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# scipy.linalg.solve_discrete_lyapunov

`scipy.linalg.``solve_discrete_lyapunov`(a, q, method=None)

  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` 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, [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 \(B=(A-I)(A+I)^{-1}\) 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 [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/Hamilton-1994-Time-Series-Analysis

  [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.