Exercise 2.37. Suppose we represent vectors v = (vi) as sequences of numbers, and matrices m = (mij) as sequences of vectors (the rows of the matrix). For example, the matrix
is represented as the sequence ((1 2 3 4) (4 5 6 6) (6 7 8 9)). With this representation, we can use sequence operations to concisely express the basic matrix and vector operations. These operations (which are described in any book on matrix algebra) are the following:
We can define the dot product as17
(define (dot-product v w)
(accumulate + 0 (map * v w)))
Fill in the missing expressions in the following procedures for computing the other matrix operations. (The procedure accumulate-n is defined in exercise 2.36.)
(define (matrix-*-vector m v)
(map <??> m))
(define (transpose mat)
(accumulate-n <??> <??> mat))
(define (matrix-*-matrix m n)
(let ((cols (transpose n)))
(map <??> m)))
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| (define nil '()) | |
| (define (accumulate op initial sequence) | |
| (if (null? sequence) | |
| initial | |
| (op (car sequence) | |
| (accumulate op initial (cdr sequence))))) | |
| (define (accumulate-n op init seqs) | |
| (if | |
| (null? (car seqs)) | |
| nil | |
| (cons | |
| (accumulate op init (map (lambda (x) (car x)) seqs)) | |
| (accumulate-n op init (map (lambda (x) (cdr x)) seqs))))) | |
| (define (dot-product v w) | |
| (accumulate + 0 (map * v w))) | |
| (define (matrix-*-vector m v) | |
| (map (lambda (x) (dot-product x v)) m)) | |
| (define (transpose m) | |
| (accumulate-n cons nil s)) | |
| (define (matrix-*-matrix m1 m2) | |
| (map (lambda (x) (matrix-*-vector m1 x)) (transpose m2))) | |
| (define w | |
| (list | |
| (list 1 2 3) | |
| (list 4 5 6) | |
| (list 10 11 12))) | |
| (define s | |
| (list | |
| (list 1 2 3) | |
| (list 4 5 6) | |
| (list 7 8 9) | |
| (list 10 11 12))) | |
| (define w2 | |
| (list | |
| (list 1 2 3 4) | |
| (list 4 5 6 7) | |
| (list 7 8 9 10) | |
| (list 10 11 12 13))) | |
| (define v1 (list 1 2 3)) | |
| (define v2 (list 4 5 6)) | |
| (accumulate-n + 0 s) | |
| (dot-product v1 v2) | |
| (matrix-*-vector w v1) | |
| (matrix-*-matrix s w2) | |