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.