ZGGEVX - compute for a pair of N-by-N complex nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors

SUBROUTINE ZGGEVX(

BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK, LWORK, RWORK, IWORK, BWORK, INFO )

CHARACTER BALANC, JOBVL, JOBVR, SENSE

INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N

DOUBLE PRECISION ABNRM, BBNRM

LOGICAL BWORK( * )

INTEGER IWORK( * )

DOUBLE PRECISION LSCALE( * ), RCONDE( * ), RCONDV( * ), RSCALE( * ), RWORK( * )

COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )

ZGGEVX computes for a pair of N-by-N complex nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors. Optionally, it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors (ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and reciprocal condition numbers for the right eigenvectors (RCONDV).

A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A - lambda*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0, and even for both being zero.

The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

A * v(j) = lambda(j) * B * v(j) .

The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

u(j)**H * A = lambda(j) * u(j)**H * B.

where u(j)**H is the conjugate-transpose of u(j).

BALANC (input) CHARACTER*1

Specifies the balance option to be performed:

= 'N': do not diagonally scale or permute;

= 'P': permute only;

= 'S': scale only;

= 'B': both permute and scale. Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. Permuting does not change condition numbers (in exact arithmetic), but balancing does.

JOBVL (input) CHARACTER*1

= 'N': do not compute the left generalized eigenvectors;

= 'V': compute the left generalized eigenvectors.

JOBVR (input) CHARACTER*1

= 'N': do not compute the right generalized eigenvectors;

= 'V': compute the right generalized eigenvectors.

SENSE (input) CHARACTER*1

Determines which reciprocal condition numbers are computed. = 'N': none are computed;

= 'E': computed for eigenvalues only;

= 'V': computed for eigenvectors only;

= 'B': computed for eigenvalues and eigenvectors.

N (input) INTEGER

The order of the matrices A, B, VL, and VR. N >= 0.

A (input/output) COMPLEX*16 array, dimension (LDA, N)

On entry, the matrix A in the pair (A,B). On exit, A has been overwritten. If JOBVL='V' or JOBVR='V' or both, then A contains the first part of the complex Schur form of the "balanced" versions of the input A and B.

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 in the pair (A,B). On exit, B has been overwritten. If JOBVL='V' or JOBVR='V' or both, then B contains the second part of the complex Schur form of the "balanced" versions of the input A and B.

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 generalized eigenvalues.

Note: the quotient 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 u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1. 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 JOBVR = 'V', the right generalized eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1. Not referenced if JOBVR = 'N'.

LDVR (input) INTEGER

The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.

ILO,IHI (output) INTEGER ILO and IHI are integer values such that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j = 1,...,ILO-1 or i = IHI+1,...,N. If BALANC = 'N' or 'S', ILO = 1 and IHI = N.

LSCALE (output) DOUBLE PRECISION array, dimension (N)

Details of the permutations and scaling factors applied to the left side of A and B. If PL(j) is the index of the row interchanged with row j, and DL(j) is the scaling factor applied to row j, then LSCALE(j) = PL(j) for j = 1,...,ILO-1 = DL(j) for j = ILO,...,IHI = PL(j) for j = IHI+1,...,N. The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1.

RSCALE (output) DOUBLE PRECISION array, dimension (N)

Details of the permutations and scaling factors applied to the right side of A and B. If PR(j) is the index of the column interchanged with column j, and DR(j) is the scaling factor applied to column j, then RSCALE(j) = PR(j) for j = 1,...,ILO-1 = DR(j) for j = ILO,...,IHI = PR(j) for j = IHI+1,...,N The order in which the interchanges are made is N to IHI+1, then 1 to ILO-1.

ABNRM (output) DOUBLE PRECISION

The one-norm of the balanced matrix A.

BBNRM (output) DOUBLE PRECISION

The one-norm of the balanced matrix B.

RCONDE (output) DOUBLE PRECISION array, dimension (N)

If SENSE = 'E' or 'B', the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. If SENSE = 'V', RCONDE is not referenced.

RCONDV (output) DOUBLE PRECISION array, dimension (N)

If JOB = 'V' or 'B', the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. If the eigenvalues cannot be reordered to compute RCONDV(j), RCONDV(j) is set to 0; this can only occur when the true value would be very small anyway. If SENSE = 'E', RCONDV is not referenced. Not referenced if JOB = 'E'.

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). If SENSE = 'N' or 'E', LWORK >= 2*N. If SENSE = 'V' or 'B', LWORK >= 2*N*N+2*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.

RWORK (workspace) DOUBLE PRECISION array, dimension (6*N)

Real workspace.

IWORK (workspace) INTEGER array, dimension (N+2)

If SENSE = 'E', IWORK is not referenced.

BWORK (workspace) LOGICAL array, dimension (N)

If SENSE = 'N', BWORK is not referenced.

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: =N+1: other than QZ iteration failed in ZHGEQZ.

=N+2: error return from ZTGEVC.

Balancing a matrix pair (A,B) includes, first, permuting rows and columns to isolate eigenvalues, second, applying diagonal similarity transformation to the rows and columns to make the rows and columns as close in norm as possible. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see section 4.11.1.2 of LAPACK Users' Guide.

An approximate error bound on the chordal distance between the i-th computed generalized eigenvalue w and the corresponding exact eigenvalue lambda is

chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)

An approximate error bound for the angle between the i-th computed eigenvector VL(i) or VR(i) is given by

EPS * norm(ABNRM, BBNRM) / DIF(i).

For further explanation of the reciprocal condition numbers RCONDE and RCONDV, see section 4.11 of LAPACK User's Guide.