# sggev3.f - Man Page

SRC/sggev3.f

## Synopsis

### Functions/Subroutines

subroutine sggev3 (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info)
SGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)

## Function/Subroutine Documentation

### subroutine sggev3 (character jobvl, character jobvr, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) alphar, real, dimension( * ) alphai, real, dimension( * ) beta, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, real, dimension( * ) work, integer lwork, integer info)

SGGEV3 computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices (blocked algorithm)

Purpose:

``` SGGEV3 computes for a pair of N-by-N real 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
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).```
Parameters

JOBVL

```          JOBVL is CHARACTER*1
= 'N':  do not compute the left generalized eigenvectors;
= 'V':  compute the left generalized eigenvectors.```

JOBVR

```          JOBVR is CHARACTER*1
= 'N':  do not compute the right generalized eigenvectors;
= 'V':  compute the right generalized eigenvectors.```

N

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

A

```          A is REAL array, dimension (LDA, N)
On entry, the matrix A in the pair (A,B).
On exit, A has been overwritten.```

LDA

```          LDA is INTEGER
The leading dimension of A.  LDA >= max(1,N).```

B

```          B is REAL array, dimension (LDB, N)
On entry, the matrix B in the pair (A,B).
On exit, B has been overwritten.```

LDB

```          LDB is INTEGER
The leading dimension of B.  LDB >= max(1,N).```

ALPHAR

`          ALPHAR is REAL array, dimension (N)`

ALPHAI

`          ALPHAI is REAL array, dimension (N)`

BETA

```          BETA is REAL array, dimension (N)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
be 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.

Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI 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

```          VL is REAL array, dimension (LDVL,N)
If JOBVL = 'V', the left eigenvectors u(j) are stored one
after another in the columns of VL, in the same order as
their eigenvalues. 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)-th 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).
Each eigenvector is scaled so the largest component has
abs(real part)+abs(imag. part)=1.
Not referenced if JOBVL = 'N'.```

LDVL

```          LDVL is INTEGER
The leading dimension of the matrix VL. LDVL >= 1, and
if JOBVL = 'V', LDVL >= N.```

VR

```          VR is REAL array, dimension (LDVR,N)
If JOBVR = 'V', the right eigenvectors v(j) are stored one
after another in the columns of VR, in the same order as
their eigenvalues. 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)-th 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).
Each eigenvector is scaled so the largest component has
abs(real part)+abs(imag. part)=1.
Not referenced if JOBVR = 'N'.```

LDVR

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

WORK

```          WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is INTEGER

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.```

INFO

```          INFO is 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 ALPHAR(j), ALPHAI(j), and BETA(j)
should be correct for j=INFO+1,...,N.
> N:  =N+1: other than QZ iteration failed in SLAQZ0.
=N+2: error return from STGEVC.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Definition at line 223 of file sggev3.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page sggev3(3) is an alias of sggev3.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK