# 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_continuous_lyapunov`

    computes the solution to the continuous-time 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, [1]. 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 [2].

  New in version 0.11.0.

  References

  [1]

  Hamilton, James D. Time Series Analysis, Princeton: Princeton University Press, 1994. 265. Print. http://doc1.lbfl.li/aca/FLMF037168.pdf

  [2]

  Gajic, Z., and M.T.J. Qureshi. 2008. Lyapunov Matrix Equation in System Stability and Control. Dover Books on Engineering Series. Dover Publications.

  Examples

  Given a and q solve for x:

      >>> import numpy as np
      >>> 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