Skip to content
Permalink
master
Switch branches/tags

Name already in use

A tag already exists with the provided branch name. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. Are you sure you want to create this branch?
Go to file
 
 
Cannot retrieve contributors at this time
function X = lyapunov_symmetric(A, C, tol)
% Solves AX + XA = C when A = A', as a pseudo-inverse.
%
% function X = lyapunov_symmetric(A, C)
% function X = lyapunov_symmetric(A, C, tol)
%
% Matrices A, C and X have size nxn. When the solution exists and is
% unique, this is equivalent to sylvester(A, A', C) or lyap(A, -C).
% This works for both real and complex inputs.
%
% If C is a 3-D array of size nxnxk, then X has size nxnxk as well, and
% each slice X(:, :, i) corresponds to the solution for the system with
% right-hand side C(:, :, i). This is more efficient then calling the
% function multiple times for each slice of C.
%
% If the solution is not unique, the smallest-norm solution is returned.
%
% If a solution does not exist, a minimum-residue solution is returned.
%
% tol is a tolerance used to determine which eigenvalues are numerically
% zero. It can be specified manually; otherwise, a default value is chosen.
%
% Overall, if A is nxn, the output is very close to:
% X = reshape(pinv(kron(A, eye(n)) + kron(eye(n), A))*C(:), [n, n]),
% but it is computed far more efficiently. The most expensive step is an
% eigendecomposition of A, whose complexity is essentially O(n^3) flops.
%
% If A is not symmetric, only its symmetric part is used: (A+A')/2.
%
% If C is (skew-)symmetric, then X is (skew-)symmetric (up to round-off),
% and similarly in the complex case.
%
% To solve one system at a time, while reusing the eigendecomposition of A,
% call lyapunov_symmetric_eig.
%
% See also: lyapunov_symmetric_eig sylvester lyap sylvester_nochecks
% This file is part of Manopt: www.manopt.org.
% Original author: Nicolas Boumal, April 17, 2018.
% Contributors:
% Change log:
% Aug. 31, 2018 (NB):
% Now works with C having multiple slices (nxnxk), and added some
% safeguards.
n = size(A, 1);
assert(ismatrix(A) && size(A, 2) == n, 'A must be square.');
assert(size(C, 1) == n && size(C, 2) == n, ...
'Each slice of C must have the same size as A.');
if ~exist('tol', 'var')
tol = [];
end
% Make sure A is numerically Hermitian (or symmetric).
% The cost of this safety step is negligible compared to eig.
A = (A+A')/2;
% V is unitary or orthogonal and lambda is real.
[V, lambda] = eig(A, 'vector');
% Solve for each slice separately.
X = zeros(size(C));
for k = 1 : size(C, 3)
X(:, :, k) = lyapunov_symmetric_eig(V, lambda, C(:, :, k), tol);
end
end