numpy.linalg.solve — NumPy v2.2 Manual (original) (raw)
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 (…, M) if b is shape (M,) and (…, M, K) if b is (…, M, K), where the “…” part is broadcasted between a and 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, uselstsq for the least-squares best “solution” of the system/equation.
Changed in version 2.0: The b array is only treated as a shape (M,) column vector if it is exactly 1-dimensional. In all other instances it is treated as a stack of (M, K) matrices. Previously b would be treated as a stack of (M,) vectors if b.ndim was equal to a.ndim - 1.
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 and3 * x0 + 5 * x1 = 2:
import numpy as np 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