Previous: cgeevx Up: ../lapack-c.html Next: cgegv

NAME CGEGS - a pair of N-by-N complex nonsymmetric matrices A, B SYNOPSIS SUBROUTINE CGEGS( 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 REAL RWORK( * ) COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( * ) PURPOSE For a pair of N-by-N complex nonsymmetric matrices A, B: compute the generalized eigenvalues (alpha, beta) compute the complex Schur form (A,B) compute the left and/or right Schur vectors (VSL and VSR) The last action is optional -- see the description of JOBVSL and JOBVSR below. (If only the generalized eigenvalues are needed, use the driver CGEGV instead.) A generalized eigenvalue for a pair of matrices (A,B) is, roughly speaking, a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpre- tation for beta=0, and even for both being zero. A good beginning reference is the book, "Matrix Computations", by G. Golub & C. van Loan (Johns Hopkins U. Press) The (generalized) Schur form of a pair of matrices is the result of multiplying both matrices on the left by one uni- tary matrix and both on the right by another unitary matrix, these two unitary matrices being chosen so as to bring the pair of matrices into upper triangular form with the diago- nal elements of B being non-negative real numbers (this is also called complex Schur form.) The left and right Schur vectors are the columns of VSL and VSR, respectively, where VSL and VSR are the unitary matrices which reduce A and B to Schur form: Schur form of (A,B) = ( (VSL)**H A (VSR), (VSL)**H B (VSR) ) ARGUMENTS JOBVSL (input) CHARACTER*1 = 'N': do not compute the left Schur vectors; = 'V': compute the left Schur vectors. JOBVSR (input) CHARACTER*1 = 'N': do not compute the right Schur vectors; = 'V': compute the right Schur vectors. N (input) INTEGER The number of rows and columns in the matrices A, B, VSL, and VSR. N >= 0. A (input/output) COMPLEX array, dimension (LDA, N) On entry, the first of the pair of matrices whose generalized eigenvalues and (optionally) Schur vec- tors are to be computed. On exit, the generalized Schur form of A. LDA (input) INTEGER The leading dimension of A. LDA >= max(1,N). B (input/output) COMPLEX array, dimension (LDB, N) On entry, the second of the pair of matrices whose generalized eigenvalues and (optionally) Schur vec- tors are to be computed. On exit, the generalized Schur form of B. LDB (input) INTEGER The leading dimension of B. LDB >= max(1,N). ALPHA (output) COMPLEX array, dimension (N) BETA (output) COMPLEX array, dimension (N) On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the gen- eralized eigenvalues. ALPHA(j), j=1,...,N and BETA(j), j=1,...,N are the diagonals of the complex Schur form (A,B) output by CGEGS. The BETA(j) will be non-negative real. Note: the quotients ALPHA(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio alpha/beta. However, ALPHA will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B). VSL (output) COMPLEX array, dimension (LDVSL,N) If JOBVSL = 'V', VSL will contain the left Schur vectors. (See "Purpose", above.) 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 array, dimension (LDVSR,N) If JOBVSR = 'V', VSR will contain the right Schur vectors. (See "Purpose", above.) 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 array, dimension (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 gen- erally be larger. To compute the optimal value of LWORK, call ILAENV to get blocksizes (for CGEQRF, CUNMQR, and CUNGQR.) Then compute: NB -- MAX of the blocksizes for CGEQRF, CUNMQR, and CUNGQR; the optimal LWORK is N*(NB+1). RWORK (workspace) REAL 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 usu- ally indicate LAPACK problems: =N+1: error return from CGGBAL =N+2: error return from CGEQRF =N+3: error return from CUNMQR =N+4: error return from CUNGQR =N+5: error return from CGGHRD =N+6: error return from CHGEQZ (other than failed iteration) =N+7: error return from CGGBAK (computing VSL) =N+8: error return from CGGBAK (computing VSR) =N+9: error return from CLASCL (various places)