iterations $nmax is reached.
vecang: angle between two vectors
SYNOPSIS:
# double p[3], q[3], vecang();
$y = vecang( $p, $q );
DESCRIPTION:
For two vectors p, q, the angle A between them is given by
p.q / (|p| |q|) = cos A .
where "." represents inner product, "|x|" the length of vector x.
If the angle is small, an expression in sin A is preferred.
Set r = q - p. Then
p.q = p.p + p.r ,
|p|^2 = p.p ,
|q|^2 = p.p + 2 p.r + r.r ,
p.p^2 + 2 p.p p.r + p.r^2
cos^2 A = ----------------------------
p.p (p.p + 2 p.r + r.r)
p.p + 2 p.r + p.r^2 / p.p
= --------------------------- ,
p.p + 2 p.r + r.r
sin^2 A = 1 - cos^2 A
r.r - p.r^2 / p.p
= --------------------
p.p + 2 p.r + r.r
= (r.r - p.r^2 / p.p) / q.q .
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE -1, 1 10^6 1.7e-16 4.2e-17
onef2: Hypergeometric function 1F2
SYNOPSIS:
# double a, b, c, x, value;
# double *err;
($value, $err) = onef2( $a, $b, $c, $x)
ACCURACY:
Not accurately characterized, but spot checked against tables.
threef0: Hypergeometric function 3F0
SYNOPSIS:
# double a, b, c, x, value;
# double *err;
($value, $err) = threef0( $a, $b, $c, $x )
ACCURACY:
Not accurately characterized, but spot checked against tables.
yv: Bessel function Yv with noninteger v
SYNOPSIS:
# double v, x;
# double yv( v, x );
$y = yv( $v, $x );
ACCURACY:
Not accurately characterized, but spot checked against tables.
tan: Circular tangent
SYNOPSIS:
# double x, y, tan();
$y = tan( $x );
DESCRIPTION:
Returns the circular tangent of the radian argument x.
Range reduction is modulo pi/4. A rational function
x + x**3 P(x**2)/Q(x**2)
is employed in the basic interval [0, pi/4].
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC +-1.07e9 44000 4.1e-17 1.0e-17
IEEE +-1.07e9 30000 2.9e-16 8.1e-17
ERROR MESSAGES:
message condition value returned
tan total loss x > 1.073741824e9 0.0
cot: Circular cotangent
SYNOPSIS:
# double x, y, cot();
$y = cot( $x );
DESCRIPTION:
Returns the circular cotangent of the radian argument x.
Range reduction is modulo pi/4. A rational function
x + x**3 P(x**2)/Q(x**2)
is employed in the basic interval [0, pi/4].
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
IEEE +-1.07e9 30000 2.9e-16 8.2e-17
ERROR MESSAGES:
message condition value returned
cot total loss x > 1.073741824e9 0.0
cot singularity x = 0 INFINITY
tandg: Circular tangent of argument in degrees
SYNOPSIS:
# double x, y, tandg();
$y = tandg( $x );
DESCRIPTION:
Returns the circular tangent of the argument x in degrees.
Range reduction is modulo pi/4. A rational function
x + x**3 P(x**2)/Q(x**2)
is employed in the basic interval [0, pi/4].
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC 0,10 8000 3.4e-17 1.2e-17
IEEE 0,10 30000 3.2e-16 8.4e-17
ERROR MESSAGES:
message condition value returned
tandg total loss x > 8.0e14 (DEC) 0.0
x > 1.0e14 (IEEE)
tandg singularity x = 180 k + 90 MAXNUM
cotdg: Circular cotangent of argument in degrees
SYNOPSIS:
# double x, y, cotdg();
$y = cotdg( $x );
DESCRIPTION:
Returns the circular cotangent of the argument x in degrees.
Range reduction is modulo pi/4. A rational function
x + x**3 P(x**2)/Q(x**2)
is employed in the basic interval [0, pi/4].
ERROR MESSAGES:
message condition value returned
cotdg total loss x > 8.0e14 (DEC) 0.0
x > 1.0e14 (IEEE)
cotdg singularity x = 180 k MAXNUM
tanh: Hyperbolic tangent
SYNOPSIS:
# double x, y, tanh();
$y = tanh( $x );
DESCRIPTION:
Returns hyperbolic tangent of argument in the range MINLOG to
MAXLOG.
A rational function is used for |x| < 0.625. The form
x + x**3 P(x)/Q(x) of Cody _& Waite is employed.
Otherwise,
tanh(x) = sinh(x)/cosh(x) = 1 - 2/(exp(2x) + 1).
ACCURACY:
Relative error:
arithmetic domain # trials peak rms
DEC -2,2 50000 3.3e-17 6.4e-18
IEEE -2,2 30000 2.5e-16 5.8e-17
unity: Relative error approximations for function arguments near unity.
SYNOPSIS:
# log1p(x) = log(1+x)
$y = log1p( $x );
# expm1(x) = exp(x) - 1
$y = expm1( $x );
# cosm1(x) = cos(x) - 1
$y = cosm1( $x );
yn: Bessel function of second kind of integer order
SYNOPSIS:
# double x, y, yn();
# int n;
$y = yn( $n, $x );
DESCRIPTION:
Returns Bessel function of order n, where n is a
(possibly negative) integer.
The function is evaluated by forward recurrence on
n, starting with values computed by the routines
y0() and y1().
If n = 0 or 1 the routine for y0 or y1 is called
directly.
ACCURACY:
Absolute error, except relative
when y > 1:
arithmetic domain # trials peak rms
DEC 0, 30 2200 2.9e-16 5.3e-17
IEEE 0, 30 30000 3.4e-15 4.3e-16
ERROR MESSAGES:
message condition value returned
yn singularity x = 0 MAXNUM
yn overflow MAXNUM
Spot checked against tables for x, n between 0 and 100.
zeta: Riemann zeta function of two arguments
SYNOPSIS:
# double x, q, y, zeta();
$y = zeta( $x, $q );
DESCRIPTION:
inf.
- -x
zeta(x,q) = > (k+q)
-
k=0
where x > 1 and q is not a negative integer or zero.
The Euler-Maclaurin summation formula is used to obtain
the expansion
n
- -x
zeta(x,q) = > (k+q)
-
k=1
1-x inf. B x(x+1)...(x+2j)
(n+q) 1 - 2j
+ --------- - ------- + > --------------------
x-1 x - x+2j+1
2(n+q) j=1 (2j)! (n+q)
where the B2j are Bernoulli numbers. Note that (see zetac.c)
zeta(x,1) = zetac(x) + 1.
ACCURACY:
REFERENCE:
Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals,
Series, and Products, p. 1073; Academic Press, 1980.
zetac: Riemann zeta function
SYNOPSIS:
# double x, y, zetac();
$y = zetac( $x );
DESCRIPTION:
inf.
- -x
zetac(x) = > k , x > 1,
-
k=2
is related to the Riemann zeta function by
Riemann zeta(x) = zetac(x) + 1.
Extension of the function definition for x < 1 is implemented.
Zero is returned for x > log2(MAXNUM).
An overflow error may occur for large negative x, due to the
gamma function in the reflection formula.
ACCURACY:
Tabulated values have full machine accuracy.
Relative error:
arithmetic domain # trials peak rms
IEEE 1,50 10000 9.8e-16 1.3e-16
DEC 1,50 2000 1.1e-16 1.9e-17
TODO
• Include more operating systems when generating mconf.h.
MAINTAINER
Shlomi Fish, <http://www.shlomifish.org/>, <https://metacpan.org/author/SHLOMIF> .
BUGS
Please report any on the rt.cpan.org interface:
<https://rt.cpan.org/Dist/Display.html?Queue=Math-Cephes>
VERSION CONTROL
This distribution is maintained in this GitHub repository:
<https://github.com/shlomif/Math-Cephes>.
SEE ALSO
For interfaces to programs which can do symbolic manipulation, see PDL, Math::Pari, and
Math::ematica. For a command line interface to the routines of Math::Cephes, see the
included "pmath" script. For a different interface to the fraction and complex number
routines, see Math::Cephes::Fraction and Math::Cephes::Complex. For an interface to some
polynomial routines, see Math::Cephes::Polynomial, and for some matrix routines, see
Math::Cephes::Matrix.
COPYRIGHT
The C code for the Cephes Math Library is Copyright 1984, 1987, 1989, 2002 by Stephen L.
Moshier, and is available at <http://www.netlib.org/cephes/>. Direct inquiries to 30
Frost Street, Cambridge, MA 02140.
The file arrays.c included here to handle passing arrays into and out of C routines comes
from the PGPLOT module of Karl Glazebrook <kgb@zzoepp.aao.gov.au>.
The perl interface is copyright 2000, 2002 by Randy Kobes. This library is free software;
you can redistribute it and/or modify it under the same terms as Perl itself.
Perl interface maintained by Shlomi Fish starting from 2012. All explicit or implicit
copyrights on the changes are disclaimed by him.