1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
|
;;; "determ.scm" Matrix Algebra
;Copyright 2002, 2004 Aubrey Jaffer
;
;Permission to copy this software, to modify it, to redistribute it,
;to distribute modified versions, and to use it for any purpose is
;granted, subject to the following restrictions and understandings.
;
;1. Any copy made of this software must include this copyright notice
;in full.
;
;2. I have made no warranty or representation that the operation of
;this software will be error-free, and I am under no obligation to
;provide any services, by way of maintenance, update, or otherwise.
;
;3. In conjunction with products arising from the use of this
;material, there shall be no use of my name in any advertising,
;promotional, or sales literature without prior written consent in
;each case.
(require 'array)
;;@code{(require 'determinant)}
;;@ftindex determinant
;;@noindent
;;A Matrix can be either a list of lists (rows) or an array.
;;Unlike linear-algebra texts, this package uses 0-based coordinates.
;;; Internal conversion routines
(define (matrix2array matrix prototype)
(let* ((dim1 (length matrix))
(dim2 (length (car matrix)))
(mat (make-array '#() dim1 dim2)))
(do ((idx 0 (+ 1 idx))
(rows matrix (cdr rows)))
((>= idx dim1) rows)
(do ((jdx 0 (+ 1 jdx))
(row (car rows) (cdr row)))
((>= jdx dim2))
(array-set! mat (car row) idx jdx)))
mat))
(define (matrix2lists matrix)
(let ((dims (array-dimensions matrix)))
(do ((idx (+ -1 (car dims)) (+ -1 idx))
(rows '()
(cons (do ((jdx (+ -1 (cadr dims)) (+ -1 jdx))
(row '() (cons (array-ref matrix idx jdx) row)))
((< jdx 0) row))
rows)))
((< idx 0) rows))))
(define (coerce-like-arg matrix arg)
(cond ((array? arg) (matrix2array matrix arg))
(else matrix)))
;;@body
;;Returns the list-of-lists form of @1.
(define (matrix->lists matrix)
(cond ((array? matrix)
(if (not (eqv? 2 (array-rank matrix)))
(slib:error 'not 'matrix matrix))
(matrix2lists matrix))
((and (pair? matrix) (list? (car matrix))) matrix)
((vector? matrix) (list (vector->list matrix)))
(else (slib:error 'not 'matrix matrix))))
;;@body
;;Returns the (ones-based) array form of @1.
(define (matrix->array matrix)
(cond ((array? matrix)
(if (not (eqv? 2 (array-rank matrix)))
(slib:error 'not 'matrix matrix))
matrix)
((and (pair? matrix) (list? (car matrix)))
(matrix2array matrix '#()))
((vector? matrix) matrix)
(else (slib:error 'not 'matrix matrix))))
(define (matrix:cofactor matrix i j)
(define mat (matrix->lists matrix))
(define (butnth n lst)
(if (<= n 1) (cdr lst) (cons (car lst) (butnth (+ -1 n) (cdr lst)))))
(define (minor matrix i j)
(map (lambda (x) (butnth j x)) (butnth i mat)))
(coerce-like-arg
(* (if (odd? (+ i j)) -1 1) (determinant (minor mat i j)))
matrix))
;;@body
;;@1 must be a square matrix.
;;@0 returns the determinant of @1.
;;
;;@example
;;(require 'determinant)
;;(determinant '((1 2) (3 4))) @result{} -2
;;(determinant '((1 2 3) (4 5 6) (7 8 9))) @result{} 0
;;@end example
(define (determinant matrix)
(define mat (matrix->lists matrix))
(let ((n (length mat)))
(if (eqv? 1 n) (caar mat)
(do ((j n (+ -1 j))
(ans 0 (+ ans (* (list-ref (car mat) (+ -1 j))
(matrix:cofactor mat 1 j)))))
((<= j 0) ans)))))
;;@body
;;Returns a copy of @1 flipped over the diagonal containing the 1,1
;;element.
(define (transpose matrix)
(if (number? matrix)
matrix
(let ((mat (matrix->lists matrix)))
(coerce-like-arg (apply map list mat)
matrix))))
;;@body
;;Returns the element-wise sum of matricies @1 and @2.
(define (matrix:sum m1 m2)
(define mat1 (matrix->lists m1))
(define mat2 (matrix->lists m2))
(coerce-like-arg (map (lambda (row1 row2) (map + row1 row2)) mat1 mat2)
m1))
;;@body
;;Returns the element-wise difference of matricies @1 and @2.
(define (matrix:difference m1 m2)
(define mat1 (matrix->lists m1))
(define mat2 (matrix->lists m2))
(coerce-like-arg (map (lambda (row1 row2) (map - row1 row2)) mat1 mat2)
m1))
(define (matrix:scale m1 scl)
(coerce-like-arg (map (lambda (row1) (map (lambda (x) (* scl x)) row1))
(matrix->lists m1))
m1))
;;@args m1 m2
;;Returns the product of matrices @1 and @2.
;;@args m1 z
;;Returns matrix @var{m1} times scalar @var{z}.
;;@args z m1
;;Returns matrix @var{m1} times scalar @var{z}.
(define (matrix:product m1 m2)
(cond ((number? m1) (matrix:scale m2 m1))
((number? m2) (matrix:scale m1 m2))
(else
(let ((mat1 (matrix->lists m1))
(mat2 (matrix->lists m2)))
(define (dot-product v1 v2) (apply + (map * v1 v2)))
(coerce-like-arg
(map (lambda (arow)
(apply map
(lambda bcol (dot-product bcol arow))
mat2))
mat1)
m1)))))
;;@body
;;@1 must be a square matrix.
;;If @1 is singlar, then @0 returns #f; otherwise @0 returns the
;;@code{matrix:product} inverse of @1.
(define (matrix:inverse matrix)
(let* ((mat (matrix->lists matrix))
(det (determinant mat))
(rank (length mat)))
(and (not (zero? det))
(do ((i rank (+ -1 i))
(inv '() (cons
(do ((j rank (+ -1 j))
(row '()
(cons (/ (matrix:cofactor mat j i) det) row)))
((<= j 0) row))
inv)))
((<= i 0)
(coerce-like-arg inv matrix))))))
|