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## man page of LAPACK-3

### LAPACK-3: routine i deprecated and has been replaced by routine ZGGES

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## NAME

LAPACK-3 - routine i deprecated and has been replaced by routine ZGGES## SYNOPSIS

SUBROUTINE ZGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, INFO ) CHARACTER JOBVSL, JOBVSR INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N DOUBLE PRECISION RWORK( * ) COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( * )## PURPOSE

This routine is deprecated and has been replaced by routine ZGGES. ZGEGS computes the eigenvalues, Schur form, and, optionally, the left and or/right Schur vectors of a complex matrix pair (A,B). Given two square matrices A and B, the generalized Schur factorization has the form A = Q*S*Z**H, B = Q*T*Z**H where Q and Z are unitary matrices and S and T are upper triangular. The columns of Q are the left Schur vectors and the columns of Z are the right Schur vectors. If only the eigenvalues of (A,B) are needed, the driver routine ZGEGV should be used instead. See ZGEGV for a description of the eigenvalues of the generalized nonsymmetric eigenvalue problem (GNEP).## ARGUMENTS

JOBVSL (input) CHARACTER*1 = 'N': do not compute the left Schur vectors; = 'V': compute the left Schur vectors (returned in VSL). JOBVSR (input) CHARACTER*1 = 'N': do not compute the right Schur vectors; = 'V': compute the right Schur vectors (returned in VSR). N (input) INTEGER The order of the matrices A, B, VSL, and VSR. N >= 0. A (input/output) COMPLEX*16 array, dimension (LDA, N) On entry, the matrix A. On exit, the upper triangular matrix S from the generalized Schur factorization. LDA (input) INTEGER The leading dimension of A. LDA >= max(1,N). B (input/output) COMPLEX*16 array, dimension (LDB, N) On entry, the matrix B. On exit, the upper triangular matrix T from the generalized Schur factorization. LDB (input) INTEGER The leading dimension of B. LDB >= max(1,N). ALPHA (output) COMPLEX*16 array, dimension (N) The complex scalars alpha that define the eigenvalues of GNEP. ALPHA(j) = S(j,j), the diagonal element of the Schur form of A. BETA (output) COMPLEX*16 array, dimension (N) The non-negative real scalars beta that define the eigenvalues of GNEP. BETA(j) = T(j,j), the diagonal element of the triangular factor T. 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. VSL (output) COMPLEX*16 array, dimension (LDVSL,N) If JOBVSL = 'V', the matrix of left Schur vectors Q. Not referenced if JOBVSL = 'N'. LDVSL (input) INTEGER The leading dimension of the matrix VSL. LDVSL >= 1, and if JOBVSL = 'V', LDVSL >= N. VSR (output) COMPLEX*16 array, dimension (LDVSR,N) If JOBVSR = 'V', the matrix of right Schur vectors Z. Not referenced if JOBVSR = 'N'. LDVSR (input) INTEGER The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N. WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= max(1,2*N). For good performance, LWORK must generally be larger. To compute the optimal value of LWORK, call ILAENV to get blocksizes (for ZGEQRF, ZUNMQR, and CUNGQR.) Then compute: NB -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and CUNGQR; the optimal LWORK is N*(NB+1). 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. RWORK (workspace) DOUBLE PRECISION array, dimension (3*N) INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. =1,...,N: The QZ iteration failed. (A,B) are not in Schur form, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N. > N: errors that usually indicate LAPACK problems: =N+1: error return from ZGGBAL =N+2: error return from ZGEQRF =N+3: error return from ZUNMQR =N+4: error return from ZUNGQR =N+5: error return from ZGGHRD =N+6: error return from ZHGEQZ (other than failed iteration) =N+7: error return from ZGGBAK (computing VSL) =N+8: error return from ZGGBAK (computing VSR) =N+9: error return from ZLASCL (various places) LAPACK driver routine (version 3.March 2011 LAPACK-3(3)

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