scipy.linalg.solve_discrete_lyapunov¶

scipy.linalg.
solve_discrete_lyapunov
(a, q, method=None)[source]¶ Solves the discrete Lyapunov equation \(AXA^H  X + Q = 0\).
Parameters: a, q : (M, M) array_like
Square matrices corresponding to A and Q in the equation above respectively. Must have the same shape.
method : {‘direct’, ‘bilinear’}, optional
Type of solver.
If not given, chosen to be
direct
ifM
is less than 10 andbilinear
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 andbilinear
otherwise.Method direct uses a direct analytical solution to the discrete Lyapunov equation. The algorithm is given in, for example, [R130]. 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 \((BX+XB'=C)\) where \(B=(AI)(A+I)^{1}\) 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 [R131].
New in version 0.11.0.
References
[R130] (1, 2) Hamilton, James D. Time Series Analysis, Princeton: Princeton University Press, 1994. 265. Print. http://www.scribd.com/doc/20577138/Hamilton1994TimeSeriesAnalysis [R131] (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.