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NAME CGEGV - a pair of N-by-N complex nonsymmetric matrices A, B SYNOPSIS SUBROUTINE CGEGV( 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 REAL RWORK( * ) COMPLEX 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 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 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 array, dimension (N) BETA (output) COMPLEX 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 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 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 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 MAX( 2*N, N*(NB+1) ). RWORK (workspace/output) REAL array, dimension (8*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. 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 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 CTGEVC =N+8: error return from CGGBAK (computing VL) =N+9: error return from CGGBAK (computing VR) =N+10: error return from CLASCL (various calls) FURTHER DETAILS Balancing --------- This driver calls CGGBAL 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, CGGBAK 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.