scipy.linalg.solve_discrete_lyapunov¶
- scipy.linalg.solve_discrete_lyapunov(a, q, method=None)[source]¶
Solves the discrete Lyapunov equation \(A'XA-X=-Q\).
Parameters: a : (M, M) array_like
A square matrix
q : (M, M) array_like
Right-hand side square matrix
method : {‘direct’, ‘bilinear’}, optional
Type of solver.
If not given, chosen to be direct if M is less than 10 and bilinear otherwise.
Returns: x : ndarray
Solution to the discrete Lyapunov equation
See also
- solve_lyapunov
- computes the solution to the continuous Lyapunov equation
Notes
This section describes the available solvers that can be selected by the ‘method’ parameter. The default method is direct if M is less than 10 and bilinear otherwise.
Method direct uses a direct analytical solution to the discrete Lyapunov equation. The algorithm is given in, for example, [R106]. However it requires the linear solution of a system with dimension \(M^2\) so that performance degrades rapidly for even moderately sized matrices.
Method bilinear uses a bilinear transformation to convert the discrete Lyapunov equation to a continuous Lyapunov equation \((B'X+XB=-C)\) where \(B=(A-I)(A+I)^{-1}\) and \(C=2(A' + I)^{-1} Q (A + I)^{-1}\). The continuous equation can be efficiently solved since it is a special case of a Sylvester equation. The transformation algorithm is from Popov (1964) as described in [R107].
New in version 0.11.0.
References
[R106] (1, 2) Hamilton, James D. Time Series Analysis, Princeton: Princeton University Press, 1994. 265. Print. http://www.scribd.com/doc/20577138/Hamilton-1994-Time-Series-Analysis [R107] (1, 2) Gajic, Z., and M.T.J. Qureshi. 2008. Lyapunov Matrix Equation in System Stability and Control. Dover Books on Engineering Series. Dover Publications.