scipy.linalg.solve_sylvester#

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

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

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