scipy.linalg.

solve#

scipy.linalg.solve(a, b, lower=None, overwrite_a=False, overwrite_b=False, check_finite=True, assume_a=None, transposed=False)[source]#

Solve the equation a @ x = b for x, where a is a square 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

diagonal	‘diagonal’
tridiagonal	‘tridiagonal’
banded	‘banded’
upper triangular	‘upper triangular’
lower triangular	‘lower triangular’
symmetric	‘symmetric’ (or ‘sym’)
hermitian	‘hermitian’ (or ‘her’)
symmetric positive definite	‘positive definite’ (or ‘pos’)
general	‘general’ (or ‘gen’)

Parameters:

aarray_like, shape (…, N, N): Square left-hand side matrix or a batch of matrices.
b(…, N, NRHS) array_like: Input data for the right hand side or a batch of right-hand sides.
lowerbool, default: False: Ignored unless assume_a is one of 'sym', 'her', or 'pos'. 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, optional: Valid entries are described above. If omitted or None, checks are performed to identify structure so the appropriate solver can be called.
transposedbool, default: False: If True, solve a.T @ x == b. Raises NotImplementedError for complex a.

Returns:

xndarray, shape (N, NRHS) or (…, N): The solution array.

Raises:

ValueError: If size mismatches detected or input a is not square.
LinAlgError: If the computation fails because of matrix singularity.
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 general, symmetric, Hermitian and positive definite solutions are obtained via calling ?GETRF/?GETRS, ?SYSV, ?HESV, and ?POTRF/?POTRS routines of LAPACK respectively.

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.

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.linalg import solve
>>> x = solve(a, b)
>>> x
array([ 2., -2.,  9.])
>>> a @ x == b
array([ True,  True,  True], dtype=bool)

Batches of matrices are supported, with and without structure detection:

>>> a = np.arange(12).reshape(3, 2, 2)   # a batch of 3 2x2 matrices
>>> A = a.transpose(0, 2, 1) @ a    # A is a batch of 3 positive definite matrices
>>> b = np.ones(2)
>>> solve(A, b)      # this automatically detects that A is pos.def.
array([[ 1. , -0.5],
       [ 3. , -2.5],
       [ 5. , -4.5]])
>>> solve(A, b, assume_a='pos')   # bypass structucture detection
array([[ 1. , -0.5],
       [ 3. , -2.5],
       [ 5. , -4.5]])