# zlaqz0.f - Man Page

SRC/zlaqz0.f

## Synopsis

### Functions/Subroutines

recursive subroutine **zlaqz0** (wants, wantq, wantz, n, ilo, ihi, a, lda, b, ldb, alpha, beta, q, ldq, z, ldz, work, lwork, rwork, rec, info)**ZLAQZ0**

## Function/Subroutine Documentation

### recursive subroutine zlaqz0 (character, intent(in) wants, character, intent(in) wantq, character, intent(in) wantz, integer, intent(in) n, integer, intent(in) ilo, integer, intent(in) ihi, complex*16, dimension( lda, * ), intent(inout) a, integer, intent(in) lda, complex*16, dimension( ldb, * ), intent(inout) b, integer, intent(in) ldb, complex*16, dimension( * ), intent(inout) alpha, complex*16, dimension( * ), intent(inout) beta, complex*16, dimension( ldq, * ), intent(inout) q, integer, intent(in) ldq, complex*16, dimension( ldz, * ), intent(inout) z, integer, intent(in) ldz, complex*16, dimension( * ), intent(inout) work, integer, intent(in) lwork, double precision, dimension( * ), intent(out) rwork, integer, intent(in) rec, integer, intent(out) info)

**ZLAQZ0**

**Purpose:**

ZLAQZ0 computes the eigenvalues of a real matrix pair (H,T), where H is an upper Hessenberg matrix and T is upper triangular, using the double-shift QZ method. Matrix pairs of this type are produced by the reduction to generalized upper Hessenberg form of a real matrix pair (A,B): A = Q1*H*Z1**H, B = Q1*T*Z1**H, as computed by ZGGHRD. If JOB='S', then the Hessenberg-triangular pair (H,T) is also reduced to generalized Schur form, H = Q*S*Z**H, T = Q*P*Z**H, where Q and Z are unitary matrices, P and S are an upper triangular matrices. 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 ZGGHRD that reduced the matrix pair (A,B) to generalized upper Hessenberg form, then the output matrices Q1*Q and Z1*Z are the unitary factors from the generalized Schur factorization of (A,B): A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H. To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently, of (A,B)) are computed as a pair of values (alpha,beta), where alpha is complex and beta real. If beta is nonzero, lambda = alpha / beta is an eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP) A*x = lambda*B*x and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the alternate form of the GNEP mu*A*y = B*y. Eigenvalues can be read directly from the generalized Schur form: alpha = S(i,i), beta = P(i,i). Ref: C.B. Moler & G.W. Stewart, 'An Algorithm for Generalized Matrix Eigenvalue Problems', SIAM J. Numer. Anal., 10(1973), pp. 241--256. Ref: B. Kagstrom, D. Kressner, 'Multishift Variants of the QZ Algorithm with Aggressive Early Deflation', SIAM J. Numer. Anal., 29(2006), pp. 199--227. Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril 'A multishift, multipole rational QZ method with aggressive early deflation'

**Parameters***WANTS*WANTS is CHARACTER*1 = 'E': Compute eigenvalues only; = 'S': Compute eigenvalues and the Schur form.

*WANTQ*WANTQ is CHARACTER*1 = 'N': Left Schur vectors (Q) are not computed; = 'I': Q is initialized to the unit matrix and the matrix Q of left Schur vectors of (A,B) is returned; = 'V': Q must contain an unitary matrix Q1 on entry and the product Q1*Q is returned.

*WANTZ*WANTZ is CHARACTER*1 = 'N': Right Schur vectors (Z) are not computed; = 'I': Z is initialized to the unit matrix and the matrix Z of right Schur vectors of (A,B) is returned; = 'V': Z must contain an unitary matrix Z1 on entry and the product Z1*Z is returned.

*N*N is INTEGER The order of the matrices A, B, Q, and Z. N >= 0.

*ILO*ILO is INTEGER

*IHI*IHI is INTEGER ILO and IHI mark the rows and columns of A which are in Hessenberg form. It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.

*A*A is COMPLEX*16 array, dimension (LDA, N) On entry, the N-by-N upper Hessenberg matrix A. On exit, if JOB = 'S', A contains the upper triangular matrix S from the generalized Schur factorization. If JOB = 'E', the diagonal blocks of A match those of S, but the rest of A is unspecified.

*LDA*LDA is INTEGER The leading dimension of the array A. LDA >= max( 1, N ).

*B*B is COMPLEX*16 array, dimension (LDB, N) On entry, the N-by-N upper triangular matrix B. On exit, if JOB = 'S', B contains the upper triangular matrix P from the generalized Schur factorization; If JOB = 'E', the diagonal blocks of B match those of P, but the rest of B is unspecified.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max( 1, N ).

*ALPHA*ALPHA is COMPLEX*16 array, dimension (N) Each scalar alpha defining an eigenvalue of GNEP.

*BETA*BETA is COMPLEX*16 array, dimension (N) The scalars beta that define the eigenvalues of GNEP. Together, the quantities alpha = ALPHA(j) and beta = BETA(j) represent the j-th eigenvalue of the matrix pair (A,B), in one of the forms lambda = alpha/beta or mu = beta/alpha. Since either lambda or mu may overflow, they should not, in general, be computed.

*Q*Q is COMPLEX*16 array, dimension (LDQ, N) On entry, if COMPQ = 'V', the unitary matrix Q1 used in the reduction of (A,B) to generalized Hessenberg form. On exit, if COMPQ = 'I', the unitary matrix of left Schur vectors of (A,B), and if COMPQ = 'V', the unitary matrix of left Schur vectors of (A,B). Not referenced if COMPQ = 'N'.

*LDQ*LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1. If COMPQ='V' or 'I', then LDQ >= N.

*Z*Z is COMPLEX*16 array, dimension (LDZ, N) On entry, if COMPZ = 'V', the unitary matrix Z1 used in the reduction of (A,B) to generalized Hessenberg form. On exit, if COMPZ = 'I', the unitary matrix of right Schur vectors of (H,T), and if COMPZ = 'V', the unitary matrix of right Schur vectors of (A,B). Not referenced if COMPZ = 'N'.

*LDZ*LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1. If COMPZ='V' or 'I', then LDZ >= N.

*WORK*WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.

*RWORK*RWORK is DOUBLE PRECISION array, dimension (N)

*LWORK*LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,N). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

*REC*REC is INTEGER REC indicates the current recursion level. Should be set to 0 on first call.

*INFO*INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value = 1,...,N: the QZ iteration did not converge. (A,B) is not in Schur form, but ALPHA(i) and BETA(i), i=INFO+1,...,N should be correct.

**Author**Thijs Steel, KU Leuven

**Date**May 2020

Definition at line **280** of file **zlaqz0.f**.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page zlaqz0(3) is an alias of zlaqz0.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK