ml_dlyapdl_smith_ldl

Discrete-time dual Lyapunov equation solver.

Contents

Syntax

[RZ, RY, LZ, LY, info] = ml_dlyapdl_smith_ldl(A, B, [], C, [])
[RZ, RY, LZ, LY, info] = ml_dlyapdl_smith_ldl(A, B, Q, C, R)
[RZ, RY, LZ, LY, info] = ml_dlyapdl_smith_ldl(A, B, Q, C, R, [])
[RZ, RY, LZ, LY, info] = ml_dlyapdl_smith_ldl(A, B, Q, C, R, [], opts)
[RZ, RY, LZ, LY, info] = ml_dlyapdl_smith_ldl(A, B, [], C, [], E)
[RZ, RY, LZ, LY, info] = ml_dlyapdl_smith_ldl(A, B, Q, C, R, E)
[RZ, RY, LZ, LY, info] = ml_dlyapdl_smith_ldl(A, B, Q, C, R, E, opts)

Description

Computes the solution matrix of the dual standard discrete-time Lyapunov equations

   A*X1*A' - X1 + B*Q*B' = 0,                                      (1)
   A'*X2*A - X2 + C'*R*C = 0,                                      (2)

or of the dual generalized Lyapunov equations

   A*X1*A' - E*X1*E' + B*Q*B' = 0,                                 (3)
   A'*X2*A - E'*X2*E + C'*R*C = 0,                                 (4)

with X1 = RZ*RY*RZ' and X2 = LZ*LY*LZ' using the Smith iteration. It is assumed that the eigenvalues of A (or s*E - A) lie inside the open unit-circle.

Input

Parameter
Meaning
AbsTol
nonnegative scalar, tolerance for the absolute error in the last iteration step
default: 0
CompTol
nonnegative scalar, tolerance for the column and row compression during the iteration
default: sqrt(n)*eps
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 error in the last iteration step
default: 1.0e+01*n*eps

Output

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

Reference

V. Simoncini, Computational methods for linear matrix equations, SIAM Rev. 38 (3) (2016) 377--441. https://doi.org/10.1137/130912839

See Also

ml_dlyap_smith | ml_dlyap_smith_ldl