This is the SPECIAL-FUNCTIONS Reference Manual, version 1.2.0, generated automatically by Declt version 4.0b2.
Copyright © 2019-2022 Steve Nunez
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The main system appears first, followed by any subsystem dependency.
Special functions in Common Lisp
Mathematical Special Functions
Steve Nunez <steve@symbolics.tech>
(GIT https://github.com/Lisp-Stat/special-functions.git)
MS-PL
Special functions written in common lisp with accuracy equal to Boost, Python and Cephes.
1.2.0
Files are sorted by type and then listed depth-first from the systems components trees.
special-functions (system).
pkgdcl.lisp (file).
special-functions (system).
utils.lisp (file).
special-functions (system).
erf.lisp (file).
special-functions (system).
gamma.lisp (file).
special-functions (system).
lanczos.lisp (file).
special-functions (system).
log-gamma (function).
log-gamma.lisp (file).
special-functions (system).
factorial (function).
Packages are listed by definition order.
common-lisp.
Definitions are sorted by export status, category, package, and then by lexicographic order.
Returns the error function of n
Return the complementary error function erfc(x) = 1-erf(x)
Return the factorial value X! for X <= MAX-FACTORIAL; DOUBLE-FLOAT-POSITIVE-INFINITY if x < 0. X must be an INTEGER.
Return gamma(x), x <= +MAXGAMD+; NAN/RTE if x is a non-positive integer
Return the normalised incomplete gamma functions P and Q, a>=0, x>=0
P(a,x) = integral(exp(-t)*t^(a-1), t=0..x )/gamma(a)
Q(a,x) = integral(exp(-t)*t^(a-1), t=x..Inf)/gamma(a))
dax = x^a*exp(-x)/gamma(a) (prefix factor)
Returns three values:
P is the first value, Q the second, DAX the third, e.g. (values p q dax)
Return the inverse function of erf: (erf (inverse-erf x)) = x, -1 < x < 1
Return the inverse function of erfc: (erfc (inverse-erfc x)) = x, 0 < x < 2
Return the Lanczos sum for x, exp(g). If UNSCALED is non-nil, return the unscaled result
Return the logarithm of gamma(x)
Return the normalised lower incomplete gamma function P(a,x), a>=0, x>=0 P(a,x) = integral(exp(-t)*t^(a-1), t=0..x)/gamma(a)
Return x^a * exp(-x) / gamma(a)
Return the normalised upper incomplete gamma function Q(a,x), a>=0, x>=0 Q(a,x) = integral(exp(-t)*t^(a-1), t=x..Inf)/gamma(a))
Maximum argument for gamma
Table of factorials for integer values up to 100
Convert the (unsigned-byte 64) bit representation into a native double-float
Returns the bit representation of the double-float X as an (unsigned-byte 64)
p/q := exp(x^2)*erfc(x), 1<=x<=128
Return 1/gamma(x) for |x| < 0.03125
Return gamma(x), |x| <= 13, x negative integer produces div by 0
Partial derivative with respect to x of the incomplete gamma function
Return value of inverse error function: erf_inv(p) if p <= 0.5, erfc_inv(q) otherwise
Temme/Gautschi code for P(a,x), dax = x^a*exp(-x)/gamma(a+1) Returns (values p q)
Incomplete gamma functions for large A and A near X
Continued fraction for Q(a,x)
Calculates normalised Q when a is a half-integer for a < min(30, x+1)
Return Q(a,x) when A is an integer, A < min(30,x+1)
Temme/Gautschi code for Q(a,x) when x < 1
Ramanujan’s original approximation of n!
Return x^a * exp(-x) / gamma(a)
Return (z^a)(e^-z)/gamma(a), the power term prefix, using Lanczos summation Most of the error occurs in this function
Modification of Ramanujan’s approximation of n! by Sidney A. Morris
Return sign(gamma(x)), invalid for 0 or negative integer
Returns (sin (* pi x))
Return gamma(x) for x > 13
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