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# dlyap

## PURPOSE

% Solve the discrete-time Lyapunov equation

## SYNOPSIS

function x = dlyap (a, b)

## DESCRIPTION

```% -*- texinfo -*-
% @deftypefn {Function File} {} dlyap (@var{a}, @var{b})
% Solve the discrete-time Lyapunov equation
%
% @strong{Inputs}
% @table @var
%   @item a
%   @var{n} by @var{n} matrix;
%   @item b
%   Matrix: @var{n} by @var{n}, @var{n} by @var{m}, or @var{p} by @var{n}.
% @end table
%
% @strong{Output}
% @table @var
% @item x
% matrix satisfying appropriate discrete time Lyapunov equation.
% @end table
%
% Options:
% @itemize @bullet
% @item @var{b} is square: solve
% @iftex
% @tex
% \$\$ axa^T - x + b = 0 \$\$
% @end tex
% @end iftex
% @ifinfo
% @code{a x a' - x + b = 0}
% @end ifinfo
% @item @var{b} is not square: @var{x} satisfies either
% @iftex
% @tex
% \$\$ axa^T - x + bb^T = 0 \$\$
% @end tex
% @end iftex
% @ifinfo
% @example
% a x a' - x + b b' = 0
% @end example
% @end ifinfo
% @noindent
% or
% @iftex
% @tex
% \$\$ a^Txa - x + b^Tb = 0, \$\$
% @end tex
% @end iftex
% @ifinfo
% @example
% a' x a - x + b' b = 0,
% @end example
% @end ifinfo
% @noindent
% whichever is appropriate.
% @end itemize
%
% @strong{Method}
% Uses Schur decomposition method as in Kitagawa,
% @cite{An Algorithm for Solving the Matrix Equation @math{X = F X F' + S}},
% International Journal of Control, Volume 25, Number 5, pages 745--753
% (1977).
%
% Column-by-column solution method as suggested in
% Hammarling, @cite{Numerical Solution of the Stable, Non-Negative
% Definite Lyapunov Equation}, @acronym{IMA} Journal of Numerical Analysis, Volume
% 2, pages 303--323 (1982).
% @end deftypefn```

## CROSS-REFERENCE INFORMATION

This function calls:
This function is called by:
• gram % @code{gram (@var{sys}, 'c')} returns the controllability gramian of
• h2norm % Computes the

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