Cephes Mathematical Library

A common lisp CFFI wrapper for the SciPy version of Cephes special functions.

For many years Cephes was considered the gold standard in cross-platform mathematical libraries, superior to the libm distributions of the time (early 1990's). Since then C99 and later ANSI C specifications have closed the gap, but there are still certain special and statistical functions available in Cephes that are not in the standard C libraries.

The SciPy version differs from standard Cephes in that it has some additional functions, improvements in accuracy, and is better documented. Only double-float versions are provided.

Installation

The ASDF file will automatically build the shared library as part of the load operation. If you need to build on a system other than MS Windows or UNIX, you will need to modify the make file to account for the linker command to create a shared library.

As delivered, the Makefile is set-up to work on MS Windows or UNIX, and you can build it manually like so:

cd scipy-cephes && make

If you build this on another platform, please drop a note into a Cephes repository issue with the build instructions so we can update the Makefile and system definition.

The init.lisp file, where CFFI loads the library, should work out of the box if libmd on the path somewhere, regardless of platform. If in doubt, place a copy in the same directory as this README.

Documentation

If you know your way around special functions, the table below should suffice to get started. For a more complete description, see the doc strings in cephes.lisp. Finally, the C source code itself, referenced in cephes.lisp, is thorough and complete from a mathematical perspective.

You can also use the scipy.special online documentation.

The API

There is no overlap between the wrapped Cephes functions and the Common Lisp numerical tower. All functions are in the cephes package.

Exported functions are:

function	description
airy	Airy function
bdtr	Binomial distribution
bdtrc	Complement of binomial distribution
bdtri	Inverse binomial distribution
besselpoly	Weighted integral of the Bessel function of the first kind
beta	Beta function
lbeta	Natural log of beta
btdtr	incomplete beta integral
cbrt	cube root
chdtr	Chi-square distribution
chdtrc	Complemented Chi-square distribution
chdtri	Inverse of complemented Chi-square distribution
dawsn	Dawson's Integral
ellik	Incomplete elliptic integral of the first kind
ellie	Incomplete elliptic integral of the second kind
ellpk	Complete elliptic integral of the first kind
ellpe	Complete elliptic integral of the second kind
jacobian-elliptic	jacobian Elliptic Functions
exp2	Base 2 exponential function
exp10	Base 10 exponential function (Common antilogarithm)
expn	Exponential integral
fdtr	F distribution
fdtrc	Complemented F distribution
fdtri	Inverse of F distribution
fresnl	Fresnel integral
gamma	Gamma function
log-gamma	Natural logarithm of Gamma function
gdtr	Gamma distribution function
gdtrc	Complemented Gamma distribution function
gdtri	Inverse Gamma distribution function (?) - not documented in src
hyp2f1	Gauss hypergeometric function
hyperg	Confluent hypergeometric function
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
igam	Regularized lower incomplete gamma function
igamc	Regularized upper incomplete gamma function
igami	Inverse of the lower incomplete gamma function
igamci	Inverse of the upper incomplete gamma function
incbet	Incomplete beta integral
incbi	Inverse of incomplete beta integral
iv	Modified Bessel function of noninteger order
j0	Bessel function of order zero
y0	Bessel function of the second kind, order zero
j1	Bessel function of order one
y1	Bessel function of second kind of order one
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 of the third kind of order one
k1e	Modified Bessel function of the third kind of order one, exponentially scaled
kn	Modified Bessel function, third kind, integer order
nbdtr	Negative binomial distribution
nbdtrc	Complemented negative binomial distribution
nbdtri	Inverse complemented negative binomial distribution
ndtr	Normal distribution function
log-ndtr	Log of the normal distribution function
erf	Error function
erfc	Complementary error function
erfinv	Inverse of the error function
erfcinv	Inverse of the complementary error function
ndtri	Inverse of Normal distribution function
pdtr	Poisson distribution
pdtrc	Complemented poisson distribution
pdtri	Inverse Poisson distribution
poch	Pochhammer symbol (a)_m = gamma(a + m) / gamma(a)
psi	Psi (digamma) function
rgamma	Reciprocal Gamma function
shichi	Hyperbolic sine and cosine integrals
sici	Sine and cosine integrals
sindg	Circular sine of angle in degrees
cosdg	Circular cosine of angle in degrees
sinpi	Compute sin(pi * x)
cospi	Compute cos(pi * x)
spence	Dilogarithm
stdtr	Student's t distribution
stdtri	Functional inverse of Student's t distribution
yv	Bessel function of noninteger order
tandg	Circular tangent of angle in degrees
cotdg	Circular cotangent of argument in degrees
log1p	log(1 + x)
log1pmx	log(1 + x) - x
expm1	exp(x) - 1
cosm1	cos(x) - 1
lgam1p	lgam(x + 1)
yn	Bessel function of second kind of integer order
zeta	Riemann zeta function of two arguments
zetac	Riemann zeta function
owens-t	Owen's T-Function

Contributing

When contributing to this repository, please first discuss major changes to the existing code you wish to make via a github issue. Minor changes and major additions are welcome. Please write good commit messages.

License

CEPHES.CL is available under the Microsoft Public License.