naginterfaces.library.matop.real_​symm_​posdef_​fac

naginterfaces.library.matop.real_symm_posdef_fac(k, a)[source]

real_symm_posdef_fac performs a decomposition of a real symmetric positive definite band matrix.

For full information please refer to the NAG Library document for f01bu

https://www.nag.com/numeric/nl/nagdoc_28.4/flhtml/f01/f01buf.html

Parameters
kint

, the change-over point in the decomposition.

afloat, array-like, shape

The upper triangle of the symmetric band matrix , with the diagonal of the matrix stored in the th row of the array, and the superdiagonals within the band stored in the first rows of the array. Each column of the matrix is stored in the corresponding column of the array. For example, if and , the storage scheme is

Elements in the top left corner of the array are not used. The matrix elements within the band can be assigned to the correct elements of the array using the following code:

for j in range(A.shape[1]):
    for i in range(max(0, j+1-m1), j+1):
        a[i-j+m1-1, j] = A[i, j]
Returns
afloat, ndarray, shape

is overwritten by the corresponding elements of , and .

Raises
NagValueError
(errno )

On entry, and .

Constraint: .

(errno )

On entry, and .

Constraint: .

(errno )

The matrix is not positive definite.

(errno )

The matrix is not positive definite.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

The symmetric positive definite matrix , of order and bandwidth , is divided into the leading principal sub-matrix of order and its complement, where . A decomposition of the latter and an decomposition of the former are obtained by means of a sequence of elementary transformations, where is unit upper triangular, is unit lower triangular and is diagonal. Thus if , an decomposition of is obtained.

This function is specifically designed to precede real_symm_posdef_geneig() for the transformation of the symmetric-definite eigenproblem by the method of Crawford where and are of band form. In this context, is chosen to be close to and the decomposition is applied to the matrix .

References

Wilkinson, J H, 1965, The Algebraic Eigenvalue Problem, Oxford University Press, Oxford

Wilkinson, J H and Reinsch, C, 1971, Handbook for Automatic Computation II, Linear Algebra, Springer–Verlag