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 : array_like, shape (M, M)

Coefficient matrix.

b : array_like, shape (M,) or (M, N)

Ordinate or “dependent variable” values.

Returns:

x : ndarray, shape (M,) or (M, N) depending on b

Solution to the system a x = b

Raises:

LinAlgError :

If a is singular or not square.

Notes

solve is a wrapper for the LAPACK routines dgesv and zgesv, the former being used if a is real-valued, the latter if it is complex-valued. The solution to the system of linear equations is computed using an LU decomposition [R50] with partial pivoting and row interchanges.

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

[R50]

(1, 2) 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.dot(a, x) == b).all()
True