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;;; I, Vytautas Jancauskas, place this code in the public domain (2010).
(require 'common-list-functions) ; butlast
(require 'combinatorics) ; for menon and factorial
;;; ***************************************************************************
;;; Lagrange interpolation
;;; ***************************************************************************
(define (interp:coeff x xj xs)
(cond ((null? xs) 1)
(else (app* $1*$2
(app* $1/$2
(app* $1-$2 x (car xs))
(app* $1-$2 xj (car xs)))
(interp:coeff x xj (cdr xs))))))
(define (interp:only-xs points j index)
;; From a given list of points remove the one with index j and only
;; return the car of each point.
(cond ((null? points) '())
((= j index) (interp:only-xs (cdr points) j (+ index 1)))
(else (cons (caar points)
(interp:only-xs (cdr points) j (+ index 1))))))
(define (interp:interp1 all-points points x index)
(cond ((null? points) 0)
(else (app* $1*$2+$3
(cadar points)
(interp:coeff x (caar points)
(interp:only-xs all-points index 0))
(interp:interp1 all-points (cdr points) x (+ index 1))))))
;;; **************************************************************************
;;; Newtons method for polynomial interpolation
;;; **************************************************************************
;;; Return a list without it's last element.
(define (interp:all-but-last lst) (butlast lst 1))
;;; Return a list without it's first element.
(define interp:all-but-first cdr)
(define (interp:last lst) (car (last-pair lst)))
(define (interp:divided-differences points)
;; The recursive divided differences function from the definition of
;; Newton's method:
;;
;; f[x(k - j), x(k - j + 1), ..., x(k)] =
;;
;; f[x(k - j + 1), ..., x(k)] - f[x(k - j), ..., x(k - 1)]
;; -------------------------------------------------------
;; x(k) - x(k - j)
;;
(let ((identical (interp:how-many-identical points)))
(if (> identical 0)
(app* $1/$2
(list-ref (car points) identical)
(factorial (- identical 1)))
(app* $1/$2
(app* $1-$2
(interp:divided-differences (interp:all-but-first points))
(interp:divided-differences (interp:all-but-last points)))
(app* $1-$2 (car (interp:last points)) (caar points))))))
(set! interp:divided-differences (memon interp:divided-differences))
(define (interp:newton-multiply var points)
(if (null? points)
1
(app* $1*$2
(app* $1-$2 var (caar points))
(interp:newton-multiply var (cdr points)))))
(define (interp:newton-interpolation var points)
(if (null? points)
0
(app* $1*$2+$3
(interp:divided-differences points)
(interp:newton-multiply var (cdr points))
(interp:newton-interpolation var (cdr points)))))
;;; ***************************************************************************
;;; Hermite interpolation
;;; ***************************************************************************
(define (interp:duplicate points)
;; Duplicate each point in the point list n times.
(define (simple-duplicate point n)
(if (= n 0)
'()
(cons point (simple-duplicate point (- n 1)))))
(if (null? points)
'()
(append (simple-duplicate (car points) (- (length (car points)) 1))
(interp:duplicate (cdr points)))))
(define (interp:how-many-identical points)
;; If points is a list of length n and contains n identical points then
;; return n, otherwise return 0.
(define (all-identical points point)
(cond ((null? points) #t)
((not (equal? (car point) (caar points))) #f)
(else (all-identical (cdr points) point))))
(let ((n (length points)))
(if (and (> n 0) (all-identical points (car points)))
n
0)))
;;; ***************************************************************************
;;; Neville algorithm
;;; ***************************************************************************
(define (interp:neville var points)
(if (= (length points) 1)
(cadar points)
(app*
$1/$2
(app* $1+$2
(app* $1*$2
(app* $1-$2
var
(car (interp:last points)))
(interp:neville var (interp:all-but-last points)))
(app* $1*$2
(app* $1-$2
(caar points)
var)
(interp:neville var (interp:all-but-first points))))
(app* $1-$2 (caar points) (car (interp:last points))))))
;;; ***************************************************************************
;;; Taylor polynomial
;;; ***************************************************************************
(define (interp:taylor expr a n)
(define $1-a (app* $1-$2 cidentity a))
(define (taylor expr $1-a^i i)
(cond ((> i n)
0)
(else
(app* $1*$2+$3
(app* $1/$2 (app* expr a) (factorial i))
$1-a^i
(taylor (diff expr $1) (app* $1*$2 $1-a^i $1-a) (+ i 1))))))
(taylor expr 1 0))
;;; ***************************************************************************
;;; Main procedure and validity checking
;;; ***************************************************************************
(define (interp:overdefined? points)
;; Returns #t if points contains multiple definitions for the same argument,
;; returns #f otherwise.
(define (overdefined? points xs)
(cond ((null? points) #f)
((member (caar points) xs) #t)
(else (overdefined? (cdr points) (cons (caar points) xs)))))
(overdefined? points '()))
(define (interp:preprocess args)
;; Preprocess a set of points given by the interp built-in
(if (matrix? (car args))
(car args)
args))
(define (interp:interp var points method)
(let ((points (interp:preprocess points)))
(cond ((interp:overdefined? points)
(math:error 'multiply-defined-points points))
((null? points)
(math:error 'point-list-is-empty))
(else
(cond ((eq? method 'lagrange)
(interp:interp1 points points var 0))
((eq? method 'newton)
(let ((dup-points (interp:duplicate points)))
(interp:newton-interpolation var dup-points)))
((eq? method 'neville)
(interp:neville var points)))))))
;(trace interp:all-but-last)
;(trace interp:all-but-first)
;(trace interp:last)
;(trace interp:divided-differences)
;(trace interp:newton-multiply)
;(trace interp:newton-interpolation)
;(trace interp:interp)
;(trace interp:duplicate)
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