## cephes.cl

2022-11-07

No Description

# 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.