scipy.linalg.solve#

scipy.linalg.solve(a, b, sym_pos=False, lower=False, overwrite_a=False, overwrite_b=False, debug=None, 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.
sym_posbool, optional: Assume a is symmetric and positive definite. This key is deprecated and assume_a = ‘pos’ keyword is recommended instead. The functionality is the same. It will be removed in the future.
lowerbool, optional: If True, only the data contained in the lower triangle of a. Default is to use upper triangle. (ignored for 'gen')
overwrite_abool, optional: Allow overwriting data in a (may enhance performance). Default is False.
overwrite_bbool, optional: Allow overwriting data in b (may enhance performance). Default is False.
check_finitebool, optional: 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 explained above.
transposed: bool, optional: If True, a^T x = b for real matrices, raises NotImplementedError for complex matrices (only for True).

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:

>>> 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)

scipy.linalg.inv

scipy.linalg.solve_banded