scipy.linalg.solve#

scipy.linalg.solve(a, b, lower=False, overwrite_a=False, overwrite_b=False, check_finite=True, assume_a='gen', transposed=False)[source]#

Solves the linear equation set a @ x == b for the unknown x for square a matrix.

If the data matrix is known to be a particular type then supplying the corresponding string to assume_a key chooses the dedicated solver. The available options are

generic matrix	‘gen’
symmetric	‘sym’
hermitian	‘her’
positive definite	‘pos’

If omitted, 'gen' is the default structure.

The datatype of the arrays define which solver is called regardless of the values. In other words, even when the complex array entries have precisely zero imaginary parts, the complex solver will be called based on the data type of the array.

Parameters:

a(N, N) array_like: Square input data
b(N, NRHS) array_like: Input data for the right hand side.
lowerbool, default: False: Ignored if assume_a == 'gen' (the default). If True, the calculation uses only the data in the lower triangle of a; entries above the diagonal are ignored. If False (default), the calculation uses only the data in the upper triangle of a; entries below the diagonal are ignored.
overwrite_abool, default: False: Allow overwriting data in a (may enhance performance).
overwrite_bbool, default: False: Allow overwriting data in b (may enhance performance).
check_finitebool, default: True: Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.
assume_astr, {‘gen’, ‘sym’, ‘her’, ‘pos’}: Valid entries are explained above.
transposedbool, default: False: If True, solve a.T @ x == b. Raises NotImplementedError for complex a.

Returns:

x(N, NRHS) ndarray: The solution array.

Raises:

ValueError: If size mismatches detected or input a is not square.
LinAlgError: If the matrix is singular.
LinAlgWarning: If an ill-conditioned input a is detected.
NotImplementedError: If transposed is True and input a is a complex matrix.

Notes

If the input b matrix is a 1-D array with N elements, when supplied together with an NxN input a, it is assumed as a valid column vector despite the apparent size mismatch. This is compatible with the numpy.dot() behavior and the returned result is still 1-D array.

The generic, symmetric, Hermitian and positive definite solutions are obtained via calling ?GESV, ?SYSV, ?HESV, and ?POSV routines of LAPACK respectively.

Examples

Given a and b, solve for x:

>>> import numpy as np
>>> a = np.array([[3, 2, 0], [1, -1, 0], [0, 5, 1]])
>>> b = np.array([2, 4, -1])
>>> from scipy import linalg
>>> x = linalg.solve(a, b)
>>> x
array([ 2., -2.,  9.])
>>> np.dot(a, x) == b
array([ True,  True,  True], dtype=bool)