111 lines
2.7 KiB
Factor
111 lines
2.7 KiB
Factor
USING: kernel math math.constants math.functions math.intervals
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math.vectors namespaces sequences ;
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IN: math.analysis
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<PRIVATE
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! http://www.rskey.org/gamma.htm "Lanczos Approximation"
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! n=6: error ~ 3 x 10^-11
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: gamma-g6 5.15 ; inline
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: gamma-p6
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{
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2.50662827563479526904 225.525584619175212544 -268.295973841304927459
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80.9030806934622512966 -5.00757863970517583837 0.0114684895434781459556
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} ; inline
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: gamma-z ( x n -- seq )
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[ + recip ] curry* map 1.0 0 pick set-nth ;
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: (gamma-lanczos6) ( x -- log[gamma[x+1]] )
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#! log(gamma(x+1)
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dup 0.5 + dup gamma-g6 + dup >r log * r> -
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swap 6 gamma-z gamma-p6 v. log + ;
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: gamma-lanczos6 ( x -- gamma[x] )
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#! gamma(x) = gamma(x+1) / x
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dup (gamma-lanczos6) exp swap / ;
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: gammaln-lanczos6 ( x -- gammaln[x] )
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#! log(gamma(x)) = log(gamma(x+1)) - log(x)
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dup (gamma-lanczos6) swap log - ;
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: gamma-neg ( gamma[abs[x]] x -- gamma[x] )
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dup pi * sin * * pi neg swap / ; inline
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PRIVATE>
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: gamma ( x -- y )
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#! gamma(x) = integral 0..inf [ t^(x-1) exp(-t) ] dt
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#! gamma(n+1) = n! for n > 0
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dup 0.0 <= over 1.0 mod zero? and [
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drop 1./0.
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] [
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dup abs gamma-lanczos6 swap dup 0 > [ drop ] [ gamma-neg ] if
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] if ;
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: gammaln ( x -- gamma[x] )
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#! gammaln(x) is an alternative when gamma(x)'s range
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#! varies too widely
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dup 0 < [
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drop 1./0.
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] [
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dup abs gammaln-lanczos6 swap dup 0 > [ drop ] [ gamma-neg ] if
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] if ;
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: nth-root ( n x -- y )
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over 0 = [ "0th root is undefined" throw ] when >r recip r> swap ^ ;
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! Forth Scientific Library Algorithm #1
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!
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! Evaluates the Real Exponential Integral,
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! E1(x) = - Ei(-x) = int_x^\infty exp^{-u}/u du for x > 0
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! using a rational approximation
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!
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! Collected Algorithms from ACM, Volume 1 Algorithms 1-220,
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! 1980; Association for Computing Machinery Inc., New York,
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! ISBN 0-89791-017-6
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!
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! (c) Copyright 1994 Everett F. Carter. Permission is granted by the
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! author to use this software for any application provided the
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! copyright notice is preserved.
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: exp-int ( x -- y )
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#! For real values of x only. Accurate to 7 decimals.
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dup 1.0 < [
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dup 0.00107857 * 0.00976004 -
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over *
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0.05519968 +
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over *
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0.24991055 -
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over *
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0.99999193 +
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over *
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0.57721566 -
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swap log -
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] [
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dup 8.5733287401 +
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over *
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18.059016973 +
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over *
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8.6347608925 +
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over *
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0.2677737343 +
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over
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dup 9.5733223454 +
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over *
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25.6329561486 +
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over *
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21.0996530827 +
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over *
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3.9584969228 +
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nip
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/
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over /
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swap -1.0 * exp
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*
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] if ;
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