Test Matrices for the Generalized Nonsymmetric Eigenvalue Drivers

• Simple Drivers
  Twenty-six different types of test matrix pairs may be generated for the simple generalized eigenvalue drivers xGGES and xGGEV. Tables 6 and 7 show the available types, along with the numbers used to refer to the matrix types.

• Expert generalized Schur form driver

  For the expert generalized Schur form driver xGGESX, two kinds of tests are executed: one is to use the built-in test matrix pairs and another one is to use a precomputed set of test problems. The test matrix generator xLATM5 generates the five types of built-in test matrix pairs 174#174 of the form:
  

  204#204
  
  
  where 205#205 matrices 206#206 and 207#207 and 208#208 matrices 209#209 and 207#207 are chosen as the following, and 210#210 and 211#211 are chosen so that the generalized Sylvester equation:
  212#212
  
  
  
  
  
  has the prescribed solutions 74#74 and 140#140.

  Type 1:

  213#213 and 214#214, where 215#215 denotes a Jordan block of dimension 198#198 with 216#216 and 52#52 as diagonal and superdiagonal elements, respectively. In the tests, 217#217, and 218#218.

  Type 2:

  219#219 and 220#220 are upper triangular with
  

  221#221

  
  

  
  

  222#222

  
  

  Type 3:

  219#219 and 220#220 are upper quasi-triangular, where the entries are first set as for Type 2. Then each second diagonal block in 206#206 and each third block in 209#209 are made 2 181#181 2 by setting 223#223 and 224#224 for appropriate values of 198#198.

  Type 4:

  219#219 and 220#220 are set as dense blocks:
  

  225#225

  
  

  
  

  226#226

  
  

  Type 5:

  174#174 has potentially close or common eigenvalues, and large or very large departure from block diagonality. First, 206#206 is chosen as the 3#3 181#181 3#3 leading submatrix of 227#227, where
  

  228#228

  
  

  then 209#209 is chosen as the 229#229) 181#181 230#230 leading submatrix of 231#231, where
  

  232#232

  
  

  207#207 and 233#233 are chosen as the identity matrices 234#234 and 235#235, respectively. The values of 236#236 and 237#237 are used for 217#217 and 218#218.

• Expert generalized eigenvalue/eigenvector driver
  For the expert generalized eigenvalue/eigenvector driver xGGEVX, two kinds of tests can be executed: one is to use the built-in test matrix pairs and another one is to use a precomputed set of test problems. The test matrix generator xLATM6 generates the two types of built-in test matrix pairs of the form
  
  238#238
  
  
  where

  Type 1:

  
  

  239#239

  
  

  Type 2:

  
  

  240#240

  
  

  The exact left and right eigenvectors of 174#174 are the rows and columns of
  
  241#241
  
  
  respectively, where 242#242 are chosen from the values 243#243, independently. For these test matrix pairs, the left and right eigenvectors are known exactly. Thus, the eigenvalue condition numbers can be computed exactly. In addition, xLATM6 will compute reciprocal condition numbers for the first and fifth eigenvectors. A total of 1250 different pencils 174#174 are generated in the tests.