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;;; I, Vytautas Jancauskas, place this code in the public domain (2010).
;;; Parts of the code below were adapted from:
;;; http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.98.8156
(define (make-matrix m n)
(let ((matrix (make-vector m)))
(define (populate i)
(if (= i 0)
(vector-set! matrix i (make-vector n))
(begin
(vector-set! matrix i (make-vector n))
(populate (- i 1)))))
(populate (- m 1))
matrix))
(define (matrix-ref matrix i j)
(vector-ref (vector-ref matrix i) j))
(define (matrix-set! matrix i j val)
(vector-set! (vector-ref matrix i) j val))
(define (calculate-c a m k)
(let ((c (make-vector (+ k 1)))
(big-a (make-matrix (+ m 1) (+ k 1))))
(vector-set! c k 1)
(vector-set! c 0 0)
(do ((i 1 (+ i 1)))
((>= i k))
(matrix-set! big-a m i (vector-ref a (- (* k m) i))))
(do ((i 1 (+ i 1)))
((>= i (+ m 1)))
(matrix-set! big-a i 0 1))
(do ((i 1 (+ i 1)))
((>= i k))
(let ((sum1 0))
(do ((l 1 (+ l 1)))
((>= l m))
(do ((j 1 (+ j 1)))
((>= j i))
(set! sum1 (app* $1+$2
sum1
(app* $1*$2
(vector-ref c (+ k (- i) j))
(matrix-ref big-a l j))))))
(vector-set! c (- k i)
(app* $1/$2
(app* $1-$2
(vector-ref a (- (* k m) i))
sum1)
m))
(do ((l (- m 1) (- l 1)))
((<= l 0))
(let ((sum2 0))
(do ((j 0 (+ j 1)))
((>= j i))
(set! sum2 (app* $1+$2
sum2
(app* $1*$2
(vector-ref c (+ k (- i) j))
(matrix-ref big-a l j)))))
(matrix-set! big-a l i (app* $1-$2
(matrix-ref big-a (+ l 1) i)
sum2))))))
c))
(define (calculate-s c)
(do ((i 1 (+ i 1)))
((or (>= i (vector-length c))
(not (equal? (vector-ref c i) 0))) i)))
(define (calculate-b a c m k)
(let ((b (make-vector (+ m 1)))
(big-b (make-matrix (+ m 1) (+ k 1)))
(s (calculate-s c)))
(do ((i 1 (+ i 1)))
((>= i k))
(matrix-set! big-b 1 i (vector-ref c i)))
(do ((l 2 (+ l 1)))
((>= l (+ m 1)))
(do ((i 1 (+ i 1)))
((>= i (+ k 1)))
(let ((sum 0))
(do ((j (max 1 (- i (* k (- l 1)))) (+ j 1)))
((>= j (+ (min k (+ (- i l) 1)) 1)))
(set! sum (app* $1+$2
sum
(app* $1*$2
(vector-ref c j)
(matrix-ref big-b (- l 1) (- i j))))))
(matrix-set! big-b l i sum))))
(do ((l 1 (+ l 1)))
((>= l m))
(let ((sum 0))
(do ((p 1 (+ p 1)))
((>= p l))
(set! sum (app* $1+$2
sum
(app* $1*$2
(vector-ref b p)
(matrix-ref big-b p (* s l))))))
(vector-set! b l (app* $1/$2
(app* $1-$2
(vector-ref a (* s l))
sum)
(poly-expt
(vector-ref c s)
l)))))
(vector-set! b 0 (vector-ref a 0))
(vector-set! b m 1)
b))
(define (make-poly var args)
(reduce (lambda (p c) (app* $1*$2+$3 p var c))
(reverse
(if (and (= (length args) 1) (bunch? (car args)))
(car args)
args))))
(define (poly-expt poly n)
(if (= n 0)
1
(app* $1*$2
poly
(poly-expt poly (- n 1)))))
(define (all-divisors n)
(let ((half (/ n 2)))
(define (helper i)
(cond ((> i half) '())
((= (remainder n i) 0)
(cons i (helper (+ i 1))))
(else (helper (+ i 1)))))
(helper 2)))
(define (compose f g var)
(let ((g-poly (make-poly var g)))
(define (helper coeffs)
(if (null? coeffs)
0
(app* $1+$2
(app* $1*$2
(car coeffs)
(poly-expt g-poly (- (length coeffs) 1)))
(helper (cdr coeffs)))))
(coeffs (helper (reverse f)) (expl->var var))))
(define (verify h f g var)
(app* $1=$2
(make-poly var (compose f g var))
(make-poly var h)))
(define (decompose-1 coeffs var m k)
(let* ((a (list->vector coeffs))
(c (calculate-c a m k))
(b (calculate-b a c m k)))
(list (vector->list b) (vector->list c))))
(define (decompose coeffs var)
(let* ((n (- (length coeffs) 1))
(divisors (reverse (all-divisors n))))
(define (helper divs)
(if (null? divs)
(list (make-poly var coeffs)
(var->expl var))
(let* ((k (car divs))
(m (/ n k))
(result (decompose-1 coeffs var m k))
(f (car result))
(g (cadr result))
(verification (verify coeffs f g var)))
(if (eqv? 0 (cdr verification))
(list (make-poly var f)
(make-poly var g))
(helper (cdr divs))))))
(helper divisors)))
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