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scipy.linalg.

# solve_sylvester

scipy.linalg.solve_sylvester(a, b, q)

  Computes a solution (X) to the Sylvester equation \(AX + XB = Q\).

  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. Note that calls with zero-size batches are unsupported and will raise a `ValueError`.

  Parameters:
    a(M, M) array_like

      Leading matrix of the Sylvester equation

    b(N, N) array_like

      Trailing matrix of the Sylvester equation

    q(M, N) array_like

      Right-hand side

  Returns:
    x(M, N) ndarray

      The solution to the Sylvester equation.

  Raises:
    LinAlgError

      If solution was not found

  Notes

  Computes a solution to the Sylvester matrix equation via the Bartels- Stewart algorithm. The A and B matrices first undergo Schur decompositions. The resulting matrices are used to construct an alternative Sylvester equation (`RY + YS^T = F`) where the R and S matrices are in quasi-triangular form (or, when R, S or F are complex, triangular form). The simplified equation is then solved using `*TRSYL` from LAPACK directly.

  Added in version 0.11.0.

  Examples

  Given a, b, and q solve for x:

      >>> import numpy as np
      >>> from scipy import linalg
      >>> a = np.array([[-3, -2, 0], [-1, -1, 3], [3, -5, -1]])
      >>> b = np.array([[1]])
      >>> q = np.array([[1],[2],[3]])
      >>> x = linalg.solve_sylvester(a, b, q)
      >>> x
      array([[ 0.0625],
             [-0.5625],
             [ 0.6875]])
      >>> np.allclose(a.dot(x) + x.dot(b), q)
      True