1zev(1) User Commands zev(1)
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6 zev - compute matrix eigenvalues
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9 zev [OPTIONS] <Matrix> [<Poly> [<Group>]]
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12 This program reads a matrix from Matrix and a list of polynomials from
13 Poly (or from the standard input). For each input polynomial, it eval‐
14 uates that function of the input matrix, calculates the nullity, and
15 puts out this nullity, divided by the degree, along with a text from
16 the input.
17 The program was specifically designed to assist in the calculation of
19 the Brauer characters of diagonalizable matrices, with the text giving
20 the complex number which is the Brauer character of the companion
21 matrix for that polynomial. Usually the polynomials have been prepared
22 in a separate data file and are fed into zev by giving the file name or
23 by redirecting its input. The preparation of the input polynomials is
24 generally a time-consuming task if it is done by hand, but there are
25 data files available for the most commonly used fields. These files
26 should be located in the library directory. They are distributed with
27 this release of the C MeatAxe. If the user is familiar with the com‐
28 puter program system GAP, he will find it easy to create his own data
29 files.
30 If the nullity is not a multiple of the degree, zev prints a warning
32 message.
33 Polynomial File Format
35 The data file contains the polynomials in text form. Several polynomi‐
36 als can be comprised in a group, and the data file can contain any num‐
37 ber of groups of polynomials. This allows several sets of polynomials
38 to be kept in one data file (for example, all polynomials for a given
39 field), the appropriate polynomials being selected through the Group
40 argument on the command line.
41 The file is read and interpreted line by line. There are three types of
43 lines:
44 · Comment lines, beginning with a "#". These lines are simply ignored
46 by zev, as are empty lines.
47 · Group headers. Each line beginning with a non-space character is
49 interpreted as the beginning of a new group of polynomials. Such
50 lines contain only one text field, the name of the group (up to 1023
51 characters).
52 · Lines beginning with a space are interpreted as polynomials. The
54 format is:
55 [space]Name Coefficients
57 where Name is any text (up to 1023 characters), and Coefficients are
59 the coefficients of the polynomial (in free format). Note that the
60 first character must be an ordinary space charcter, a TAB is not
61 allowed! The coefficients must use the names as specified by the
62 arithmetic — usually 0, 1, ..., q-1. The one exception is -1, which
63 the program treats specially as "0-1" so that the cyclotomic polyno‐
64 mials can be used over all fields. The coefficients are in decreas‐
65 ing degree, starting with the coefficient of the highest power of x
66 and continuing, ending up with the constant term.
67 Here is an example:
69 # Sample input file for zev
71 # Some polynomials over GF(5)
72 #
73 p11b11
74 1 1 4
75 b11 1 4 4 1 3 4
76 -1-b11 1 2 4 1 1 4
77 p13c13
78 1 1 4
79 c13 1 3 0 3 1
80 c13*3 1 1 4 1 1
81 c13*9 1 2 1 2 1
82 This file contains 7 polynomials in two groups. The polynomial "b11" in
84 group "p11b11" is x^5+4x^4+4x^3+x^2+3x+4.
85 Output Format
87 There are two output formats. By default the nullities are printed in
88 tabular form giving group, name, degree and multiplicity (i.e., nullity
89 divided by degree) for each polynomial. If the -G option is given, zev
90 prints an algebraic expression which can be read from GAP. Here is an
91 example with an 8 by 8 matrix over GF(3), polynomials being read from
92 "poly3":
93 $ zev mat poly3 p8i2
95 p8i2 1 1 1
96 p8i2 -1 1 4
97 p8i2 0 2 2
98 p8i2 i2 2 0
99 p8i2 -i2 2 1
100 $ zev -G mat poly3 p8i2
101 MeatAxe.BrauerChar := 1*(1) + 4*(-1) + 2*(0) + 1*(-i2);
102 Note that "i2" does not appear in the expression because its coeffi‐
104 cient is zero.
105
107 -Q Quiet, no messages.
108 -V Verbose, more messages.
110 -T <MaxTime>
112 Set CPU time limit
113 -G, --gap
115 Produce output in GAP format. This option implies -Q.
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118 There must be enough memory to hold the input matrix and two more
119 matrices of the same size. Lines in the polynomial input file must not
120 be longer than 1023 characters.
121 It is not checked that the input file is a matrix. TAB characters at
123 the beginning of a line are not interpreted as space.
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126 Matrix A square matrix.
127 Poly Polynomial definition file; read from standard input by default.
129MeatAxe 2.4.24 zev(1)