hgeqz Interface

public interface hgeqz

HGEQZ computes the eigenvalues of a complex matrix pair (H,T), where H is an upper Hessenberg matrix and T is upper triangular, using the single-shift QZ method. Matrix pairs of this type are produced by the reduction to generalized upper Hessenberg form of a complex matrix pair (A,B): A = Q1HZ1H, B = Q1TZ1H, as computed by CGGHRD. If JOB='S', then the Hessenberg-triangular pair (H,T) is also reduced to generalized Schur form, H = QSZH, T = QPZH, where Q and Z are unitary matrices and S and P are upper triangular. Optionally, the unitary matrix Q from the generalized Schur factorization may be postmultiplied into an input matrix Q1, and the unitary matrix Z may be postmultiplied into an input matrix Z1. If Q1 and Z1 are the unitary matrices from CGGHRD that reduced the matrix pair (A,B) to generalized Hessenberg form, then the output matrices Q1Q and Z1Z are the unitary factors from the generalized Schur factorization of (A,B): A = (Q1Q)S(Z1Z)H, B = (Q1Q)P(Z1Z)H. To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently, of (A,B)) are computed as a pair of complex values (alpha,beta). If beta is nonzero, lambda = alpha / beta is an eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP) Ax = lambdaBx and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the alternate form of the GNEP muAy = By. The values of alpha and beta for the i-th eigenvalue can be read directly from the generalized Schur form: alpha = S(i,i), beta = P(i,i). Ref: C.B. Moler Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973), pp. 241--256.

Subroutines

Module Procedures