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## Arguments

A
(input/output) REAL or COMPLEX square array, shape .
On entry, the matrix .
On exit, A has been overwritten. If the left, the right or both generalized eigenvectors are computed, then A contains the first part of the real/complex Schur form of the "balanced" versions of the matrix pair ().

B
(input/output) REAL or COMPLEX square array, shape with .
On entry, the matrix .
On exit, B has been overwritten. If the left, the right or both generalized eigenvectors are computed, then B contains the second part of the real/complex Schur form of the "balanced" versions of the matrix pair ().

alpha
(output) REAL or COMPLEX array, shape with .
The values of .
::= ALPHAR(:), ALPHAI(:) ALPHA(:),
where
ALPHAR(:), ALPHAI(:) are of REAL type (for the real and imaginary parts) and ALPHA(:) is of COMPLEX type.

BETA
(output) REAL or COMPLEX array, shape with (BETA) (A,1).
The values of .
Note: The generalized eigenvalues of the pair are the scalars . These quotients may easily over- or underflow, and may even be zero. Thus, the user should avoid computing them naively.
Note: If A and B are real then complex eigenvalues occur in complex conjugate pairs. Each pair is stored consecutively. Thus a complex conjugate pair is given by

where

VL
Optional (output) REAL or COMPLEX square array, shape with (VL, 1) (A, 1).
The left generalized eigenvectors are stored in the columns of VL in the order of their eigenvalues. Each eigenvector is scaled so the largest component has , except that for eigenvalues with , a zero vector is returned as the corresponding eigenvector.
Note: If and are real then complex eigenvectors, like their eigenvalues, occur in complex conjugate pairs. The real and imaginary parts of the first eigenvector of the pair are stored in VL and VL. Thus a complex conjugate pair is given by

VR
Optional (output) REAL or COMPLEX square array, shape with (VR, 1) (A, 1).
The right generalized eigenvectors are stored in the columns of VR in the order of their eigenvalues. Each eigenvector is scaled so the largest component has , except that for eigenvalues with , a zero vector is returned as the corresponding eigenvector.
Note: If and are real then complex eigenvectors, like their eigenvalues, occur in complex conjugate pairs. The real and imaginary parts of the first eigenvector of the pair are stored in VR and VR. Thus a complex conjugate pair is given by

BALANC
Optional (input) CHARACTER(LEN=1).
Specifies the balance option to be performed.

Default value: 'N'.
Note: Computed reciprocal condition numbers will be for the matrices after balancing. Permuting does not change condition numbers (in exact arithmetic), but scaling does.

ILO,IHI
Optional (output) INTEGER.
ILO and IHI are integer values such that on exit and if and or .
If BALANC = 'N' or 'S', then and .

LSCALE
Optional (output) REAL array, shape with .
Details of the permutations and scaling factors applied to the left side of and . If is the index of the row interchanged with row , and is the scaling factor applied to row , then

and

.

RSCALE
Optional (output) REAL array, shape , .
Details of the permutations and scaling factors applied to the right side of and . If is the index of the column interchanged with column , and is the scaling factor applied to column , then

and

.

ABNRM
Optional (output) REAL.
The norm of after balancing.

BBNRM
Optional (output) REAL.
The norm of after balancing.

RCONDE
Optional (output) REAL array, shape with .
The reciprocal condition numbers of the eigenvalues.

RCONDV
Optional (output) REAL array, shape with .
The estimated reciprocal condition numbers of the right eigenvectors. If the eigenvalues cannot be reordered to compute RCONDV then RCONDV is set to . This can only occur when the true value would be very small.

INFO
Optional (output) INTEGER.

If INFO is not present and an error occurs, then the program is terminated with an error message.
References: [1] and [17,9,20,21].

Next: Example (from Program LA_GGEVX_EXAMPLE) Up: Generalized Nonsymmetric Eigenvalue Problems Previous: Purpose   Contents   Index
Susan Blackford 2001-08-19