scipy.linalg.solveh_banded#
- scipy.linalg.solveh_banded(ab, b, overwrite_ab=False, overwrite_b=False, lower=False, check_finite=True)[source]#
Solve equation a x = b. a is Hermitian positive-definite banded matrix.
The matrix a is stored in ab either in lower diagonal or upper diagonal ordered form:
ab[u + i - j, j] == a[i,j] (if upper form; i <= j) ab[ i - j, j] == a[i,j] (if lower form; i >= j)
Example of ab (shape of a is (6, 6), u =2):
upper form: * * a02 a13 a24 a35 * a01 a12 a23 a34 a45 a00 a11 a22 a33 a44 a55 lower form: a00 a11 a22 a33 a44 a55 a10 a21 a32 a43 a54 * a20 a31 a42 a53 * *
Cells marked with * are not used.
- Parameters
- ab(u + 1, M) array_like
Banded matrix
- b(M,) or (M, K) array_like
Right-hand side
- overwrite_abbool, optional
Discard data in ab (may enhance performance)
- overwrite_bbool, optional
Discard data in b (may enhance performance)
- lowerbool, optional
Is the matrix in the lower form. (Default is upper form)
- 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.
- Returns
- x(M,) or (M, K) ndarray
The solution to the system a x = b. Shape of return matches shape of b.
Examples
Solve the banded system A x = b, where:
[ 4 2 -1 0 0 0] [1] [ 2 5 2 -1 0 0] [2] A = [-1 2 6 2 -1 0] b = [2] [ 0 -1 2 7 2 -1] [3] [ 0 0 -1 2 8 2] [3] [ 0 0 0 -1 2 9] [3]
>>> from scipy.linalg import solveh_banded
ab contains the main diagonal and the nonzero diagonals below the main diagonal. That is, we use the lower form:
>>> ab = np.array([[ 4, 5, 6, 7, 8, 9], ... [ 2, 2, 2, 2, 2, 0], ... [-1, -1, -1, -1, 0, 0]]) >>> b = np.array([1, 2, 2, 3, 3, 3]) >>> x = solveh_banded(ab, b, lower=True) >>> x array([ 0.03431373, 0.45938375, 0.05602241, 0.47759104, 0.17577031, 0.34733894])
Solve the Hermitian banded system H x = b, where:
[ 8 2-1j 0 0 ] [ 1 ] H = [2+1j 5 1j 0 ] b = [1+1j] [ 0 -1j 9 -2-1j] [1-2j] [ 0 0 -2+1j 6 ] [ 0 ]
In this example, we put the upper diagonals in the array hb:
>>> hb = np.array([[0, 2-1j, 1j, -2-1j], ... [8, 5, 9, 6 ]]) >>> b = np.array([1, 1+1j, 1-2j, 0]) >>> x = solveh_banded(hb, b) >>> x array([ 0.07318536-0.02939412j, 0.11877624+0.17696461j, 0.10077984-0.23035393j, -0.00479904-0.09358128j])