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slicot_mb03rd

Reduction of a real Schur form matrix to a block-diagonal form.

Syntax

[A_OUT, X_OUT, NBLCKS, BLSIZE, WR, WI, INFO] = slicot_mb03rd(JOBX, SORT, PMAX, A_IN, X_IN, TOL)

Input argument

JOBX

Specifies whether or not the transformations are accumulated, as follows: = 'N': The transformations are not accumulated; = 'U': The transformations are accumulated in X (the given matrix X is updated)

SORT

Specifies whether or not the diagonal blocks of the real Schur form are reordered, as follows: = 'N': The diagonal blocks are not reordered; = 'S': The diagonal blocks are reordered before each step of reduction, so that clustered eigenvalues appear in the same block; = 'C': The diagonal blocks are not reordered, but the "closest-neighbour" strategy is used instead of the standard "closest to the mean" strategy. = 'B': The diagonal blocks are reordered before each step of reduction, and the "closest-neighbour" strategy is used.

PMAX

An upper bound for the infinity norm of elementary submatrices of the individual transformations used for reduction

A_IN

the leading N-by-N part of this array must contain the matrix A to be block-diagonalized, in real Schur form.

X_IN

if JOBX = 'U', the leading N-by-N part of this array must contain a given matrix X.

TOL

The tolerance to be used in the ordering of the diagonal blocks of the real Schur form matrix.

Output argument

A_OUT

the leading N-by-N part of this array contains the computed block-diagonal matrix, in real Schur canonical form. The non-diagonal blocks are set to zero.

X_OUT

if JOBX = 'U', the leading N-by-N part of this array contains the product of the given matrix X and the transformation matrix that reduced A to block-diagonal form. The transformation matrix is itself a product of non-orthogonal similarity transformations having elements with magnitude less than or equal to PMAX. If JOBX = 'N', this array is not referenced

NBLCKS

The number of diagonal blocks of the matrix A.

BLSIZE

The first NBLCKS elements of this array contain the orders of the resulting diagonal blocks of the matrix A.

WR

real parts of the eigenvalues of the matrix A.

WI

imaginary parts of the eigenvalues of the matrix A.

INFO

= 0: successful exit;

Description

To reduce a matrix A in real Schur form to a block-diagonal form using well-conditioned non-orthogonal similarity transformations. The condition numbers of the transformations used for reduction are roughly bounded by PMAX*PMAX, where PMAX is a given value. The transformations are optionally postmultiplied in a given matrix X. The real Schur form is optionally ordered, so that clustered eigenvalues are grouped in the same block.

Used function(s)

MB03RD

Bibliography

http://slicot.org/objects/software/shared/doc/MB03RD.html

Example

N = 8;
PMAX = 1.D03;
TOL = 1.D-2;
JOBX = 'U';
SORT = 'S';
A_IN = [1.   -1.    1.    2.    3.    1.    2.    3.;
   1.    1.    3.    4.    2.    3.    4.    2.;
   0.    0.    1.   -1.    1.    5.    4.    1.;
   0.    0.    0.    1.   -1.    3.    1.    2.;
   0.    0.    0.    1.    1.    2.    3.   -1.;
   0.    0.    0.    0.    0.    1.    5.    1.;
   0.    0.    0.    0.    0.    0.    0.99999999   -0.99999999;
   0.    0.    0.    0.    0.    0.    0.99999999    0.99999999];
X_IN = zeros(N, N);
[A_OUT, X_OUT, NBLCKS, BLSIZE, WR, WI, INFO] = slicot_mb03rd(JOBX, SORT, PMAX, A_IN, X_IN, TOL)

History

Version Description
1.0.0 initial version

Author

SLICOT Documentation

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