WOLFRAM SYSTEM MODELER

# dtrevc

Compute the right and/or left eigenvectors of a real upper quasi-triangular matrix T

# Wolfram Language

In[1]:=
`SystemModel["Modelica.Math.Matrices.LAPACK.dtrevc"]`
Out[1]:=

# Information

This information is part of the Modelica Standard Library maintained by the Modelica Association.

```Lapack documentation
Purpose
=======

DTREVC 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**H)*T = w*(y**H)

where y**H denotes the conjugate transpose of 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.

Arguments
=========

SIDE    (input) CHARACTER*1
= 'R':  compute right eigenvectors only;
= 'L':  compute left eigenvectors only;
= 'B':  compute both right and left eigenvectors.

HOWMNY  (input) 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  (input/output) 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       (input) INTEGER
The order of the matrix T. N >= 0.

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

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

VL      (input/output) 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    (input) INTEGER
The leading dimension of the array VL.  LDVL >= 1, and if
SIDE = 'L' or 'B', LDVL >= N.

VR      (input/output) 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    (input) INTEGER
The leading dimension of the array VR.  LDVR >= 1, and if
SIDE = 'R' or 'B', LDVR >= N.

MM      (input) INTEGER
The number of columns in the arrays VL and/or VR. MM >= M.

M       (output) 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    (workspace) DOUBLE PRECISION array, dimension (3*N)

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value

Further Details
===============

The algorithm used in this program is basically backward (forward)
substitution, with scaling to make 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|.```

# Syntax

(lEigenVectors, rEigenVectors, info) = dtrevc(T, side, howmny, Q)

# Inputs (4)

T Type: Real[:,size(T, 1)] Description: Upper quasi triangular matrix Default Value: "R" Type: String Description: Specify which eigenvectors Default Value: "B" Type: String Description: Specify how many eigenvectors Type: Real[size(T, 1),size(T, 1)] Description: Orthogonal matrix Q of Schur vectors returned by DHSEQR

# Outputs (3)

lEigenVectors Default Value: Q Type: Real[size(T, 1),size(T, 1)] Description: Left eigenvectors of matrix T Default Value: Q Type: Real[size(T, 1),size(T, 1)] Description: Right eigenvectors of matrix T Type: Integer