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NAME ZGEGV - a pair of N-by-N complex nonsymmetric matrices A, B SYNOPSIS SUBROUTINE ZGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO ) CHARACTER JOBVL, JOBVR INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N DOUBLE PRECISION RWORK( * ) COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * ) PURPOSE For a pair of N-by-N complex nonsymmetric matrices A, B: compute the generalized eigenvalues (alpha, beta) compute the left and/or right generalized eigenvectors (VL and VR) The second action is optional -- see the description of JOBVL and JOBVR below. 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) A right generalized eigenvector corresponding to a general- ized eigenvalue w for a pair of matrices (A,B) is a vector r such that (A - w B) r = 0 . A left generalized eigen- vector is a vector l such that (A - w B)**H l = 0 . Note: this routine performs "full balancing" on A and B -- see "Further Details", below. ARGUMENTS JOBVL (input) CHARACTER*1 = 'N': do not compute the left generalized eigen- vectors; = 'V': compute the left generalized eigenvectors. JOBVR (input) CHARACTER*1 = 'N': do not compute the right generalized eigenvectors; = 'V': compute the right generalized eigenvectors. N (input) INTEGER The number of rows and columns in the matrices A, B, VL, and VR. N >= 0. A (input/workspace) COMPLEX*16 array, dimension (LDA, N) On entry, the first of the pair of matrices whose generalized eigenvalues and (optionally) generalized eigenvectors are to be computed. On exit, the con- tents will have been destroyed. (For a description of the contents of A on exit, see "Further Details", below.) LDA (input) INTEGER The leading dimension of A. LDA >= max(1,N). B (input/workspace) COMPLEX*16 array, dimension (LDB, N) On entry, the second of the pair of matrices whose generalized eigenvalues and (optionally) generalized eigenvectors are to be computed. On exit, the con- tents will have been destroyed. (For a description of the contents of B on exit, see "Further Details", below.) LDB (input) INTEGER The leading dimension of B. LDB >= max(1,N). ALPHA (output) COMPLEX*16 array, dimension (N) BETA (output) COMPLEX*16 array, dimension (N) On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the gen- eralized eigenvalues. 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). VL (output) COMPLEX*16 array, dimension (LDVL,N) If JOBVL = 'V', the left generalized eigenvectors. (See "Purpose", above.) Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1, *except* that for eigenvalues with alpha=beta=0, a zero vector will be returned as the corresponding eigenvector. Not referenced if JOBVL = 'N'. LDVL (input) INTEGER The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N. VR (output) COMPLEX*16 array, dimension (LDVR,N) If JOBVL = 'V', the right generalized eigenvectors. (See "Purpose", above.) Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1, *except* that for eigenvalues with alpha=beta=0, a zero vector will be returned as the corresponding eigenvector. Not referenced if JOBVR = 'N'. LDVR (input) INTEGER The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N. WORK (workspace/output) COMPLEX*16 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 ZGEQRF, ZUNMQR, and CUNGQR.) Then compute: NB -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and CUNGQR; The optimal LWORK is MAX( 2*N, N*(NB+1) ). (8*N) RWORK (workspace/output) DOUBLE PRECISION array, dimension INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. =1,...,N: The QZ iteration failed. No eigenvectors have been calculated, 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 ZTGEVC =N+8: error return from ZGGBAK (computing VL) =N+9: error return from ZGGBAK (computing VR) =N+10: error return from ZLASCL (various calls) FURTHER DETAILS Balancing --------- This driver calls ZGGBAL to both permute and scale rows and columns of A and B. The permutations PL and PR are chosen so that PL*A*PR and PL*B*R will be upper triangular except for the diagonal blocks A(i:j,i:j) and B(i:j,i:j), with i and j as close together as possible. The diagonal scaling matrices DL and DR are chosen so that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have entries close to one (except for the entries that start out zero.) After the eigenvalues and eigenvectors of the balanced matrices have been computed, ZGGBAK transforms the eigenvec- tors back to what they would have been (in perfect arith- metic) if they had not been balanced. Contents of A and B on Exit -------- -- - --- - -- ---- If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or both), then on exit the arrays A and B will contain the complex Schur form[*] of the "balanced" versions of A and B. If no eigenvectors are computed, then only the diagonal blocks will be correct. [*] In other words, upper triangular form.