`scipy.linalg._basic:solveh_banded`

source: /scipy/linalg/_basic.py :894

Signature

   def     solveh_banded (    ab    ,    b    ,    overwrite_ab    =  False   ,    overwrite_b    =  False   ,    lower    =  False   ,    check_finite    =  True     )

Summary

Solve the equation a @ x = b for x, where a is the Hermitian positive-definite banded matrix defined by ab.

Extended Summary

Uses Thomas' Algorithm, which is more efficient than standard LU factorization, but should only be used for Hermitian positive-definite matrices.

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), number of upper diagonals, 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.

The documentation is written assuming array arguments are of specified "core" shapes. However, array argument(s) of this function may have additional "batch" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see linalg_batch for details. Note that calls with zero-size batches are unsupported and will raise a ValueError.

Parameters

ab : (``u`` + 1, M) array_like: Banded matrix
b : (M,) or (M, K) array_like: Right-hand side
overwrite_ab : bool, optional: Discard data in ab (may enhance performance)
overwrite_b : bool, optional: Discard data in b (may enhance performance)
lower : bool, optional: Is the matrix in the lower form. (Default is upper form)
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.

Returns

x : (M,) or (M, K) ndarray: The solution to the system a x = b. Shape of return matches shape of b.

Notes

In the case of a non-positive definite matrix a, the solver solve_banded may be used.

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]

import numpy as np
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)

✓

✗

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)

✓

✗

Aliases

scipy.linalg.solveh_banded