realGEeigen(3) real

## Functions

subroutine sgegs (JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, INFO)
SGEGS computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine sgegv (JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)
SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine sgees (JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI, VS, LDVS, WORK, LWORK, BWORK, INFO)
SGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine sgeesx (JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO)
SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine sgeev (JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)
SGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine sgeevx (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, INFO)
SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine sgges (JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, BWORK, INFO)
SGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine sgges3 (JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, BWORK, INFO)
SGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)
subroutine sggesx (JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO)
SGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine sggev (JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)
SGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
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)
subroutine sggevx (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, BWORK, INFO)
SGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

## Detailed Description

This is the group of real eigenvalue driver functions for GE matrices

## subroutine sgees (character JOBVS, character SORT, external SELECT, integer N, real, dimension( lda, * ) A, integer LDA, integer SDIM, real, dimension( * ) WR, real, dimension( * ) WI, real, dimension( ldvs, * ) VS, integer LDVS, real, dimension( * ) WORK, integer LWORK, logical, dimension( * ) BWORK, integer INFO)

SGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

``` SGEES computes for an N-by-N real nonsymmetric matrix A, the
eigenvalues, the real Schur form T, and, optionally, the matrix of
Schur vectors Z.  This gives the Schur factorization A = Z*T*(Z**T).
Optionally, it also orders the eigenvalues on the diagonal of the
real Schur form so that selected eigenvalues are at the top left.
The leading columns of Z then form an orthonormal basis for the
invariant subspace corresponding to the selected eigenvalues.
A matrix is in real Schur form if it is upper quasi-triangular with
1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the
form
[  a  b  ]
[  c  a  ]
where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
```

Parameters:

JOBVS

```          JOBVS is CHARACTER*1
= 'N': Schur vectors are not computed;
= 'V': Schur vectors are computed.
```

SORT

```          SORT is CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the Schur form.
= 'N': Eigenvalues are not ordered;
= 'S': Eigenvalues are ordered (see SELECT).
```

SELECT

```          SELECT is LOGICAL FUNCTION of two REAL arguments
SELECT must be declared EXTERNAL in the calling subroutine.
If SORT = 'S', SELECT is used to select eigenvalues to sort
to the top left of the Schur form.
If SORT = 'N', SELECT is not referenced.
An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex
conjugate pair of eigenvalues is selected, then both complex
eigenvalues are selected.
Note that a selected complex eigenvalue may no longer
satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
ordering may change the value of complex eigenvalues
(especially if the eigenvalue is ill-conditioned); in this
case INFO is set to N+2 (see INFO below).
```

N

```          N is INTEGER
The order of the matrix A. N >= 0.
```

A

```          A is REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten by its real Schur form T.
```

LDA

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

SDIM

```          SDIM is INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting)
for which SELECT is true. (Complex conjugate
pairs for which SELECT is true for either
eigenvalue count as 2.)
```

WR

```          WR is REAL array, dimension (N)
```

WI

```          WI is REAL array, dimension (N)
WR and WI contain the real and imaginary parts,
respectively, of the computed eigenvalues in the same order
that they appear on the diagonal of the output Schur form T.
Complex conjugate pairs of eigenvalues will appear
consecutively with the eigenvalue having the positive
imaginary part first.
```

VS

```          VS is REAL array, dimension (LDVS,N)
If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
vectors.
If JOBVS = 'N', VS is not referenced.
```

LDVS

```          LDVS is INTEGER
The leading dimension of the array VS.  LDVS >= 1; if
JOBVS = 'V', LDVS >= N.
```

WORK

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

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.  LWORK >= max(1,3*N).
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.
```

BWORK

```          BWORK is LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.
```

INFO

```          INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, and i is
<= N: the QR algorithm failed to compute all the
eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
contain those eigenvalues which have converged; if
JOBVS = 'V', VS contains the matrix which reduces A
to its partially converged Schur form.
= N+1: the eigenvalues could not be reordered because some
eigenvalues were too close to separate (the problem
is very ill-conditioned);
= N+2: after reordering, roundoff changed values of some
complex eigenvalues so that leading eigenvalues in
the Schur form no longer satisfy SELECT=.TRUE.  This
could also be caused by underflow due to scaling.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

November 2011

## subroutine sgeesx (character JOBVS, character SORT, external SELECT, character SENSE, integer N, real, dimension( lda, * ) A, integer LDA, integer SDIM, real, dimension( * ) WR, real, dimension( * ) WI, real, dimension( ldvs, * ) VS, integer LDVS, real RCONDE, real RCONDV, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, logical, dimension( * ) BWORK, integer INFO)

SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

``` SGEESX computes for an N-by-N real nonsymmetric matrix A, the
eigenvalues, the real Schur form T, and, optionally, the matrix of
Schur vectors Z.  This gives the Schur factorization A = Z*T*(Z**T).
Optionally, it also orders the eigenvalues on the diagonal of the
real Schur form so that selected eigenvalues are at the top left;
computes a reciprocal condition number for the average of the
selected eigenvalues (RCONDE); and computes a reciprocal condition
number for the right invariant subspace corresponding to the
selected eigenvalues (RCONDV).  The leading columns of Z form an
orthonormal basis for this invariant subspace.
For further explanation of the reciprocal condition numbers RCONDE
and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
these quantities are called s and sep respectively).
A real matrix is in real Schur form if it is upper quasi-triangular
with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in
the form
[  a  b  ]
[  c  a  ]
where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
```

Parameters:

JOBVS

```          JOBVS is CHARACTER*1
= 'N': Schur vectors are not computed;
= 'V': Schur vectors are computed.
```

SORT

```          SORT is CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the Schur form.
= 'N': Eigenvalues are not ordered;
= 'S': Eigenvalues are ordered (see SELECT).
```

SELECT

```          SELECT is a LOGICAL FUNCTION of two REAL arguments
SELECT must be declared EXTERNAL in the calling subroutine.
If SORT = 'S', SELECT is used to select eigenvalues to sort
to the top left of the Schur form.
If SORT = 'N', SELECT is not referenced.
An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
SELECT(WR(j),WI(j)) is true; i.e., if either one of a
complex conjugate pair of eigenvalues is selected, then both
are.  Note that a selected complex eigenvalue may no longer
satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
ordering may change the value of complex eigenvalues
(especially if the eigenvalue is ill-conditioned); in this
case INFO may be set to N+3 (see INFO below).
```

SENSE

```          SENSE is CHARACTER*1
Determines which reciprocal condition numbers are computed.
= 'N': None are computed;
= 'E': Computed for average of selected eigenvalues only;
= 'V': Computed for selected right invariant subspace only;
= 'B': Computed for both.
If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
```

N

```          N is INTEGER
The order of the matrix A. N >= 0.
```

A

```          A is REAL array, dimension (LDA, N)
On entry, the N-by-N matrix A.
On exit, A is overwritten by its real Schur form T.
```

LDA

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

SDIM

```          SDIM is INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting)
for which SELECT is true. (Complex conjugate
pairs for which SELECT is true for either
eigenvalue count as 2.)
```

WR

```          WR is REAL array, dimension (N)
```

WI

```          WI is REAL array, dimension (N)
WR and WI contain the real and imaginary parts, respectively,
of the computed eigenvalues, in the same order that they
appear on the diagonal of the output Schur form T.  Complex
conjugate pairs of eigenvalues appear consecutively with the
eigenvalue having the positive imaginary part first.
```

VS

```          VS is REAL array, dimension (LDVS,N)
If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
vectors.
If JOBVS = 'N', VS is not referenced.
```

LDVS

```          LDVS is INTEGER
The leading dimension of the array VS.  LDVS >= 1, and if
JOBVS = 'V', LDVS >= N.
```

RCONDE

```          RCONDE is REAL
If SENSE = 'E' or 'B', RCONDE contains the reciprocal
condition number for the average of the selected eigenvalues.
Not referenced if SENSE = 'N' or 'V'.
```

RCONDV

```          RCONDV is REAL
If SENSE = 'V' or 'B', RCONDV contains the reciprocal
condition number for the selected right invariant subspace.
Not referenced if SENSE = 'N' or 'E'.
```

WORK

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

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.  LWORK >= max(1,3*N).
Also, if SENSE = 'E' or 'V' or 'B',
LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of
selected eigenvalues computed by this routine.  Note that
N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only
returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or
'B' this may not be large enough.
For good performance, LWORK must generally be larger.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates upper bounds on the optimal sizes of the
arrays WORK and IWORK, returns these values as the first
entries of the WORK and IWORK arrays, and no error messages
related to LWORK or LIWORK are issued by XERBLA.
```

IWORK

```          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
```

LIWORK

```          LIWORK is INTEGER
The dimension of the array IWORK.
LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is
only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this
may not be large enough.
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates upper bounds on the optimal sizes of
the arrays WORK and IWORK, returns these values as the first
entries of the WORK and IWORK arrays, and no error messages
related to LWORK or LIWORK are issued by XERBLA.
```

BWORK

```          BWORK is LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.
```

INFO

```          INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, and i is
<= N: the QR algorithm failed to compute all the
eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
contain those eigenvalues which have converged; if
JOBVS = 'V', VS contains the transformation which
reduces A to its partially converged Schur form.
= N+1: the eigenvalues could not be reordered because some
eigenvalues were too close to separate (the problem
is very ill-conditioned);
= N+2: after reordering, roundoff changed values of some
complex eigenvalues so that leading eigenvalues in
the Schur form no longer satisfy SELECT=.TRUE.  This
could also be caused by underflow due to scaling.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

June 2016

## subroutine sgeev (character JOBVL, character JOBVR, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) WR, real, dimension( * ) WI, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension( ldvr, * ) VR, integer LDVR, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGEEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

``` SGEEV computes for an N-by-N real nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors.
The right eigenvector v(j) of A satisfies
A * v(j) = lambda(j) * v(j)
where lambda(j) is its eigenvalue.
The left eigenvector u(j) of A satisfies
u(j)**H * A = lambda(j) * u(j)**H
where u(j)**H denotes the conjugate-transpose of u(j).
The computed eigenvectors are normalized to have Euclidean norm
equal to 1 and largest component real.
```

Parameters:

JOBVL

```          JOBVL is CHARACTER*1
= 'N': left eigenvectors of A are not computed;
= 'V': left eigenvectors of A are computed.
```

JOBVR

```          JOBVR is CHARACTER*1
= 'N': right eigenvectors of A are not computed;
= 'V': right eigenvectors of A are computed.
```

N

```          N is INTEGER
The order of the matrix A. N >= 0.
```

A

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

LDA

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

WR

```          WR is REAL array, dimension (N)
```

WI

```          WI is REAL array, dimension (N)
WR and WI contain the real and imaginary parts,
respectively, of the computed eigenvalues.  Complex
conjugate pairs of eigenvalues appear consecutively
with the eigenvalue having the positive imaginary part
first.
```

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 JOBVL = 'N', VL is not referenced.
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).
```

LDVL

```          LDVL is INTEGER
The leading dimension of the array VL.  LDVL >= 1; 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 JOBVR = 'N', VR is not referenced.
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).
```

LDVR

```          LDVR is INTEGER
The leading dimension of the array VR.  LDVR >= 1; 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
The dimension of the array WORK.  LWORK >= max(1,3*N), and
if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N.  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.
```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.
> 0:  if INFO = i, the QR algorithm failed to compute all the
eigenvalues, and no eigenvectors have been computed;
elements i+1:N of WR and WI contain eigenvalues which
have converged.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

June 2016

## subroutine sgeevx (character BALANC, character JOBVL, character JOBVR, character SENSE, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) WR, real, dimension( * ) WI, real, dimension( ldvl, * ) VL, integer LDVL, real, dimension( ldvr, * ) VR, integer LDVR, integer ILO, integer IHI, real, dimension( * ) SCALE, real ABNRM, real, dimension( * ) RCONDE, real, dimension( * ) RCONDV, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO)

SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

``` SGEEVX computes for an N-by-N real nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors.
Optionally also, it computes a balancing transformation to improve
the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
(RCONDE), and reciprocal condition numbers for the right
eigenvectors (RCONDV).
The right eigenvector v(j) of A satisfies
A * v(j) = lambda(j) * v(j)
where lambda(j) is its eigenvalue.
The left eigenvector u(j) of A satisfies
u(j)**H * A = lambda(j) * u(j)**H
where u(j)**H denotes the conjugate-transpose of u(j).
The computed eigenvectors are normalized to have Euclidean norm
equal to 1 and largest component real.
Balancing a matrix means permuting the rows and columns to make it
more nearly upper triangular, and applying a diagonal similarity
transformation D * A * D**(-1), where D is a diagonal matrix, to
make its rows and columns closer in norm and the condition numbers
of its eigenvalues and eigenvectors smaller.  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 4.10.2 of the LAPACK
Users' Guide.
```

Parameters:

BALANC

```          BALANC is CHARACTER*1
Indicates how the input matrix should be diagonally scaled
and/or permuted to improve the conditioning of its
eigenvalues.
= 'N': Do not diagonally scale or permute;
= 'P': Perform permutations to make the matrix more nearly
upper triangular. Do not diagonally scale;
= 'S': Diagonally scale the matrix, i.e. replace A by
D*A*D**(-1), where D is a diagonal matrix chosen
to make the rows and columns of A more equal in
norm. Do not permute;
= 'B': Both diagonally scale and permute A.
Computed reciprocal condition numbers will be for the matrix
after balancing and/or permuting. Permuting does not change
condition numbers (in exact arithmetic), but balancing does.
```

JOBVL

```          JOBVL is CHARACTER*1
= 'N': left eigenvectors of A are not computed;
= 'V': left eigenvectors of A are computed.
If SENSE = 'E' or 'B', JOBVL must = 'V'.
```

JOBVR

```          JOBVR is CHARACTER*1
= 'N': right eigenvectors of A are not computed;
= 'V': right eigenvectors of A are computed.
If SENSE = 'E' or 'B', JOBVR must = 'V'.
```

SENSE

```          SENSE is CHARACTER*1
Determines which reciprocal condition numbers are computed.
= 'N': None are computed;
= 'E': Computed for eigenvalues only;
= 'V': Computed for right eigenvectors only;
= 'B': Computed for eigenvalues and right eigenvectors.
If SENSE = 'E' or 'B', both left and right eigenvectors
must also be computed (JOBVL = 'V' and JOBVR = 'V').
```

N

```          N is INTEGER
The order of the matrix A. N >= 0.
```

A

```          A is REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten.  If JOBVL = 'V' or
JOBVR = 'V', A contains the real Schur form of the balanced
version of the input matrix A.
```

LDA

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

WR

```          WR is REAL array, dimension (N)
```

WI

```          WI is REAL array, dimension (N)
WR and WI contain the real and imaginary parts,
respectively, of the computed eigenvalues.  Complex
conjugate pairs of eigenvalues will appear consecutively
with the eigenvalue having the positive imaginary part
first.
```

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 JOBVL = 'N', VL is not referenced.
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).
```

LDVL

```          LDVL is INTEGER
The leading dimension of the array VL.  LDVL >= 1; 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 JOBVR = 'N', VR is not referenced.
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).
```

LDVR

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

ILO

```          ILO is INTEGER
```

IHI

```          IHI is INTEGER
ILO and IHI are integer values determined when A was
balanced.  The balanced A(i,j) = 0 if I > J and
J = 1,...,ILO-1 or I = IHI+1,...,N.
```

SCALE

```          SCALE is REAL array, dimension (N)
Details of the permutations and scaling factors applied
when balancing A.  If P(j) is the index of the row and column
interchanged with row and column j, and D(j) is the scaling
factor applied to row and column j, then
SCALE(J) = P(J),    for J = 1,...,ILO-1
= D(J),    for J = ILO,...,IHI
= P(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

```          ABNRM is REAL
The one-norm of the balanced matrix (the maximum
of the sum of absolute values of elements of any column).
```

RCONDE

```          RCONDE is REAL array, dimension (N)
RCONDE(j) is the reciprocal condition number of the j-th
eigenvalue.
```

RCONDV

```          RCONDV is REAL array, dimension (N)
RCONDV(j) is the reciprocal condition number of the j-th
right eigenvector.
```

WORK

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

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.   If SENSE = 'N' or 'E',
LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',
LWORK >= 3*N.  If SENSE = 'V' or 'B', LWORK >= N*(N+6).
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.
```

IWORK

```          IWORK is INTEGER array, dimension (2*N-2)
If SENSE = 'N' or 'E', not referenced.
```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value.
> 0:  if INFO = i, the QR algorithm failed to compute all the
eigenvalues, and no eigenvectors or condition numbers
have been computed; elements 1:ILO-1 and i+1:N of WR
and WI contain eigenvalues which have converged.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

June 2016

## subroutine sgegs (character JOBVSL, character JOBVSR, 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( ldvsl, * ) VSL, integer LDVSL, real, dimension( ldvsr, * ) VSR, integer LDVSR, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGEGS computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

``` This routine is deprecated and has been replaced by routine SGGES.
SGEGS computes the eigenvalues, real Schur form, and, optionally,
left and or/right Schur vectors of a real matrix pair (A,B).
Given two square matrices A and B, the generalized real Schur
factorization has the form

A = Q*S*Z**T,  B = Q*T*Z**T
where Q and Z are orthogonal matrices, T is upper triangular, and S
is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal
blocks, the 2-by-2 blocks corresponding to complex conjugate pairs
of eigenvalues of (A,B).  The columns of Q are the left Schur vectors
and the columns of Z are the right Schur vectors.

If only the eigenvalues of (A,B) are needed, the driver routine
SGEGV should be used instead.  See SGEGV for a description of the
eigenvalues of the generalized nonsymmetric eigenvalue problem
(GNEP).
```

Parameters:

JOBVSL

```          JOBVSL is CHARACTER*1
= 'N':  do not compute the left Schur vectors;
= 'V':  compute the left Schur vectors (returned in VSL).
```

JOBVSR

```          JOBVSR is CHARACTER*1
= 'N':  do not compute the right Schur vectors;
= 'V':  compute the right Schur vectors (returned in VSR).
```

N

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

A

```          A is REAL array, dimension (LDA, N)
On entry, the matrix A.
On exit, the upper quasi-triangular matrix S from the
generalized real Schur factorization.
```

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.
On exit, the upper triangular matrix T from the generalized
real Schur factorization.
```

LDB

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

ALPHAR

```          ALPHAR is REAL array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue
of GNEP.
```

ALPHAI

```          ALPHAI is REAL array, dimension (N)
The imaginary parts of each scalar alpha defining an
eigenvalue of GNEP.  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) = -ALPHAI(j).
```

BETA

```          BETA is REAL array, dimension (N)
The scalars beta that define the eigenvalues of GNEP.
Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the matrix
pair (A,B), in one of the forms lambda = alpha/beta or
mu = beta/alpha.  Since either lambda or mu may overflow,
they should not, in general, be computed.
```

VSL

```          VSL is REAL array, dimension (LDVSL,N)
If JOBVSL = 'V', the matrix of left Schur vectors Q.
Not referenced if JOBVSL = 'N'.
```

LDVSL

```          LDVSL is INTEGER
The leading dimension of the matrix VSL. LDVSL >=1, and
if JOBVSL = 'V', LDVSL >= N.
```

VSR

```          VSR is REAL array, dimension (LDVSR,N)
If JOBVSR = 'V', the matrix of right Schur vectors Z.
Not referenced if JOBVSR = 'N'.
```

LDVSR

```          LDVSR is INTEGER
The leading dimension of the matrix VSR. LDVSR >= 1, and
if JOBVSR = 'V', LDVSR >= 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
The dimension of the array WORK.  LWORK >= max(1,4*N).
For good performance, LWORK must generally be larger.
To compute the optimal value of LWORK, call ILAENV to get
blocksizes (for SGEQRF, SORMQR, and SORGQR.)  Then compute:
NB  -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR
The optimal LWORK is  2*N + N*(NB+1).
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.  (A,B) are not in Schur
form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
be correct for j=INFO+1,...,N.
> N:  errors that usually indicate LAPACK problems:
=N+1: error return from SGGBAL
=N+2: error return from SGEQRF
=N+3: error return from SORMQR
=N+4: error return from SORGQR
=N+5: error return from SGGHRD
=N+6: error return from SHGEQZ (other than failed
iteration)
=N+7: error return from SGGBAK (computing VSL)
=N+8: error return from SGGBAK (computing VSR)
=N+9: error return from SLASCL (various places)
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

November 2011

## subroutine sgegv (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)

SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

``` This routine is deprecated and has been replaced by routine SGGEV.
SGEGV computes the eigenvalues and, optionally, the left and/or right
eigenvectors of a real matrix pair (A,B).
Given two square matrices A and B,
the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
eigenvalues lambda and corresponding (non-zero) eigenvectors x such
that
A*x = lambda*B*x.
An alternate form is to find the eigenvalues mu and corresponding
eigenvectors y such that
mu*A*y = B*y.
These two forms are equivalent with mu = 1/lambda and x = y if
neither lambda nor mu is zero.  In order to deal with the case that
lambda or mu is zero or small, two values alpha and beta are returned
for each eigenvalue, such that lambda = alpha/beta and
mu = beta/alpha.
The vectors x and y in the above equations are right eigenvectors of
the matrix pair (A,B).  Vectors u and v satisfying
u**H*A = lambda*u**H*B  or  mu*v**H*A = v**H*B
are left eigenvectors of (A,B).
Note: this routine performs "full balancing" on A and B
```

Parameters:

JOBVL

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

JOBVR

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

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.
If JOBVL = 'V' or JOBVR = 'V', then on exit A
contains the real Schur form of A from the generalized Schur
factorization of the pair (A,B) after balancing.
If no eigenvectors were computed, then only the diagonal
blocks from the Schur form will be correct.  See SGGHRD and
SHGEQZ for details.
```

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.
If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
upper triangular matrix obtained from B in the generalized
Schur factorization of the pair (A,B) after balancing.
If no eigenvectors were computed, then only those elements of
B corresponding to the diagonal blocks from the Schur form of
A will be correct.  See SGGHRD and SHGEQZ for details.
```

LDB

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

ALPHAR

```          ALPHAR is REAL array, dimension (N)
The real parts of each scalar alpha defining an eigenvalue of
GNEP.
```

ALPHAI

```          ALPHAI is REAL array, dimension (N)
The imaginary parts of each scalar alpha defining an
eigenvalue of GNEP.  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) = -ALPHAI(j).
```

BETA

```          BETA is REAL array, dimension (N)
The scalars beta that define the eigenvalues of GNEP.

Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
beta = BETA(j) represent the j-th eigenvalue of the matrix
pair (A,B), in one of the forms lambda = alpha/beta or
mu = beta/alpha.  Since either lambda or mu may overflow,
they should not, in general, be computed.
```

VL

```          VL is REAL array, dimension (LDVL,N)
If JOBVL = 'V', the left eigenvectors u(j) are stored
in the columns of VL, in the same order as their eigenvalues.
If the j-th eigenvalue is real, then u(j) = VL(:,j).
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).
Each eigenvector is scaled so that its largest component has
abs(real part) + abs(imag. part) = 1, except for eigenvectors
corresponding to an eigenvalue with alpha = beta = 0, which
are set to zero.
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 x(j) are stored
in the columns of VR, in the same order as their eigenvalues.
If the j-th eigenvalue is real, then x(j) = VR(:,j).
If the j-th and (j+1)-st eigenvalues form a complex conjugate
pair, then
x(j) = VR(:,j) + i*VR(:,j+1)
and
x(j+1) = VR(:,j) - i*VR(:,j+1).
Each eigenvector is scaled so that its largest component has
abs(real part) + abs(imag. part) = 1, except for eigenvalues
corresponding to an eigenvalue with alpha = beta = 0, which
are set to zero.
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
The dimension of the array WORK.  LWORK >= max(1,8*N).
For good performance, LWORK must generally be larger.
To compute the optimal value of LWORK, call ILAENV to get
blocksizes (for SGEQRF, SORMQR, and SORGQR.)  Then compute:
NB  -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR;
The optimal LWORK is:
2*N + MAX( 6*N, N*(NB+1) ).
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:  errors that usually indicate LAPACK problems:
=N+1: error return from SGGBAL
=N+2: error return from SGEQRF
=N+3: error return from SORMQR
=N+4: error return from SORGQR
=N+5: error return from SGGHRD
=N+6: error return from SHGEQZ (other than failed
iteration)
=N+7: error return from STGEVC
=N+8: error return from SGGBAK (computing VL)
=N+9: error return from SGGBAK (computing VR)
=N+10: error return from SLASCL (various calls)
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

November 2011

Further Details:

```  Balancing
---------
This driver calls SGGBAL 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 elements close to
one (except for the elements that start out zero.)
After the eigenvalues and eigenvectors of the balanced matrices
have been computed, SGGBAK transforms the eigenvectors back to what
they would have been (in perfect arithmetic) 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 real Schur
form[*] of the "balanced" versions of A and B.  If no eigenvectors
are computed, then only the diagonal blocks will be correct.
[*] See SHGEQZ, SGEGS, or read the book "Matrix Computations",
by Golub & van Loan, pub. by Johns Hopkins U. Press.
```

## subroutine sgges (character JOBVSL, character JOBVSR, character SORT, external SELCTG, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, integer SDIM, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldvsl, * ) VSL, integer LDVSL, real, dimension( ldvsr, * ) VSR, integer LDVSR, real, dimension( * ) WORK, integer LWORK, logical, dimension( * ) BWORK, integer INFO)

SGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

``` SGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B),
the generalized eigenvalues, the generalized real Schur form (S,T),
optionally, the left and/or right matrices of Schur vectors (VSL and
VSR). This gives the generalized Schur factorization
(A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
Optionally, it also orders the eigenvalues so that a selected cluster
of eigenvalues appears in the leading diagonal blocks of the upper
quasi-triangular matrix S and the upper triangular matrix T.The
leading columns of VSL and VSR then form an orthonormal basis for the
corresponding left and right eigenspaces (deflating subspaces).
(If only the generalized eigenvalues are needed, use the driver
A generalized eigenvalue for a pair of matrices (A,B) is 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 interpretation for beta=0 or both being zero.
A pair of matrices (S,T) is in generalized real Schur form if T is
upper triangular with non-negative diagonal and S is block upper
triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
to real generalized eigenvalues, while 2-by-2 blocks of S will be
"standardized" by making the corresponding elements of T have the
form:
[  a  0  ]
[  0  b  ]
and the pair of corresponding 2-by-2 blocks in S and T will have a
complex conjugate pair of generalized eigenvalues.
```

Parameters:

JOBVSL

```          JOBVSL is CHARACTER*1
= 'N':  do not compute the left Schur vectors;
= 'V':  compute the left Schur vectors.
```

JOBVSR

```          JOBVSR is CHARACTER*1
= 'N':  do not compute the right Schur vectors;
= 'V':  compute the right Schur vectors.
```

SORT

```          SORT is CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the generalized Schur form.
= 'N':  Eigenvalues are not ordered;
= 'S':  Eigenvalues are ordered (see SELCTG);
```

SELCTG

```          SELCTG is a LOGICAL FUNCTION of three REAL arguments
SELCTG must be declared EXTERNAL in the calling subroutine.
If SORT = 'N', SELCTG is not referenced.
If SORT = 'S', SELCTG is used to select eigenvalues to sort
to the top left of the Schur form.
An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
one of a complex conjugate pair of eigenvalues is selected,
then both complex eigenvalues are selected.
Note that in the ill-conditioned case, a selected complex
eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
in this case.
```

N

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

A

```          A is REAL array, dimension (LDA, N)
On entry, the first of the pair of matrices.
On exit, A has been overwritten by its generalized Schur
form S.
```

LDA

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

B

```          B is REAL array, dimension (LDB, N)
On entry, the second of the pair of matrices.
On exit, B has been overwritten by its generalized Schur
form T.
```

LDB

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

SDIM

```          SDIM is INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting)
for which SELCTG is true.  (Complex conjugate pairs for which
SELCTG is true for either eigenvalue count as 2.)
```

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.  ALPHAR(j) + ALPHAI(j)*i,
and  BETA(j),j=1,...,N are the diagonals of the complex Schur
form (S,T) that would result if the 2-by-2 diagonal blocks of
the real Schur form of (A,B) were further reduced to
triangular form using 2-by-2 complex unitary transformations.
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.
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).
```

VSL

```          VSL is REAL array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur vectors.
Not referenced if JOBVSL = 'N'.
```

LDVSL

```          LDVSL is INTEGER
The leading dimension of the matrix VSL. LDVSL >=1, and
if JOBVSL = 'V', LDVSL >= N.
```

VSR

```          VSR is REAL array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur vectors.
Not referenced if JOBVSR = 'N'.
```

LDVSR

```          LDVSR is INTEGER
The leading dimension of the matrix VSR. LDVSR >= 1, and
if JOBVSR = 'V', LDVSR >= 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
The dimension of the array WORK.
If N = 0, LWORK >= 1, else LWORK >= max(8*N,6*N+16).
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.
```

BWORK

```          BWORK is LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.
```

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.  (A,B) are not in Schur
form, 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 SHGEQZ.
=N+2: after reordering, roundoff changed values of
some complex eigenvalues so that leading
eigenvalues in the Generalized Schur form no
longer satisfy SELCTG=.TRUE.  This could also
be caused due to scaling.
=N+3: reordering failed in STGSEN.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

November 2011

## subroutine sgges3 (character JOBVSL, character JOBVSR, character SORT, external SELCTG, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, integer SDIM, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldvsl, * ) VSL, integer LDVSL, real, dimension( ldvsr, * ) VSR, integer LDVSR, real, dimension( * ) WORK, integer LWORK, logical, dimension( * ) BWORK, integer INFO)

SGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm)

Purpose:

``` SGGES3 computes for a pair of N-by-N real nonsymmetric matrices (A,B),
the generalized eigenvalues, the generalized real Schur form (S,T),
optionally, the left and/or right matrices of Schur vectors (VSL and
VSR). This gives the generalized Schur factorization
(A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
Optionally, it also orders the eigenvalues so that a selected cluster
of eigenvalues appears in the leading diagonal blocks of the upper
quasi-triangular matrix S and the upper triangular matrix T.The
leading columns of VSL and VSR then form an orthonormal basis for the
corresponding left and right eigenspaces (deflating subspaces).
(If only the generalized eigenvalues are needed, use the driver
A generalized eigenvalue for a pair of matrices (A,B) is 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 interpretation for beta=0 or both being zero.
A pair of matrices (S,T) is in generalized real Schur form if T is
upper triangular with non-negative diagonal and S is block upper
triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
to real generalized eigenvalues, while 2-by-2 blocks of S will be
"standardized" by making the corresponding elements of T have the
form:
[  a  0  ]
[  0  b  ]
and the pair of corresponding 2-by-2 blocks in S and T will have a
complex conjugate pair of generalized eigenvalues.
```

Parameters:

JOBVSL

```          JOBVSL is CHARACTER*1
= 'N':  do not compute the left Schur vectors;
= 'V':  compute the left Schur vectors.
```

JOBVSR

```          JOBVSR is CHARACTER*1
= 'N':  do not compute the right Schur vectors;
= 'V':  compute the right Schur vectors.
```

SORT

```          SORT is CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the generalized Schur form.
= 'N':  Eigenvalues are not ordered;
= 'S':  Eigenvalues are ordered (see SELCTG);
```

SELCTG

```          SELCTG is a LOGICAL FUNCTION of three REAL arguments
SELCTG must be declared EXTERNAL in the calling subroutine.
If SORT = 'N', SELCTG is not referenced.
If SORT = 'S', SELCTG is used to select eigenvalues to sort
to the top left of the Schur form.
An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
one of a complex conjugate pair of eigenvalues is selected,
then both complex eigenvalues are selected.
Note that in the ill-conditioned case, a selected complex
eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
in this case.
```

N

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

A

```          A is REAL array, dimension (LDA, N)
On entry, the first of the pair of matrices.
On exit, A has been overwritten by its generalized Schur
form S.
```

LDA

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

B

```          B is REAL array, dimension (LDB, N)
On entry, the second of the pair of matrices.
On exit, B has been overwritten by its generalized Schur
form T.
```

LDB

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

SDIM

```          SDIM is INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting)
for which SELCTG is true.  (Complex conjugate pairs for which
SELCTG is true for either eigenvalue count as 2.)
```

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.  ALPHAR(j) + ALPHAI(j)*i,
and  BETA(j),j=1,...,N are the diagonals of the complex Schur
form (S,T) that would result if the 2-by-2 diagonal blocks of
the real Schur form of (A,B) were further reduced to
triangular form using 2-by-2 complex unitary transformations.
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.
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).
```

VSL

```          VSL is REAL array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur vectors.
Not referenced if JOBVSL = 'N'.
```

LDVSL

```          LDVSL is INTEGER
The leading dimension of the matrix VSL. LDVSL >=1, and
if JOBVSL = 'V', LDVSL >= N.
```

VSR

```          VSR is REAL array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur vectors.
Not referenced if JOBVSR = 'N'.
```

LDVSR

```          LDVSR is INTEGER
The leading dimension of the matrix VSR. LDVSR >= 1, and
if JOBVSR = 'V', LDVSR >= 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
The dimension of the array WORK.
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.
```

BWORK

```          BWORK is LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.
```

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.  (A,B) are not in Schur
form, 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 SHGEQZ.
=N+2: after reordering, roundoff changed values of
some complex eigenvalues so that leading
eigenvalues in the Generalized Schur form no
longer satisfy SELCTG=.TRUE.  This could also
be caused due to scaling.
=N+3: reordering failed in STGSEN.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

January 2015

## subroutine sggesx (character JOBVSL, character JOBVSR, character SORT, external SELCTG, character SENSE, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, integer SDIM, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldvsl, * ) VSL, integer LDVSL, real, dimension( ldvsr, * ) VSR, integer LDVSR, real, dimension( 2 ) RCONDE, real, dimension( 2 ) RCONDV, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, logical, dimension( * ) BWORK, integer INFO)

SGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

``` SGGESX computes for a pair of N-by-N real nonsymmetric matrices
(A,B), the generalized eigenvalues, the real Schur form (S,T), and,
optionally, the left and/or right matrices of Schur vectors (VSL and
VSR).  This gives the generalized Schur factorization
(A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T )
Optionally, it also orders the eigenvalues so that a selected cluster
of eigenvalues appears in the leading diagonal blocks of the upper
quasi-triangular matrix S and the upper triangular matrix T; computes
a reciprocal condition number for the average of the selected
eigenvalues (RCONDE); and computes a reciprocal condition number for
the right and left deflating subspaces corresponding to the selected
eigenvalues (RCONDV). The leading columns of VSL and VSR then form
an orthonormal basis for the corresponding left and right eigenspaces
(deflating subspaces).
A generalized eigenvalue for a pair of matrices (A,B) is 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 interpretation for beta=0 or for both being zero.
A pair of matrices (S,T) is in generalized real Schur form if T is
upper triangular with non-negative diagonal and S is block upper
triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
to real generalized eigenvalues, while 2-by-2 blocks of S will be
"standardized" by making the corresponding elements of T have the
form:
[  a  0  ]
[  0  b  ]
and the pair of corresponding 2-by-2 blocks in S and T will have a
complex conjugate pair of generalized eigenvalues.
```

Parameters:

JOBVSL

```          JOBVSL is CHARACTER*1
= 'N':  do not compute the left Schur vectors;
= 'V':  compute the left Schur vectors.
```

JOBVSR

```          JOBVSR is CHARACTER*1
= 'N':  do not compute the right Schur vectors;
= 'V':  compute the right Schur vectors.
```

SORT

```          SORT is CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the generalized Schur form.
= 'N':  Eigenvalues are not ordered;
= 'S':  Eigenvalues are ordered (see SELCTG).
```

SELCTG

```          SELCTG is procedure) LOGICAL FUNCTION of three REAL arguments
SELCTG must be declared EXTERNAL in the calling subroutine.
If SORT = 'N', SELCTG is not referenced.
If SORT = 'S', SELCTG is used to select eigenvalues to sort
to the top left of the Schur form.
An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
one of a complex conjugate pair of eigenvalues is selected,
then both complex eigenvalues are selected.
Note that a selected complex eigenvalue may no longer satisfy
SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering,
since ordering may change the value of complex eigenvalues
(especially if the eigenvalue is ill-conditioned), in this
case INFO is set to N+3.
```

SENSE

```          SENSE is CHARACTER*1
Determines which reciprocal condition numbers are computed.
= 'N' : None are computed;
= 'E' : Computed for average of selected eigenvalues only;
= 'V' : Computed for selected deflating subspaces only;
= 'B' : Computed for both.
If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.
```

N

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

A

```          A is REAL array, dimension (LDA, N)
On entry, the first of the pair of matrices.
On exit, A has been overwritten by its generalized Schur
form S.
```

LDA

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

B

```          B is REAL array, dimension (LDB, N)
On entry, the second of the pair of matrices.
On exit, B has been overwritten by its generalized Schur
form T.
```

LDB

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

SDIM

```          SDIM is INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting)
for which SELCTG is true.  (Complex conjugate pairs for which
SELCTG is true for either eigenvalue count as 2.)
```

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.  ALPHAR(j) + ALPHAI(j)*i
and BETA(j),j=1,...,N  are the diagonals of the complex Schur
form (S,T) that would result if the 2-by-2 diagonal blocks of
the real Schur form of (A,B) were further reduced to
triangular form using 2-by-2 complex unitary transformations.
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.
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).
```

VSL

```          VSL is REAL array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur vectors.
Not referenced if JOBVSL = 'N'.
```

LDVSL

```          LDVSL is INTEGER
The leading dimension of the matrix VSL. LDVSL >=1, and
if JOBVSL = 'V', LDVSL >= N.
```

VSR

```          VSR is REAL array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur vectors.
Not referenced if JOBVSR = 'N'.
```

LDVSR

```          LDVSR is INTEGER
The leading dimension of the matrix VSR. LDVSR >= 1, and
if JOBVSR = 'V', LDVSR >= N.
```

RCONDE

```          RCONDE is REAL array, dimension ( 2 )
If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
reciprocal condition numbers for the average of the selected
eigenvalues.
Not referenced if SENSE = 'N' or 'V'.
```

RCONDV

```          RCONDV is REAL array, dimension ( 2 )
If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
reciprocal condition numbers for the selected deflating
subspaces.
Not referenced if SENSE = 'N' or 'E'.
```

WORK

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

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.
If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else
LWORK >= max( 8*N, 6*N+16 ).
Note that 2*SDIM*(N-SDIM) <= N*N/2.
Note also that an error is only returned if
LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B'
this may not be large enough.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the bound on the optimal size of the WORK
array and the minimum size of the IWORK array, returns these
values as the first entries of the WORK and IWORK arrays, and
no error message related to LWORK or LIWORK is issued by
XERBLA.
```

IWORK

```          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
```

LIWORK

```          LIWORK is INTEGER
The dimension of the array IWORK.
If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
LIWORK >= N+6.
If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the bound on the optimal size of the
WORK array and the minimum size of the IWORK array, returns
these values as the first entries of the WORK and IWORK
arrays, and no error message related to LWORK or LIWORK is
issued by XERBLA.
```

BWORK

```          BWORK is LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.
```

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.  (A,B) are not in Schur
form, 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 SHGEQZ
=N+2: after reordering, roundoff changed values of
some complex eigenvalues so that leading
eigenvalues in the Generalized Schur form no
longer satisfy SELCTG=.TRUE.  This could also
be caused due to scaling.
=N+3: reordering failed in STGSEN.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

November 2011

Further Details:

```  An approximate (asymptotic) bound on the average absolute error of
the selected eigenvalues is
EPS * norm((A, B)) / RCONDE( 1 ).
An approximate (asymptotic) bound on the maximum angular error in
the computed deflating subspaces is
EPS * norm((A, B)) / RCONDV( 2 ).
```

## subroutine sggev (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)

SGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

``` SGGEV 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
The dimension of the array WORK.  LWORK >= max(1,8*N).
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.
```

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 SHGEQZ.
=N+2: error return from STGEVC.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

April 2012

## 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 SHGEQZ.
=N+2: error return from STGEVC.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

January 2015

## subroutine sggevx (character BALANC, character JOBVL, character JOBVR, character SENSE, 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, integer ILO, integer IHI, real, dimension( * ) LSCALE, real, dimension( * ) RSCALE, real ABNRM, real BBNRM, real, dimension( * ) RCONDE, real, dimension( * ) RCONDV, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, logical, dimension( * ) BWORK, integer INFO)

SGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

``` SGGEVX 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.
Optionally also, it 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).
```

Parameters:

BALANC

```          BALANC is 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

```          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.
```

SENSE

```          SENSE is 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

```          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. If JOBVL='V' or JOBVR='V'
or both, then A contains the first part of the real Schur
form of the "balanced" versions of the input A and B.
```

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. If JOBVL='V' or JOBVR='V'
or both, then B contains the second part of the real Schur
form of the "balanced" versions of the input A and B.
```

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 will be scaled so the largest component have
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 will be scaled so the largest component have
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.
```

ILO

```          ILO is INTEGER
```

IHI

```          IHI is 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

```          LSCALE is REAL 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

```          RSCALE is REAL 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

```          ABNRM is REAL
The one-norm of the balanced matrix A.
```

BBNRM

```          BBNRM is REAL
The one-norm of the balanced matrix B.
```

RCONDE

```          RCONDE is REAL array, dimension (N)
If SENSE = 'E' or 'B', the reciprocal condition numbers of
the eigenvalues, stored in consecutive elements of the array.
For a complex conjugate pair of eigenvalues two consecutive
elements of RCONDE are set to the same value. Thus RCONDE(j),
RCONDV(j), and the j-th columns of VL and VR all correspond
to the j-th eigenpair.
If SENSE = 'N' or 'V', RCONDE is not referenced.
```

RCONDV

```          RCONDV is REAL array, dimension (N)
If SENSE = 'V' or 'B', the estimated reciprocal condition
numbers of the eigenvectors, stored in consecutive elements
of the array. For a complex eigenvector two consecutive
elements of RCONDV are set to the same value. 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 = 'N' or 'E', RCONDV is not referenced.
```

WORK

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

LWORK

```          LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,2*N).
If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
LWORK >= max(1,6*N).
If SENSE = 'E', LWORK >= max(1,10*N).
If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.
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.
```

IWORK

```          IWORK is INTEGER array, dimension (N+6)
If SENSE = 'E', IWORK is not referenced.
```

BWORK

```          BWORK is LOGICAL array, dimension (N)
If SENSE = 'N', BWORK is not referenced.
```

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 SHGEQZ.
=N+2: error return from STGEVC.
```

Author:

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date:

April 2012

Further Details:

```  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.
```

## Author

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