Tests performed on the Generalized Nonsymmetric Eigenvalue Routines

Finding the eigenvalues and eigenvectors of a pair of nonsymmetric matrices 174#174 is done in the following stages:

1. 174#174 is decomposed as 244#244, where 132#132 and 245#245 are unitary (orthogonal in the real case), 135#135 is upper Hessenberg, 74#74 is upper triangular, and 246#246 is the conjugate transpose of 132#132.

2. 247#247 is decomposed as 248#248, where 64#64 and 88#88 are unitary (orthogonal in the real case), 249#249 is in generalized Schur form, where 250#250 is upper (quasi)-triangular and 85#85 is upper triangular with non-negative real diagonal entries and 250#250 is in Schur form; this also gives the generalized eigenvalues 138#138, which are expressed as pairs 251#251, where 252#252.

3. The left and right generalized eigenvectors 253#253 and 254#254 for the pair 249#249 are computed, and from them the back-transformed eigenvectors 255#255 and 256#256 for the matrix pair 247#247. The eigenvectors are normalized so that their largest element has absolute value 1³. (Note that eigenvectors corresponding to singular eigenvalues, i.e., eigenvalues for which 259#259, are not well defined, these are not tested in the eigenvector tests described below.)

To check these calculations, the following test ratios are computed:

260#260

All norms are 261#261. The scalings in the test ratios assure that the ratios will be 124#124, independent of 262#262 and 203#203, and nearly independent of 4#4.

When the test program is run, these test ratios will be compared with a user-specified threshhold THRESH, and for each test ratio that exceeds THRESH, a message is printed specifying the test matrix, the ratio that failed, and its value. A sample message is

Matrix order=   25, type=18, seed=2548,1429,1713,1411, result  8 is   11.33

In this example, the test matrix was of order 146#146 and of type 18 from Table 6, ``seed'' is the initial 4-integer seed of the random number generator used to generate 16#16 and 97#97, and ``result'' specifies that test ratio 147#147 failed to pass the threshhold, and its value was 148#148.

The normalization of the eigenvectors will also be checked. If the absolute value of the largest entry in an eigenvector is not within 203#203 181#181 THRESH of 1, then a message is printed specifying the error. A sample message is

SCHK51: Right Eigenvectors from STGEVC(JOB=B) incorrectly normalized.
Error/precision=0.103E+05, n=    25, type=  18, seed=2548,1429,1713,1411.