numpy.linalg.solve#

linalg.solve(a, b)[source]#

Solve a linear matrix equation, or system of linear scalar equations.

Computes the “exact” solution, x, of the well-determined, i.e., full rank, linear matrix equation ax = b.

Parameters:

a(…, M, M) array_like: Coefficient matrix.
b{(…, M,), (…, M, K)}, array_like: Ordinate or “dependent variable” values.

Returns:

x{(…, M,), (…, M, K)} ndarray: Solution to the system a x = b. Returned shape is identical to b.

Raises:

LinAlgError: If a is singular or not square.

See also

scipy.linalg.solve: Similar function in SciPy.

Notes

New in version 1.8.0.

Broadcasting rules apply, see the numpy.linalg documentation for details.

The solutions are computed using LAPACK routine _gesv.

a must be square and of full-rank, i.e., all rows (or, equivalently, columns) must be linearly independent; if either is not true, use lstsq for the least-squares best “solution” of the system/equation.

References

[1]

G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pg. 22.

Examples

Solve the system of equations x0 + 2 * x1 = 1 and 3 * x0 + 5 * x1 = 2:

>>> a = np.array([[1, 2], [3, 5]])
>>> b = np.array([1, 2])
>>> x = np.linalg.solve(a, b)
>>> x
array([-1.,  1.])

Check that the solution is correct:

>>> np.allclose(np.dot(a, x), b)
True