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Twentysix 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
20010813