scipy.linalg.

solve_sylvester#

scipy.linalg.solve_sylvester(a, b, q)[source]#

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.

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