# ?ggev

Computes the generalized eigenvalues, and the left and/or right generalized eigenvectors for a pair of nonsymmetric matrices.

## Syntax

FORTRAN 77:

call sggev(jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info)

call dggev(jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info)

call cggev(jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info)

call zggev(jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info)

Fortran 95:

call ggev(a, b, alphar, alphai, beta [,vl] [,vr] [,info])

call ggev(a, b, alpha, beta [, vl] [,vr] [,info])

## Description

This routine computes for a pair of n-by-n real/complex nonsymmetric matrices (A,B), the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.

A generalized eigenvalue for a pair of matrices (A,B) is a scalar λ or a ratio alpha / `beta = λ`, such that `A - λ*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 generalized eigenvector `v(j)` corresponding to the generalized eigenvalue `λ(j)` of (A,B) satisfies

`A*v(j) = λ(j)*B*v(j)`.

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

`u(j)H*A = λ(j)*u(j)H*B`

where `u(j)H` denotes the conjugate transpose of `u(j)`.

## Input Parameters

jobvl

  CHARACTER*1. Must be 'N' or 'V'.

  If `jobvl = 'N'`, the left generalized eigenvectors are not computed;

  If `jobvl = 'V'`, the left generalized eigenvectors are computed.

jobvr

  CHARACTER*1. Must be 'N' or 'V'.

  If `jobvr = 'N'`, the right generalized eigenvectors are not computed;

  If `jobvr = 'V'`, the right generalized eigenvectors are computed.

n

  INTEGER. The order of the matrices A, B, vl, and vr (`n ≥ 0`).

a, b, work

  REAL for sggev

  DOUBLE PRECISION for dggev

  COMPLEX for cggev

  DOUBLE COMPLEX for zggev.

  Arrays:

  a(lda,*) is an array containing the n-by-n matrix A (first of the pair of matrices).

  The second dimension of a must be at least max(1, n).

  b(ldb,*) is an array containing the n-by-n matrix B (second of the pair of matrices).

  The second dimension of b must be at least max(1, n).

  work is a workspace array, its dimension `max(1, lwork)`.

lda

  INTEGER. The first dimension of the array a. Must be at least max(1, n).

ldb

  INTEGER. The first dimension of the array b. Must be at least max(1, n).

ldvl, ldvr

  INTEGER. The first dimensions of the output matrices vl and vr, respectively.

  Constraints:

  `ldvl ≥ 1`. If `jobvl = 'V'`, `ldvl ≥ max(1, n)`.

  `ldvr ≥ 1`. If `jobvr = 'V'`, `ldvr ≥ max(1, n)`.

lwork

  INTEGER.

  The dimension of the array work.

  `lwork ≥ max(1, 8n+16)` for real flavors;

  `lwork ≥ max(1, 2n)` for complex flavors.

  For good performance, lwork must generally be larger.

  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

  REAL for cggev

  DOUBLE PRECISION for zggev

  Workspace array, DIMENSION at least max(1, 8n).

  This array is used in complex flavors only.

## Output Parameters

a, b

  On exit, these arrays have been overwritten.

alphar, alphai

  REAL for sggev;

  DOUBLE PRECISION for dggev.

  Arrays, DIMENSION at least max(1, n) each. Contain values that form generalized eigenvalues in real flavors.

  See beta.

alpha

  COMPLEX for cggev;

  DOUBLE COMPLEX for zggev.

  Array, DIMENSION at least max(1, n). Contain values that form generalized eigenvalues in complex flavors. See beta.

beta

  REAL for sggev

  DOUBLE PRECISION for dggev

  COMPLEX for cggev

  DOUBLE COMPLEX for zggev.

  Array, DIMENSION at least max(1, n).

  For real flavors:

  On exit, (alphar(j)+ alphai(j)*i)/beta(j), j=1,..., n, are the generalized eigenvalues.

  If alphai(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with alphai(j+1) negative.

  For complex flavors:

  On exit, alpha(j)/beta(j), j=1,..., n, are the generalized eigenvalues.

  See also Application Notes below.

vl, vr

  REAL for sggev

  DOUBLE PRECISION for dggev

  COMPLEX for cggev

  DOUBLE COMPLEX for zggev.

  Arrays:

  vl(ldvl,*); the second dimension of vl must be at least max(1, 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 is scaled so the largest component has `abs(Re) + abs(Im) = 1`.

  If `jobvl = 'N'`, vl is not referenced.

  For real flavors:

  If the j-th eigenvalue is real, then `u(j) = vl(:,j)`, the j-th column of vl.

  If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then `u(j) = vl(:,j) + i*vl(:,j+1)` and `u(j+1) = vl(:,j) - i*vl(:,j+1)`, where `i = sqrt(-1)`.

  For complex flavors:

  `u(j) = vl(:,j)`, the j-th column of vl.

  vr(ldvr,*); the second dimension of vr must be at least max(1, 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 is scaled so the largest component has abs(Re) + abs(Im) = 1.

  If `jobvr = 'N'`, vr is not referenced.

  For real flavors:

  If the j-th eigenvalue is real, then `v(j) = vr(:,j)`, the j-th column of vr.

  If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then `v(j) = vr(:,j) + i*vr(:,j+1)` and `v(j+1) = vr(:,j) - i*vr(:,j+1)`.

  For complex flavors:

  `v(j) = vr(:,j)`, the j-th column of vr.

work(1)

  On exit, if `info = 0`, then work(1) returns the required minimal size of lwork.

info

  INTEGER.

  If `info = 0`, the execution is successful.

  If `info = -i`, the i-th parameter had an illegal value.

  If `info = i`, and

  `i ≤ n`: the QZ iteration failed. No eigenvectors have been calculated, but alphar(j), alphai(j) (for real flavors), or alpha(j) (for complex flavors), and beta(j), `j=info+1,..., n` should be correct.

  `i > n`: errors that usually indicate LAPACK problems:

  `i = n+1`: other than QZ iteration failed in ?hgeqz;

  `i = n+2`: error return from ?tgevc.

## Fortran 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see Fortran 95 Interface Conventions.

Specific details for the routine ggev interface are the following:

a

  Holds the matrix A of size (n, n).

b

  Holds the matrix B of size (n, n).

alphar

  Holds the vector of length (n). Used in real flavors only.

alphai

  Holds the vector of length (n). Used in real flavors only.

alpha

  Holds the vector of length (n). Used in complex flavors only.

beta

  Holds the vector of length (n).

vl

  Holds the matrix VL of size (n, n).

vr

  Holds the matrix VR of size (n, n).

jobvl

  Restored based on the presence of the argument vl as follows:

  `jobvl = 'V'`, if vl is present,

  `jobvl = 'N'`, if vl is omitted.

jobvr

  Restored based on the presence of the argument vr as follows:

  `jobvr = 'V'`, if vr is present,

  `jobvr = 'N'`, if vr is omitted.

## Application Notes

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set `lwork = -1`.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (`work(1)`) for subsequent runs.

If you set `lwork = -1`, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.

The quotients alphar(j)/beta(j) and alphai(j)/beta(j) may easily over- or underflow, and beta(j) may even be zero. Thus, you should avoid simply computing the ratio. However, alphar and alphai (for real flavors) or alpha (for complex flavors) will be always less than and usually comparable with norm(A) in magnitude, and beta always less than and usually comparable with norm(B).

Parent topic: Generalized Nonsymmetric Eigenproblems