# scipy.linalg.solve

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

  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.

    lower bool, 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_a bool, default: False

      Allow overwriting data in a (may enhance performance).

    overwrite_b bool, default: False

      Allow overwriting data in b (may enhance performance).

    check_finite bool, 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_a str, {‘gen’, ‘sym’, ‘her’, ‘pos’}

      Valid entries are explained above.

    transposed bool, 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)