Numerical integration/Adaptive Simpson's method: Difference between revisions
Numerical integration/Adaptive Simpson's method (view source)
Revision as of 18:43, 20 May 2023
, 11 months agoAdded Shen before: →{{header|Sidef}}
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<pre>$ gsi adaptive_simpson_task.scm
estimated definite integral of sin(x) for x from 0 to 1: .4596976941317858</pre>
=={{header|Shen}}==
<syntaxhighlight lang="shen">
(define simpson-rule
F A FA B FB ->
(let M (* 0.5 (+ A B))
FM (F M)
QuadVal (* (/ (- B A) 6.0) (+ FA (* 4.0 FM) FB))
[M FM QuadVal]))
(define recursive-simpson
F A FA B FB Tol Whole M FM Depth ->
(let TheLeftStuff (simpson-rule F A FA M FM)
TheRightStuff (simpson-rule F M FM B FB)
(recursive-simpson-united F A FA B FB Tol Whole M FM Depth
TheLeftStuff TheRightStuff)))
(define recursive-simpson-united
F A FA B FB Tol Whole M FM Depth
[LM FLM Left] [RM FRM Right] ->
(let Delta (- (+ Left Right) Whole)
Tol_ (* 0.5 Tol)
(if (or (<= Depth 0)
(= Tol_ Tol)
(<= (abs Delta) (* 15.0 Tol)))
(+ Left Right (/ Delta 15.0))
(+ (recursive-simpson F A FA M FM Tol_
Left LM FLM (- Depth 1))
(recursive-simpson F M FM B FB Tol_
Right RM FRM (- Depth 1))))))
(define quad-asr
F A B Tol Depth ->
(let FA (F A)
FB (F B)
TheWholeStuff (simpson-rule F A FA B FB)
(quad-asr-part-two F A FA B FB Tol Depth TheWholeStuff)))
(define quad-asr-part-two
F A FA B FB Tol Depth [M FM Whole] ->
(recursive-simpson F A FA B FB Tol Whole M FM Depth))
\\ The Shen standard library contains only an unserious
\\ implementation of sin(x), so I might as well toss together
\\ my own! Thus: an uneconomized Maclaurin expansion. Gotten
\\ via "taylor(sin(x),x,0,20);" in Maxima. Using up to x**13
\\ should give an error bound on the order of 1e-12, for
\\ 0 <= x <= 1. That is much better than we need.
(define sin
X -> (compute-sin X (* X X)))
(define compute-sin
X XX ->
(* X (+ 1.0
(* XX (+ (/ -1.0 6.0)
(* XX (+ (/ 1.0 120.0)
(* XX (+ (/ -1.0 5040.0)
(* XX (+ (/ 1.0 362880.0)
(* XX (+ (/ -1.0 39916800.0)
(* XX (/ 1.0 6227020800.0)))))))))))))))
(define abs
\\ Like sin, abs is in the standard library, but it is very
\\ easy to write.
X -> (if (< X 0) (- 0 X) X))
(output "estimate of the definite integral of sin x dx from 0 to 1: ")
(print (quad-asr (lambda x (sin x)) 0.0 1.0 1.0e-9 100))
(nl)
\\ local variables:
\\ mode: indented-text
\\ tab-width: 2
\\ end:
</syntaxhighlight>
{{out}}
<pre>$ shen-scheme script adaptive_simpson_task.shen
estimate of the definite integral of sin x dx from 0 to 1: 0.4596976941318334</pre>
=={{header|Sidef}}==
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