numpy.linalg.solve

numpy.linalg.solve(a, b)

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.

Notes

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

[R43]

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

Examples

Solve the system of equations 3 * x0 + x1 = 9 and x0 + 2 * x1 = 8:

>>> a = np.array([[3,1], [1,2]])
>>> b = np.array([9,8])
>>> x = np.linalg.solve(a, b)
>>> x
array([ 2.,  3.])

Check that the solution is correct:

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