solve_sylvester — SciPy v1.15.2 Manual (original) (raw)

scipy.linalg.

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