115 lines
2.9 KiB
Text
115 lines
2.9 KiB
Text
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\ elip Complete Elliptic Integral ACM Algorithm #149
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\ Forth Scientific Library Algorithm #2
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\ Evaluates the Complete Elliptic Integral,
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\ Elip[a, b] = int_0^{\pi/2} 1/Sqrt{a^2 cos^2(t) + b^2 sin^2(t)} dt
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\ This function can be used to evaluate the complete elliptic integral
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\ of the first kind, by using the relation K[m] = a Elip[a,b], m = 1 - b^2/a^2
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\ This code conforms with ANS requiring:
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\ 1. The Floating-Point word set
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\ 2. The immediate word '%' which takes the next token
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\ and converts it to a floating-point literal
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\ 3. The FCONSTANT PI (3.1415926536...)
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\
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\ Both a recursive form and an iterative form are given, but because of the
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\ large stack consumption the recursive form is probably not of much
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\ practical use.
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\ Caution: this code can potentially go into an endless loop
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\ for certain values of the parameters.
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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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\ (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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CR .( ELIP V1.2 21 September 1994 EFC )
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: elip1 ( --, f: a b -- elip[a,b] ) \ recursive form
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FOVER FOVER FOVER F- FABS
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FSWAP % 1.0e-8 F*
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F< IF
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FDROP
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pi % 2.0 F/ FSWAP F/
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ELSE
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FOVER FOVER F+ % 2.0 F/
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FROT FROT F* FSQRT
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RECURSE
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THEN
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;
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: elip2 ( --, f: a b -- elip[a,b] ) \ nonrecursive version
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BEGIN
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FOVER FOVER F+ % 2.0 F/
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FROT FROT F* FSQRT
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FOVER FOVER FOVER F- FABS
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FSWAP % 1.0e-8 F*
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F< UNTIL
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FDROP
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pi % 2.0 F/ FSWAP F/
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;
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\ test driver, calculates the complete elliptic integral of the first
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\ kind (K(m)) using the relation: K[m] = a Elip[a,b], m = 1 - b^2/a^2
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\ compare with Abramowitz & Stegun, Handbook of Mathematical Functions,
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\ Table 17.1
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: elip_test ( -- )
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CR
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." m K(m) exact a Elip1[a,b] a Elip2[a,b] " CR
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." 0.0 1.57079633 "
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% 1000.0 % 1000.0 elip1 % 1000.0 F* F. ." "
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% 1000.0 % 1000.0 elip2 % 1000.0 F* F. CR
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." 0.44 1.80632756 "
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% 400.0 % 299.33259 elip1 % 400.0 F* F. ." "
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% 400.0 % 299.33259 elip2 % 400.0 F* F. CR
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." 0.75 2.15651565 "
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% 1000.0 % 500.0 elip1 % 1000.0 F* F. ." "
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% 1000.0 % 500.0 elip2 % 1000.0 F* F. CR
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." 0.96 3.01611249 "
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% 500.0 % 100.0 elip1 % 500.0 F* F. ." "
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% 500.0 % 100.0 elip2 % 500.0 F* F. CR
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;
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# The Code
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~~~
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:elip1 (-,f:ab-elip[a,b])
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f:over f:over f:over f:- f:abs
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f:swap .1.0e-8 f:*
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f:lt? [ f:drop
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f:PI .2.0 f:/ f:swap f:/
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]
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[ f:over f:over f:+ .2.0 f:/
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f:rot f:rot f:* f:sqrt
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elip1
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] choose
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;
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~~~
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# Test
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```
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.1000 .1000 elip1 .1000 f:* f:put nl
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```
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