1zmu(1) User Commands zmu(1)
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6 zmu - multiply matrices or permutations
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9 zmu [OPTIONS] <A> <B> <Result>
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12 This program reads two matrices or permutations and writes their prod‐
13 uct to Result.
14 The input files must contain two compatible objects, i.e., their prod‐
16 uct must be defined. Currently, zmu can handle the following data
17 types:
18 · Both files are matrices over the same field, and the number of col‐
20 umns of A equals the number of rows of B. In this case, zmu calcu‐
21 lates the standard matrix product.
22 · One of the operands is a one by one matrix, and the other is any
24 matrix over the same field. In this case, the one by one matrix is
25 interpreted as a scalar, and the program calculates the corresponding
26 multiple of the matrix.
27 · Both input files are permutations of degree a and b, respectively.
29 The result is a permutation C of degree max(a,b), which is defined by
30 C(x) = B(A(x)). If the permutations are of different degrees, the
31 smaller permutation is extended to the larger degree by adding fixed
32 points.
33 · A is a matrix, B is a permutation and the degree of the permutation
35 equals the number of columns of the matrix. The result is a matrix
36 of the same size which is calculated from the input matrix by permut‐
37 ing the marks of each row in the following way: The ith mark of the
38 row is stored as the kth mark of the result if the permutation maps i
39 to k.
40 · A is a permutation of degree m, and B is an m by n matrix. The
42 result is again an m by n matrix which consists of the rows of the
43 input matrix, rearranged according to the permutation. If the permu‐
44 tation maps i to k, then the kth row of the input matrix becomes the
45 ith row of the output matrix. Here is an example:
46 | 1 1 | | 2 2 |
48 (1 2 3) * | 2 2 | = | 3 3 |
49 | 3 3 | | 1 1 |
50 With these conventions, products between matrices and permutations are
52 defined in a consistent way. The associative law a(bc)=(ab)c holds
53 whenever ab and bc are defined (a,b,c being matrices or permutations).
54 A permutation matrix created with zcv or zcf, if multiplied with
55 another matrix, produces the same result as the original permutation.
56 Blockwise Matrix Multiplication
58 In the case of two matrices, a blockwise multiplication can be per‐
59 formed using the -r and -c options. If one or both of these options
60 are specified on the command line, zmu will read only some rows of A
61 and/or some columns of B. Multiplying the two pieces together yields a
62 rectangular piece of the result. By default the result is divided into
63 4 pieces of (almost) equal size. To calculate the 4 pieces succes‐
64 sively, type
65 zmu -r 1 -c 1 m1 m2 tmp11
67 zmu -r 1 -c 2 m1 m2 tmp12
68 zmu -r 2 -c 1 m1 m2 tmp21
69 zmu -r 2 -c 2 m1 m2 tmp22
70 The resulting matrices `tmpXX' can then be pasted together using zpt:
72 zpt -R 2 -C 2 result tmp
74 This procedure can be used in a multi-processor environment where each
76 piece of the result is computed on a separate machine.
77 By adding an additional parameter to -r and/or -c you can control the
79 number of vertical or horizontal slices. For example,
80 zmu -r 3/5
82 means to cut A horizontally into five slices and use the third slice
84 for multiplication. The number of slice must not be greater than the
85 number of rows.
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88 -Q Quiet, no messages.
89 -V Verbose, more messages.
91 -T <MaxTime>
93 Set CPU time limit
94 -r <Row>[.<NRows>]
96 Divide the matrix A horizontally into NRows (default: 2) slices
97 and use the Rowth slice as the left factor. NRows must not be
98 larger than the number of rows of A.
99 -c <Col>[.<NCols>]
101 Divide the matrix B vertically into NCols (default: 2) slices
102 and use the Colth slice as the right factor. NCols must not be
103 larger than the number of columns of B.
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106 A Left factor.
107 B Right factor.
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111 Result The product.
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114 zcf(1), zcv(1), zpt(1)
115MeatAxe 2.4.24 zmu(1)