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

The matrix a is stored in :None:None:`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 :None:None:`ab` (shape of a is (6, 6), :None:None:`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_ab : bool, optional

Discard data in :None:None:`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.

Solve equation a x = b. a is Hermitian positive-definite banded matrix.

Examples

[ 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

:None:None:`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])

[ 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])

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.linalg._basic.solveh_banded

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