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solve_discrete_lyapunov(a, q, method=None)

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, . 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 $B=(A-I)(A+I)^{-1}$ 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 .

versionadded

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 if M is less than 10 and bilinear otherwise.

Returns

x : ndarray: Solution to the discrete Lyapunov equation

Solves the discrete Lyapunov equation $AXA^H - X + Q = 0$ .

Examples

Given a and q solve for x:

>>> from scipy import linalg
... a = np.array([[0.2, 0.5],[0.7, -0.9]])
... q = np.eye(2)
... x = linalg.solve_discrete_lyapunov(a, q)
... x
array([[ 0.70872893,  1.43518822],
       [ 1.43518822, -2.4266315 ]])

>>> np.allclose(a.dot(x).dot(a.T)-x, -q)
True

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.linalg._solvers._solve_discrete_lyapunov_bilinear scipy.linalg._solvers.solve_discrete_lyapunov scipy.linalg._solvers._solve_discrete_lyapunov_direct scipy.linalg._solvers.solve_continuous_lyapunov

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