Tests performed on the Generalized Nonsymmetric Eigenvalue Drivers

The four driver routines have slightly different tests applied to them.

• For SGGES the following tests are computed:
  
  263#263
  
  
  where 264#264 and 265#265 are the 2 181#181 2 diagonal blocks of 250#250 and 85#85 corresponding to the 266#266th and 267#267th eigenvalues.

• For SGGESX the following tests are computed:
  
  268#268
  
  
  where 264#264 and 265#265 are the 2 181#181 2 diagonal blocks of 250#250 and 85#85 corresponding to the 266#266th and 267#267th eigenvalues.
  
  269#269
  
  
  The test 160#160 is for precomputed test problems. DIFtru is the true value of DIF obtained by computing the minimum singular value of the matrix generated in solving the generalized Sylvester equation and DIFest is the estimated value of DIF obtained from xTGSEN, and where PLtru is the true value of PL obtained by computing the norm of the matrix L resulting from the solution of the generalized Sylvester equation and PLest is the estimated value of PL obtained from xTGSEN.
  Test 8 is compared to 10*THRESH instead of THRESH.
  Test 9 is a check for very ill-conditioned problems. It is possible with ill-conditioned problems for reordering to be (nearly) impossible which corresponds to very large condition numbers. So if reordering fails, it is checked that the true reciprocal condition number DIFtru is small (i.e. the condition is large) and it is checked that the estimated reciprocal condition number DIFest is zero (when reordering fails, the subroutine is supposed to return DIFest = 0).

  These test ratios are compared to the input parameter THRESH. If a ratio exceeds THRESH, a message is printed specifying the test matrix, the ratio that failed and its value, just like the tests performed on the nonsymmetric eigenvalue problem computational routines.
  

• For SGGEV the following tests are computed:
  
  
  270#270
  
  
  where 151#151 are eigenvalues 271#271. 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.

• For SGGEVX the following tests are computed:
  
  272#272
  
  
  where Stru273#273 is the exact reciprocal condition number for eigenvalue 274#274 and Sest273#273 is the estimated reciprocal condition number computed by xTGSNA, and DIFtru273#273 is the exact reciprocal condition number for eigenvector 274#274 and DIFest273#273 is the estimated reciprocal condition number computed by xTGSNA.
  Test 4 is compared to 10*THRESH instead of THRESH.
  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.