Timing the Generalized Nonsymmetric Eigenproblem

A separate input file drives the timing codes for the generalized nonsymmetric eigenproblem. The input file specifies

• N, the matrix size,
• LDA, the leading dimension,
• the test matrix types,
• the routines or sequences of routines from LAPACK or EISPACK to be timed.

The number and size of the input values are limited by certain program maximums which are defined in PARAMETER statements in the main timing program:

562#562

The computations that may be timed for the REAL version are

1. SGGHRD(N) (LAPACK reduction to generalized upper Hessenberg form, without computing 132#132 or 245#245, including a call to SGEQRF and SORMQR to reduce 97#97 to upper triangular form.)
2. SGGHRD(Q) (LAPACK reduction to generalized upper Hessenberg form, computing 132#132 but not 245#245, including a call to SGEQRF, SORGQR, and SORMQR to reduce 97#97 to upper triangular form.)
3. SGGHRD(Z) (LAPACK reduction to generalized upper Hessenberg form, computing 245#245 but not 132#132, including a call to SGEQRF and SORMQR to reduce 97#97 to upper triangular form.)
4. SGGHRD(Q,Z) (LAPACK reduction to generalized upper Hessenberg form, computing 132#132 and 245#245, including a call to SGEQRF, SORGQR, and SORMQR to reduce 97#97 to upper triangular form.)
5. SHGEQZ(E) (LAPACK computation of generalized eigenvalues only of a pair of matrices in generalized Hessenberg form)
6. SHGEQZ(S) (LAPACK computation of generalized Schur form of a pair of matrices in generalized Hessenberg form)
7. SHGEQZ(Q) (LAPACK computation of generalized Schur form of a pair of matrices in generalized Hessenberg form and Q)
8. SHGEQZ(Z) (LAPACK computation of generalized Schur form of a pair of matrices in generalized Hessenberg form and Z)
9. SHGEQZ(Q,Z) (LAPACK computation of generalized Schur form of a pair of matrices in generalized Hessenberg form and Q and Z)
10. STGEVC(L,A) (LAPACK computation of the the left generalized eigenvectors of a matrix pair in generalized Schur form)
11. STGEVC(L,B) (LAPACK computation of the the left generalized eigenvectors of a matrix pair in generalized Schur form, back transformed by Q)
12. STGEVC(R,A) (LAPACK computation of the the right generalized eigenvectors of a matrix pair in generalized Schur form)
13. STGEVC(R,B) (LAPACK computation of the the right generalized eigenvectors of a matrix pair in generalized Schur form, back transformed by Z)
14. QZHES(F) (EISPACK reduction to generalized upper Hessenberg form, with MATZ=.FALSE., so 245#245 is not computed.)
15. QZHES(T) (EISPACK reduction to generalized upper Hessenberg form, with MATZ=.TRUE., so 245#245 is computed.)
16. QZIT(F) (QZIT followed by QZVAL with MATZ=.FALSE.: EISPACK computation of generalized eigenvalues only of a pair of matrices in generalized Hessenberg form)
17. QZIT(T) (QZIT followed by QZVAL with MATZ=.TRUE.: EISPACK computation of generalized Schur form of a pair of matrices in generalized Hessenberg form and Z)
18. QZVEC (EISPACK computation of the the right generalized eigenvectors of a matrix pair in generalized Schur form, back transformed by Z)

Note that SGGHRD is timed along with the QR routines that reduce 97#97 to upper-triangular form; this is to allow a fair comparison with the EISPACK routine QZHES.

Four different matrix types are provided for timing the generalized nonsymmetric eigenvalue routines. A variety of matrix types is allowed because the number of iterations to compute the eigenvalues, and hence the timing, can depend on the type of matrix whose eigendecomposition is desired. The matrices used for timing have at least one zero, one infinite, and one singular ( 259#259) generalized eigenvalue. The remaining eigenvalues are sometimes real and sometimes complex, distributed in magnitude as follows:

• ``clustered'' entries 557#557 with random signs;
• evenly spaced entries from 1 down to 9#9 with random signs;
• geometrically spaced entries from 1 down to 9#9 with random signs;
• eigenvalues randomly chosen from the interval 558#558.