ml_gdlyap_smith_fac

Nilpotent discrete-time Lyapunov equation solver.

Contents

Syntax

[Z, info] = ml_gdlyap_smith_fac(A, C, E)
[Z, info] = ml_gdlyap_smith_fac(A, C, E, opts)

Description

Computes the full-rank solution X = Z'*Z of the generalized
discrete-time Lyapunov equation
    A'*X*A - E'*X*E = C'*C,                                         (1)
where E is nilpotent, A is invertible, i.e., the matrix pencil s*E - A
has only infinite eigenvalues, via the Smith method.

Input

A    - matrix with dimensions n x n in (1)
C    - matrix with dimensions p x n in (1)
E    - matrix with dimensions n x n in (1)
opts - structure, containing the following optional entries:

Parameter
Meaning
AbsTol
nonnegative scalar, tolerance for the absolute gain in the last iteration step
default: 0
Index
nonnegative integer, index of nilpotency of the matrix E used to set the exact number of iteration steps, if the index is unknown Inf is set
default: Inf
Info
{0, 1}, used to disable/enable display of verbose status information during iteration steps
default: 0
MaxIter
positive integer, maximum number of iteration steps
default: 100
RelTol
nonnegative scalar, tolerance for the relative gain in the last iteration step
default: 1.0e+01*n*eps

Output

Z    - full-rank factor of (1), such that X = Z'*Z
info - structure, containing the following information

Entry
Meaning
AbsErr
vector, containing the absolute change of the iteration matrix in each iteration step
IterationSteps
number of performed iteration steps
RelErr
vector, containing the relative change of the iteration matrix in each iteration step

Reference

T. Stykel, Low-rank iterative methods for projected generalized Lyapunov equations, Electron. Trans. Numer. Anal. 30 (2008) 187--202.

See Also

ml_lyap_sgn | ml_lyap_sgn_fac