SciPy

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_pos : bool, 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.

lower : bool, optional

If True, only the data contained in the lower triangle of a. Default is to use upper triangle. (ignored for 'gen')

overwrite_a : bool, optional

Allow overwriting data in a (may enhance performance). Default is False.

overwrite_b : bool, optional

Allow overwriting data in b (may enhance performance). Default is False.

check_finite : bool, 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_a : str, 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 1D 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 1D 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)

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