trevc3 - Man Page

trevc3: eigenvectors of triangular Schur form, blocked

Synopsis

Functions

subroutine ctrevc3 (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork, rwork, lrwork, info)
CTREVC3
subroutine dtrevc3 (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork, info)
DTREVC3
subroutine strevc3 (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork, info)
STREVC3
subroutine ztrevc3 (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork, rwork, lrwork, info)
ZTREVC3

Function Documentation

subroutine ctrevc3 (character side, character howmny, logical, dimension( * ) select, integer n, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer lrwork, integer info)

CTREVC3

Purpose:

``` CTREVC3 computes some or all of the right and/or left eigenvectors of
a complex upper triangular matrix T.
Matrices of this type are produced by the Schur factorization of
a complex general matrix:  A = Q*T*Q**H, as computed by CHSEQR.

The right eigenvector x and the left eigenvector y of T corresponding
to an eigenvalue w are defined by:

T*x = w*x,     (y**H)*T = w*(y**H)

where y**H denotes the conjugate transpose of the vector y.
The eigenvalues are not input to this routine, but are read directly
from the diagonal of T.

This routine returns the matrices X and/or Y of right and left
eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
input matrix. If Q is the unitary factor that reduces a matrix A to
Schur form T, then Q*X and Q*Y are the matrices of right and left
eigenvectors of A.

This uses a Level 3 BLAS version of the back transformation.```
Parameters

SIDE

```          SIDE is CHARACTER*1
= 'R':  compute right eigenvectors only;
= 'L':  compute left eigenvectors only;
= 'B':  compute both right and left eigenvectors.```

HOWMNY

```          HOWMNY is CHARACTER*1
= 'A':  compute all right and/or left eigenvectors;
= 'B':  compute all right and/or left eigenvectors,
backtransformed using the matrices supplied in
VR and/or VL;
= 'S':  compute selected right and/or left eigenvectors,
as indicated by the logical array SELECT.```

SELECT

```          SELECT is LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenvectors to be
computed.
The eigenvector corresponding to the j-th eigenvalue is
computed if SELECT(j) = .TRUE..
Not referenced if HOWMNY = 'A' or 'B'.```

N

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

T

```          T is COMPLEX array, dimension (LDT,N)
The upper triangular matrix T.  T is modified, but restored
on exit.```

LDT

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

VL

```          VL is COMPLEX array, dimension (LDVL,MM)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
contain an N-by-N matrix Q (usually the unitary matrix Q of
Schur vectors returned by CHSEQR).
On exit, if SIDE = 'L' or 'B', VL contains:
if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
if HOWMNY = 'B', the matrix Q*Y;
if HOWMNY = 'S', the left eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VL, in the same order as their
eigenvalues.
Not referenced if SIDE = 'R'.```

LDVL

```          LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N.```

VR

```          VR is COMPLEX array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
contain an N-by-N matrix Q (usually the unitary matrix Q of
Schur vectors returned by CHSEQR).
On exit, if SIDE = 'R' or 'B', VR contains:
if HOWMNY = 'A', the matrix X of right eigenvectors of T;
if HOWMNY = 'B', the matrix Q*X;
if HOWMNY = 'S', the right eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VR, in the same order as their
eigenvalues.
Not referenced if SIDE = 'L'.```

LDVR

```          LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.```

MM

```          MM is INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.```

M

```          M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors.
If HOWMNY = 'A' or 'B', M is set to N.
Each selected eigenvector occupies one column.```

WORK

`          WORK is COMPLEX array, dimension (MAX(1,LWORK))`

LWORK

```          LWORK is INTEGER
The dimension of array WORK. LWORK >= max(1,2*N).
For optimum performance, LWORK >= N + 2*N*NB, where NB is
the optimal blocksize.

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

`          RWORK is REAL array, dimension (LRWORK)`

LRWORK

```          LRWORK is INTEGER
The dimension of array RWORK. LRWORK >= max(1,N).

If LRWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the RWORK array, returns
this value as the first entry of the RWORK array, and no error
message related to LRWORK is issued by XERBLA.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

```  The algorithm used in this program is basically backward (forward)
substitution, with scaling to make the the code robust against
possible overflow.

Each eigenvector is normalized so that the element of largest
magnitude has magnitude 1; here the magnitude of a complex number
(x,y) is taken to be |x| + |y|.```

Definition at line 242 of file ctrevc3.f.

subroutine dtrevc3 (character side, character howmny, logical, dimension( * ) select, integer n, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, double precision, dimension( * ) work, integer lwork, integer info)

DTREVC3

Purpose:

``` DTREVC3 computes some or all of the right and/or left eigenvectors of
a real upper quasi-triangular matrix T.
Matrices of this type are produced by the Schur factorization of
a real general matrix:  A = Q*T*Q**T, as computed by DHSEQR.

The right eigenvector x and the left eigenvector y of T corresponding
to an eigenvalue w are defined by:

T*x = w*x,     (y**T)*T = w*(y**T)

where y**T denotes the transpose of the vector y.
The eigenvalues are not input to this routine, but are read directly
from the diagonal blocks of T.

This routine returns the matrices X and/or Y of right and left
eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
input matrix. If Q is the orthogonal factor that reduces a matrix
A to Schur form T, then Q*X and Q*Y are the matrices of right and
left eigenvectors of A.

This uses a Level 3 BLAS version of the back transformation.```
Parameters

SIDE

```          SIDE is CHARACTER*1
= 'R':  compute right eigenvectors only;
= 'L':  compute left eigenvectors only;
= 'B':  compute both right and left eigenvectors.```

HOWMNY

```          HOWMNY is CHARACTER*1
= 'A':  compute all right and/or left eigenvectors;
= 'B':  compute all right and/or left eigenvectors,
backtransformed by the matrices in VR and/or VL;
= 'S':  compute selected right and/or left eigenvectors,
as indicated by the logical array SELECT.```

SELECT

```          SELECT is LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenvectors to be
computed.
If w(j) is a real eigenvalue, the corresponding real
eigenvector is computed if SELECT(j) is .TRUE..
If w(j) and w(j+1) are the real and imaginary parts of a
complex eigenvalue, the corresponding complex eigenvector is
computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
.FALSE..
Not referenced if HOWMNY = 'A' or 'B'.```

N

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

T

```          T is DOUBLE PRECISION array, dimension (LDT,N)
The upper quasi-triangular matrix T in Schur canonical form.```

LDT

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

VL

```          VL is DOUBLE PRECISION array, dimension (LDVL,MM)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by DHSEQR).
On exit, if SIDE = 'L' or 'B', VL contains:
if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
if HOWMNY = 'B', the matrix Q*Y;
if HOWMNY = 'S', the left eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VL, in the same order as their
eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part, and the second the imaginary part.
Not referenced if SIDE = 'R'.```

LDVL

```          LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N.```

VR

```          VR is DOUBLE PRECISION array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by DHSEQR).
On exit, if SIDE = 'R' or 'B', VR contains:
if HOWMNY = 'A', the matrix X of right eigenvectors of T;
if HOWMNY = 'B', the matrix Q*X;
if HOWMNY = 'S', the right eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VR, in the same order as their
eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part and the second the imaginary part.
Not referenced if SIDE = 'L'.```

LDVR

```          LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.```

MM

```          MM is INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.```

M

```          M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors.
If HOWMNY = 'A' or 'B', M is set to N.
Each selected real eigenvector occupies one column and each
selected complex eigenvector occupies two columns.```

WORK

`          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))`

LWORK

```          LWORK is INTEGER
The dimension of array WORK. LWORK >= max(1,3*N).
For optimum performance, LWORK >= N + 2*N*NB, where NB is
the optimal blocksize.

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```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

```  The algorithm used in this program is basically backward (forward)
substitution, with scaling to make the the code robust against
possible overflow.

Each eigenvector is normalized so that the element of largest
magnitude has magnitude 1; here the magnitude of a complex number
(x,y) is taken to be |x| + |y|.```

Definition at line 235 of file dtrevc3.f.

subroutine strevc3 (character side, character howmny, logical, dimension( * ) select, integer n, real, dimension( ldt, * ) t, integer ldt, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, real, dimension( * ) work, integer lwork, integer info)

STREVC3

Purpose:

``` STREVC3 computes some or all of the right and/or left eigenvectors of
a real upper quasi-triangular matrix T.
Matrices of this type are produced by the Schur factorization of
a real general matrix:  A = Q*T*Q**T, as computed by SHSEQR.

The right eigenvector x and the left eigenvector y of T corresponding
to an eigenvalue w are defined by:

T*x = w*x,     (y**T)*T = w*(y**T)

where y**T denotes the transpose of the vector y.
The eigenvalues are not input to this routine, but are read directly
from the diagonal blocks of T.

This routine returns the matrices X and/or Y of right and left
eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
input matrix. If Q is the orthogonal factor that reduces a matrix
A to Schur form T, then Q*X and Q*Y are the matrices of right and
left eigenvectors of A.

This uses a Level 3 BLAS version of the back transformation.```
Parameters

SIDE

```          SIDE is CHARACTER*1
= 'R':  compute right eigenvectors only;
= 'L':  compute left eigenvectors only;
= 'B':  compute both right and left eigenvectors.```

HOWMNY

```          HOWMNY is CHARACTER*1
= 'A':  compute all right and/or left eigenvectors;
= 'B':  compute all right and/or left eigenvectors,
backtransformed by the matrices in VR and/or VL;
= 'S':  compute selected right and/or left eigenvectors,
as indicated by the logical array SELECT.```

SELECT

```          SELECT is LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenvectors to be
computed.
If w(j) is a real eigenvalue, the corresponding real
eigenvector is computed if SELECT(j) is .TRUE..
If w(j) and w(j+1) are the real and imaginary parts of a
complex eigenvalue, the corresponding complex eigenvector is
computed if either SELECT(j) or SELECT(j+1) is .TRUE., and
on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to
.FALSE..
Not referenced if HOWMNY = 'A' or 'B'.```

N

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

T

```          T is REAL array, dimension (LDT,N)
The upper quasi-triangular matrix T in Schur canonical form.```

LDT

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

VL

```          VL is REAL array, dimension (LDVL,MM)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by SHSEQR).
On exit, if SIDE = 'L' or 'B', VL contains:
if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
if HOWMNY = 'B', the matrix Q*Y;
if HOWMNY = 'S', the left eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VL, in the same order as their
eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part, and the second the imaginary part.
Not referenced if SIDE = 'R'.```

LDVL

```          LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N.```

VR

```          VR is REAL array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by SHSEQR).
On exit, if SIDE = 'R' or 'B', VR contains:
if HOWMNY = 'A', the matrix X of right eigenvectors of T;
if HOWMNY = 'B', the matrix Q*X;
if HOWMNY = 'S', the right eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VR, in the same order as their
eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part and the second the imaginary part.
Not referenced if SIDE = 'L'.```

LDVR

```          LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.```

MM

```          MM is INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.```

M

```          M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors.
If HOWMNY = 'A' or 'B', M is set to N.
Each selected real eigenvector occupies one column and each
selected complex eigenvector occupies two columns.```

WORK

`          WORK is REAL array, dimension (MAX(1,LWORK))`

LWORK

```          LWORK is INTEGER
The dimension of array WORK. LWORK >= max(1,3*N).
For optimum performance, LWORK >= N + 2*N*NB, where NB is
the optimal blocksize.

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```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

```  The algorithm used in this program is basically backward (forward)
substitution, with scaling to make the the code robust against
possible overflow.

Each eigenvector is normalized so that the element of largest
magnitude has magnitude 1; here the magnitude of a complex number
(x,y) is taken to be |x| + |y|.```

Definition at line 235 of file strevc3.f.

subroutine ztrevc3 (character side, character howmny, logical, dimension( * ) select, integer n, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer lrwork, integer info)

ZTREVC3

Purpose:

``` ZTREVC3 computes some or all of the right and/or left eigenvectors of
a complex upper triangular matrix T.
Matrices of this type are produced by the Schur factorization of
a complex general matrix:  A = Q*T*Q**H, as computed by ZHSEQR.

The right eigenvector x and the left eigenvector y of T corresponding
to an eigenvalue w are defined by:

T*x = w*x,     (y**H)*T = w*(y**H)

where y**H denotes the conjugate transpose of the vector y.
The eigenvalues are not input to this routine, but are read directly
from the diagonal of T.

This routine returns the matrices X and/or Y of right and left
eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
input matrix. If Q is the unitary factor that reduces a matrix A to
Schur form T, then Q*X and Q*Y are the matrices of right and left
eigenvectors of A.

This uses a Level 3 BLAS version of the back transformation.```
Parameters

SIDE

```          SIDE is CHARACTER*1
= 'R':  compute right eigenvectors only;
= 'L':  compute left eigenvectors only;
= 'B':  compute both right and left eigenvectors.```

HOWMNY

```          HOWMNY is CHARACTER*1
= 'A':  compute all right and/or left eigenvectors;
= 'B':  compute all right and/or left eigenvectors,
backtransformed using the matrices supplied in
VR and/or VL;
= 'S':  compute selected right and/or left eigenvectors,
as indicated by the logical array SELECT.```

SELECT

```          SELECT is LOGICAL array, dimension (N)
If HOWMNY = 'S', SELECT specifies the eigenvectors to be
computed.
The eigenvector corresponding to the j-th eigenvalue is
computed if SELECT(j) = .TRUE..
Not referenced if HOWMNY = 'A' or 'B'.```

N

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

T

```          T is COMPLEX*16 array, dimension (LDT,N)
The upper triangular matrix T.  T is modified, but restored
on exit.```

LDT

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

VL

```          VL is COMPLEX*16 array, dimension (LDVL,MM)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
contain an N-by-N matrix Q (usually the unitary matrix Q of
Schur vectors returned by ZHSEQR).
On exit, if SIDE = 'L' or 'B', VL contains:
if HOWMNY = 'A', the matrix Y of left eigenvectors of T;
if HOWMNY = 'B', the matrix Q*Y;
if HOWMNY = 'S', the left eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VL, in the same order as their
eigenvalues.
Not referenced if SIDE = 'R'.```

LDVL

```          LDVL is INTEGER
The leading dimension of the array VL.
LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N.```

VR

```          VR is COMPLEX*16 array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
contain an N-by-N matrix Q (usually the unitary matrix Q of
Schur vectors returned by ZHSEQR).
On exit, if SIDE = 'R' or 'B', VR contains:
if HOWMNY = 'A', the matrix X of right eigenvectors of T;
if HOWMNY = 'B', the matrix Q*X;
if HOWMNY = 'S', the right eigenvectors of T specified by
SELECT, stored consecutively in the columns
of VR, in the same order as their
eigenvalues.
Not referenced if SIDE = 'L'.```

LDVR

```          LDVR is INTEGER
The leading dimension of the array VR.
LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.```

MM

```          MM is INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.```

M

```          M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors.
If HOWMNY = 'A' or 'B', M is set to N.
Each selected eigenvector occupies one column.```

WORK

`          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))`

LWORK

```          LWORK is INTEGER
The dimension of array WORK. LWORK >= max(1,2*N).
For optimum performance, LWORK >= N + 2*N*NB, where NB is
the optimal blocksize.

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

`          RWORK is DOUBLE PRECISION array, dimension (LRWORK)`

LRWORK

```          LRWORK is INTEGER
The dimension of array RWORK. LRWORK >= max(1,N).

If LRWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the RWORK array, returns
this value as the first entry of the RWORK array, and no error
message related to LRWORK is issued by XERBLA.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

```  The algorithm used in this program is basically backward (forward)
substitution, with scaling to make the the code robust against
possible overflow.

Each eigenvector is normalized so that the element of largest
magnitude has magnitude 1; here the magnitude of a complex number
(x,y) is taken to be |x| + |y|.```

Definition at line 242 of file ztrevc3.f.

Author

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Tue Nov 28 2023 12:08:43 Version 3.12.0 LAPACK