Math-Cephes
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'radian' => '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.
',
'igamc' => '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.
',
'lgam' => '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.
',
'nbdtri' => '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.
',
'yn' => '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.
',
'igami' => '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.
Starting with the approximate value
3
then
w = r cos y + i r sin y.
',
'ellpe' => '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.
',
'chdtr' => '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.
',
'zetac' => 'zetac: Riemann zeta function
SYNOPSIS:
# double x, y, zetac();
$y = zetac( $x );
DESCRIPTION:
inf.
- -x
zetac(x) = > k , x > 1,
-
k=2
is related to the Riemann zeta function by
Riemann zeta(x) = zetac(x) + 1.
Extension of the function definition for x < 1 is implemented.
Zero is returned for x > log2(MAXNUM).
An overflow error may occur for large negative x, due to the
gamma function in the reflection formula.
',
'ellpj' => 'ellpj: Jacobian Elliptic Functions
SYNOPSIS:
# double u, m, sn, cn, dn, phi;
# int ellpj();
($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.
',
'jn' => '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.
',
'ellpk' => '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.
',
'chdtrc' => '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.
',
'beta' => '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.
',
'ceil' => '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 );
',
'spence' => '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.
',
'chdtri' => '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 );
',
'jv' => '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.
',
'btdtr' => '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
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
',
'zeta' => '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.
',
'pow' => '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.
',
'kn' => '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.
',
'cabs' => 'cabs: Complex absolute value
SYNOPSIS:
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.
',
'i0e' => '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 ).
',
'floor' => '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 );
',
'struve' => '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.
',
'plancki' => 'plancki: Integral of Planck 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 the Planck 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.
',
'polylog' => 'polylog: polylogarithm function
SYNOPSIS:
# double x, y, polylog();
# int n;
$y = polylog( $n, $x );
The polylogarithm of order n is defined by the series
inf k
- x
Li (x) = > --- .
n - n
k=1 k
For x = 1,
inf
- 1
Li (1) = > --- = Riemann zeta function (n) .
n - n
k=1 k
When n = 2, the function is the dilogarithm, related to the Spence integral:
x 1-x
- -
| | -ln(1-t) | | ln t
Li (x) = | -------- dt = | ------ dt = spence(1-x) .
2 | | t | | 1 - t
- -
0 1
',
'bernum' => 'bernum: Bernoulli numbers
SYNOPSIS:
($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.
',
y = igamc( b, ax ).
',
'incbet' => '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.
',
'nbdtr' => '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.
',
'fabs' => 'fabs: Absolute value
SYNOPSIS:
# double x, y;
$y = fabs( $x );
DESCRIPTION:
Returns the absolute value of the argument.
',
'powi' => '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.
',
'i1e' => '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 ).
',
'exp2' => 'exp2: Base 2 exponential function
SYNOPSIS:
# double x, y, exp2();
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.
',
'tandg' => '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].
',
'cosdg' => '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).
',
'fdtr' => '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.
',
'rgamma' => 'rgamma: Reciprocal gamma function
SYNOPSIS:
# double x, y, rgamma();
$y = rgamma( $x );
DESCRIPTION:
Returns one divided by the gamma function of the argument.
The function is approximated by a Chebyshev expansion in
the interval [0,1]. Range reduction is by recurrence
for arguments between -34.034 and +34.84425627277176174.
1/MAXNUM is returned for positive arguments outside this
range. For arguments less than -34.034 the cosecant
reflection formula is applied; lograrithms are employed
to avoid unnecessary overflow.
The reciprocal gamma function has no singularities,
but overflow and underflow may occur for large arguments.
These conditions return either MAXNUM or 1/MAXNUM with
appropriate sign.
',
'shichi' => 'shichi: Hyperbolic sine and cosine integrals
SYNOPSIS:
# double x, Chi, Shi, shichi();
($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.
',
'ndtr' => '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
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.
',
'exp' => 'exp: Exponential function
SYNOPSIS:
# double x, y, exp();
$y = exp( $x );
DESCRIPTION:
Returns e (2.71828...) raised to the x power.
Range reduction is accomplished by separating the argument
into an integer k and fraction f such that
x k f
e = 2 e.
A Pade\' form 1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
of degree 2/3 is used to approximate exp(f) in the basic
interval [-0.5, 0.5].
',
'threef0' => 'threef0: Hypergeometric function 3F0
SYNOPSIS:
# double a, b, c, x, value;
# double *err;
($value, $err) = threef0( $a, $b, $c, $x )
',
'hyperg' => '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.
',
'log2' => 'log2: Base 2 logarithm
SYNOPSIS:
# double x, y, log2();
$y = log2( $x );
DESCRIPTION:
Returns the base 2 logarithm of x.
The argument is separated into its exponent and fractional
parts. If the exponent is between -1 and +1, the base e
logarithm of the fraction is approximated by
log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x).
Otherwise, setting z = 2(x-1)/x+1),
log(x) = z + z**3 P(z)/Q(z).
',
'airy' => 'airy: Airy function
SYNOPSIS:
# double x, ai, aiprime, bi, biprime;
# int airy();
($flag, $ai, $aiprime, $bi, $biprime) = airy( $x );
DESCRIPTION:
Solution of the differential equation
y"(x) = xy.
The function returns the two independent solutions Ai, Bi
and their first derivatives Ai\'(x), Bi\'(x).
Evaluation is by power series summation for small x,
by rational minimax approximations for large x.
',
'onef2' => 'onef2: Hypergeometric function 1F2
SYNOPSIS:
# double a, b, c, x, value;
# double *err;
($value, $err) = onef2( $a, $b, $c, $x)
',
'ei' => '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.
',
'expn' => '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.
',
'dawsn' => '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.
',
'clog' => 'clog: Complex natural logarithm
SYNOPSIS:
# void clog();
# cmplx z, w;
$z = cmplx(2, 3); # $z = 2 + 3 i
$w = $z->clog;
print $w->{r}, \' \', $w->{i}; # prints real and imaginary parts of $w
print $w->as_string; # prints $w as Re(w) + i Im(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.
',
'acos' => 'acos: Inverse circular cosine
SYNOPSIS:
# double x, y, acos();
$y = acos( $x );
DESCRIPTION:
',
'fresnl' => 'fresnl: Fresnel integral
SYNOPSIS:
# double x, S, C;
# void fresnl();
($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 )
',
'psi' => '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.
',
'csinh' => 'csinh: Complex hyperbolic sine
SYNOPSIS:
# void csinh();
# cmplx z, w;
$z = cmplx(2, 3); # z = 2 + 3 i
$w = $z->csinh;
print $w->{r}, " ", $w->{i}; # prints real and imaginary parts of $w
print $w->as_string; # prints $w as Re(w) + i Im(w)
DESCRIPTION:
csinh z = (cexp(z) - cexp(-z))/2
= sinh x * cos y + i cosh x * sin y .
',
'casinh' => 'casinh: Complex inverse hyperbolic sine
SYNOPSIS:
# void casinh();
# cmplx z, w;
$z = cmplx(2, 3); # $z = 2 + 3 i
$w = $z->casinh;
print $w->{r}, " ", $w->{i}; # prints real and imaginary parts of $w
print $w->as_string; # prints $w as Re(w) + i Im(w)
DESCRIPTION:
casinh z = -i casin iz .
',
'ccosh' => 'ccosh: Complex hyperbolic cosine
SYNOPSIS:
# void ccosh();
# cmplx z, w;
$z = cmplx(2, 3); # $z = 2 + 3 i
'k0e' => '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).
',
'erfc' => '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.
',
'gamma' => '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.
',
'incbi' => '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.
',
'round' => 'round: Round double to nearest or even integer valued double
SYNOPSIS:
# double x, y, round();
$y = round( $x );
DESCRIPTION:
Returns the nearest integer to x as a double precision
floating point result. If x ends in 0.5 exactly, the
nearest even integer is chosen.
',
'drand' => 'drand: Pseudorandom number generator
SYNOPSIS:
# double y, drand();
($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
SYNOPSIS:
# void casin();
# cmplx z, w;
$z = cmplx(2, 3); # $z = 2 + 3 i
$w = $z->casin;
print $w->{r}, \' \', $w->{i}; # prints real and imaginary parts of $w
print $w->as_string; # prints $w as Re(w) + i Im(w)
DESCRIPTION:
Inverse complex sine:
2
w = -i clog( iz + csqrt( 1 - z ) ).
',
'acosh' => '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) ).
',
'bdtrc' => '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.
',
'gdtr' => '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 ).
',
'lrand' => '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.
',
'sinh' => '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.
',
'fdtrc' => '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) ).
',
'bdtri' => 'bdtri: Inverse binomial distribution
SYNOPSIS:
# int k, n;
# double p, y, bdtri();
$p = bdtri( $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 ).
',
'atan2' => '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).
',
'lsqrt' => '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.
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.
',
'j1' => '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.
',
'ldexp' => 'ldexp: multiplies x by 2**n.
SYNOPSIS:
# double x, y, ldexp();
# int n;
$y = ldexp( $x, $n );
',
'pdtrc' => '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.
',
'igam' => '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.
',
'machconst' => '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)
series near these points.
',
'cot' => '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].
',
'asin' => '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 ) ).
',
'bdtr' => '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.
',
'cosh' => '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.
',
'sindg' => '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).
',
'k0' => '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.
',
'k1' => '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.
',
'nbdtrc' => '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.
',
'iv' => '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.
'
);
}
sub get_descs {
$topics{'trigs'} = "Help is available on the following functions: \n\n";
foreach (sort qw(asin acos atan atan2 sin cos tan cot hypot
tandg cotdg sindg cosdg radian unity)) {
(my $desc = $topics{$_}) =~ s!^(.*?\n).*!$1!s;
$topics{'trigs'} .= $desc;
}
$topics{'hypers'} = "Help is available on the following functions: \n\n";
foreach (sort qw(acosh asinh atanh sinh cosh tanh) ) {
(my $desc = $topics{$_}) =~ s!^(.*?\n).*!$1!s;
$topics{'hypers'} .= $desc;
}
$topics{'explog'} = "Help is available on the following functions: \n\n";
foreach (sort qw(unity exp exp10 exp2 log log10 log2 expxx)) {
(my $desc = $topics{$_}) =~ s!^(.*?\n).*!$1!s;
$topics{'explog'} .= $desc;
}
$topics{'complex'} = "Help is available on the following functions: \n\n";
foreach (sort qw(clog cexp csin ccos ctan ccot casin cabs csqrt
cacos catan cadd csub cmul cdiv cmov cneg cmplx
csinh ccosh ctanh cpow casinh cacosh catanh) ) {
(my $desc = $topics{$_}) =~ s!^(.*?\n).*!$1!s;
$topics{'complex'} .= $desc unless $desc =~ /^\s*$/;;
}
$topics{'utils'} = "Help is available on the following functions: \n\n";
foreach (sort qw(ceil floor frexp ldexp fabs fac cbrt
round sqrt lrand pow powi drand lsqrt ) ) {
(my $desc = $topics{$_}) =~ s!^(.*?\n).*!$1!s;
$topics{'utils'} .= $desc;
}
$topics{'bessels'} = "Help is available on the following functions: \n\n";
foreach (sort qw( i0 i0e i1 i1e iv j0 j1 jn jv k0 k1 kn yn
yv k0e k1e y0 y1) ) {
(my $desc = $topics{$_}) =~ s!^(.*?\n).*!$1!s;
$topics{'bessels'} .= $desc;
}
$topics{'dists'} = "Help is available on the following functions: \n\n";
foreach (sort qw(bdtr bdtrc bdtri btdtr chdtr chdtrc chdtri
fdtr fdtrc fdtri gdtr gdtrc nbdtr nbdtrc nbdtri
ndtr ndtri pdtr pdtrc pdtri stdtr stdtri) ) {
(my $desc = $topics{$_}) =~ s!^(.*?\n).*!$1!s;
$topics{'dists'} .= $desc;
}
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