# ztrevc3.f - Man Page

SRC/ztrevc3.f

## Synopsis

### Functions/Subroutines

subroutine ztrevc3 (side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork, rwork, lrwork, info)
ZTREVC3

## Function/Subroutine Documentation

### 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

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page ztrevc3(3) is an alias of ztrevc3.f(3).

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