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streams.scm
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streams.scm
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;(define-syntax cons-stream
; (syntax-rules ()
; ((cons-stream head tail)
; (cons head (delay tail)))))
(define stream-car car)
(define (stream-cdr stream) (force (cdr stream)))
(define (stream-filter pred stream)
(cond ((null? stream) '())
((pred (stream-car stream))
(cons-stream (stream-car stream)
(stream-filter pred (stream-cdr stream))))
(else (stream-filter pred (stream-cdr stream)))))
;****************************************************************
; procedure that prints the first n elements of
; stream, each on a separate line.
(define (display-n data n)
(let loop ((index n) (st data))
(if (null? st)
(newline)
(if (> index 0)
(begin (display (stream-car st))
(newline)
(loop (- index 1) (stream-cdr st)))
(newline)
)
)
)
)
;Generalizes stream-map to allow procedures that take
;multiple arguments.
(define (stream-map proc . argstreams)
(cons-stream
(apply proc (map stream-car argstreams))
(apply stream-map(cons proc
(map stream-cdr argstreams))))
)
(define ones (cons-stream 1 ones))
(define integers (cons-stream 1 (add-streams ones integers)))
(define notdiv-235
(stream-filter
(lambda (x)
(and (not (equal? 0 (remainder x 2)))
(not (equal? 0 (remainder x 3)))
(not (equal? 0 (remainder x 5))))) integers)
)
; Computing the digits in PI
; convert an integer to a list of its digits
; avoiding all the pitfalls and nastiness
; of convertion to and from strings.
; It uses some neat math. This is the
; same method employed by the C Standard
; library in itoa (integer to array).
(define (digits n . args)
(let ((b 10 ))
(let loop ((n n) (d '()))
(if (zero? n) d
(loop (quotient n b)
(cons (modulo n b) d))))))
; mult-stream procedure.
; simple test -
; (mult-stream 87 (list->stream '(9 8 7 4 3 6 9 1 7)))
(define (mult-stream m stream)
;;f(n) -> y normal
; power function, except when y <= 0,
; in which case the function returns 1
(define (fix-power lst)
(if(> (expt 10 (- (length lst) 1)) 0)
(expt 10 (- (length lst) 1))
1
)
)
(let loop ((a 0) (a-list '()) (st stream))
(if (stream-null? st)
(if (null? a-list)
'()
(cons-stream (car a-list)
(loop a (cdr a-list) st))
; here we are done once we append the rest
; of a-list to our output.
)
(if (not (equal? a-list '()))
(if (< (+ m (modulo a (fix-power a-list)))
(fix-power a-list))
(cons-stream (car a-list)
(loop (modulo a (fix-power a-list))
(cdr a-list) st)) ;;produce
(loop (+ (* a 10) (* m (stream-car st)))
(pad-list (digits (+ (* a 10) (* m (stream-car st))))
a-list
0
)
(stream-cdr st)) ;consume
)
(loop (+ (* a 10) (* m (stream-car st)))
(pad-list (digits (+ (* a 10) (* m (stream-car st))))
a-list
0
)
(stream-cdr st));consume
)
)
)
)
; Inputs:
; to-pad: the list to pad
; to-comp: the list we want to exceed in length (by 1).
; pad-data: the data we would like to use as padding,
; (in mult-stream this is 0)
; Outputs:
; pad-list with pad-data prepended until it exceeds the length
; of to-comp by 1.
(define (pad-list to-pad to-comp pad-data)
(let loop ((tp to-pad))
(if (<= (length tp) (length to-comp))
(loop (cons pad-data tp))
tp
)
)
)
; Inputs:
; M: matrix given as list of rows
; n: the index of the desired column
; Ouputs:
; A list containing the elements of column n in matrix M.
(define (matrix-col M n)
(let loop ((index (length M)) (column-data '()))
(if (equal? index 0)
column-data
(loop (- index 1)
(cons (matrix-row (matrix-row M (- index 1)) n) column-data)))))
; Inputs:
; M: matrix given as list of rows
; n: the index of the desired row
; Ouputs:
; A list containing the elements of row n in matrix M.
(define (matrix-row list n)
(let loop ((row-index n) (row list))
(if (equal? row-index 0)
(car row)
(loop (- row-index 1) (cdr row))
)
)
)
(define (reduce op lst)
(let loop ((res (car lst)) (lst (cdr lst)))
(if (null? lst)
res
(loop (op res (car lst)) (cdr lst)))))
; Function to multiply two matricies.
; Inputs:
; M: matrix given as list of rows.
; N: matrix given as list of rows.
; Outputs:
; The dot product - N.M
(define (matrix-composition N M)
(let row-loop ((row-index (length N)) (result '()))
(if (equal? row-index 0)
result
(row-loop (- row-index 1)
(cons
(let col-loop ((col-index (length (car M))) (row '()))
(if (equal? col-index 0)
row
(col-loop
(- col-index 1)
(cons (reduce + (map *
(matrix-row N (- row-index 1))
(matrix-col M (- col-index 1))))
row)
)
)
)
result)
)
)
)
)
; Function to add two matricies.
; Inputs:
; M: matrix given as list of rows.
; N: matrix given as list of rows.
; Outputs:
; The sum of matrix N and matrix M.
(define (matrix-sum N M)
(let loop ((N N) (M M) (mat-sum '()))
(if (or (null? N) (null? M))
(reverse mat-sum)
(loop (cdr N)
(cdr M)
(cons (map + (car N) (car M)) mat-sum)
)
)
)
)
; Returns element at row i, column j in matrix M
(define (get-ele i j M)
(matrix-row (matrix-row M i) j)
)
; self explanatory.
(define m1 '((1 6) (0 3)))
(define to-add '((1 4) (0 2)))
(define increase-stream (cons-stream to-add increase-stream))
(define fractional-transformation-stream
(cons-stream
m1
(stream-map matrix-sum increase-stream fractional-transformation-stream)
)
)
; Uses integer division.
; Returns the floor of:
; m[0,0]*x + m[0,1]
; divided by
; m[1,0]*x + m[1,1]
(define (linear-transform x M)
(quotient (+ (* (get-ele 0 0 M) x) (get-ele 0 1 M))
(+ (* (get-ele 1 0 M) x) (get-ele 1 1 M))
)
)
;Nice and concise generation of the pi stream.
(define pi
(let loop ((mat m1) (st (stream-cdr fractional-transformation-stream)))
(let ((lt1 (linear-transform 3 mat)) (lt2 (linear-transform 4 mat)))
(if (equal? lt1 lt2)
(cons-stream lt2
(loop (matrix-composition (list (list 10 (* -10 lt2)) (list 0 1)) mat) st))
(loop (matrix-composition mat (stream-car st)) (stream-cdr st))
)
)
)
)