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### Test Matrices for the Generalized Nonsymmetric Eigenvalue Routines

Twenty-six different types of test matrix pairs may be generated for the generalized eigenvalue routines. Tables 6 and 7 show the available types, along with the numbers used to refer to the matrix types. Except as noted, all matrices have 124#124 entries.

Table 6: Sparse test matrices for the generalized nonsymmetric eigenvalue problem
 Matrix 97#97: 0 176#176 177#177 178#178 179#179 180#180 Matrix 16#16: 181#181 1 182#182 183#183 181#181 1 182#182 183#183 0 1 3 176#176 2 4 8 184#184 12 185#185 11 177#177 5 186#186 6 179#179 7 187#187 14 10 188#188 9 13 189#189 15

The following symbols and abbreviations are used:

Table 7: Dense test matrices for the generalized nonsymmetric eigenvalue problem
 Magnitude of 16#16, 97#97 Distribution of 190#190, 191#191, 192#192, 191#191, 192#192, Eigenvalues 193#193, 194#194 194#194 195#195 195#195 All Ones 16 (Same as type 15) 17 Arithmetic 19 22 24 25 23 Geometric 20 Clustered 18 Random 21 Random Entries 26

0: The zero matrix.
176#176: The identity matrix.
196#196: Generally, the underflow threshhold times the order of the matrix divided by the machine precision. In other words, this is a very small number, useful for testing the sensitivity to underflow and division by small numbers. Its reciprocal tests for overflow problems.
177#177: Transposed Jordan block, i.e., matrix with ones on the first subdiagonal and zeros elsewhere. (Note that the diagonal is zero.)
197#197: A (198#198 + 1) 181#181 (198#198 + 1) transposed Jordan block which is a diagonal block within a (2198#198 + 1) 181#181 (2198#198 + 1) matrix. Thus, 199#199 has all zero entries except for the last 198#198 diagonal entries and the first 198#198 entries on the first subdiagonal. (Note that the matrices 199#199 and 200#200 have odd order; if an even order matrix is needed, a zero row and column are added at the end.)
179#179: A diagonal matrix with the entries 0, 1, 2, 12#12, 17#17 on the diagonal, where 4#4 is the order of the matrix.
189#189: A diagonal matrix with the entries 0, 0, 1, 2, 12#12, 201#201, 0 on the diagonal, where 4#4 is the order of the matrix.
180#180: A diagonal matrix with the entries 0, 201#201, 202#202, 12#12, 1, 0, 0 on the diagonal, where 4#4 is the order of the matrix.

Except for matrices with random entries, all the matrix pairs include at least one infinite, one zero, and one singular eigenvalue (0/0, which is not well defined). For arithmetic, geometric, and clustered eigenvalue distributions, the eigenvalues lie between 203#203 (the machine precision) and 1 in absolute value. The eigenvalue distributions have the following meanings:

Arithmetic: Difference between adjacent eigenvalues is a constant.

Geometric: Ratio of adjacent eigenvalues is a constant.

Clustered: One eigenvalue is 1 and the rest are 203#203 in absolute value.

Random: Eigenvalues are logarithmically distributed.

Random entries: Matrix entries are uniformly distributed random numbers.

Next: Test Matrices for the Up: Testing the Generalized Nonsymmetric Previous: The Generalized Nonsymmetric Eigenvalue   Contents
Susan Blackford 2001-08-13