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scipy.linalg.

# solve_discrete_lyapunov

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

  Solves the discrete Lyapunov equation \(AXA^H - X + Q = 0\).

  The documentation is written assuming array arguments are of specified “core” shapes. However, array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details.

  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].

  Added in version 0.11.0.

  References

  [1]

  “Lyapunov equation”, Wikipedia, https://en.wikipedia.org/wiki/Lyapunov_equation#Discrete_time

  [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