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DESCRIPTION Three Fortran subroutines are provided that implement a complex form of the QZ algorithm for finding ~ and z such that Az ffi kBz, where A and B are complex N by N matrices. The subroutines are complex analogs of those for the corresponding real problem included in the EISPACK eigensystem package [1]; the original QZ algorithm is described in [3] and [4]. The complex QZ algorithm shares the fundamental property of the real QZ algorithm in being unaffected by singularity or near singularity of B. Subroutine CQZHES implements the first step of the algorithm wherein A and B are simultaneously reduced by unitary transformations to upper Hessenberg and upper triangular form, respectively. Subroutine CQZVAL implements an iterative process that reduces A to upper triangular form while maintaining the triangular form of B. At the completion of this step, the eigenvalues are derivable from the corresponding diagonal elements of the reduced A and B. Finally, if eigenvectors are desired, subroutine CQZVEC applies the accumulated transformations from the two earlier steps onto the eigenvectors of the triangular problem. No facility is provided for obtaining just a few eigenvectors; a subroutine employing inverse iteration might be included for this purpose, proceeding from the Hessenberg A, the triangular B, and an approximate eigenvalue of the problem. Inverse iteration could be expected to save time, but it might be somewhat less accurate and reliable, especially in the presence of close eigenvalues. Also, no facility is included for balancing A and B. Optimally scaled matrices

Algorithm 535: The QZ algorithm to solve the generalized eigenvalue problem for complex matrices [F2]