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

naginterfaces.library.matop.real_symm_posdef_fac(k, a)

real_symm_posdef_fac performs a $ULD{L}^T{U}^T$ decomposition of a real symmetric positive definite band matrix.

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

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f01/f01buf.html

Parameters

kint

$k$, the change-over point in the decomposition.

afloat, array-like, shape $(\text{m1},n)$

The upper triangle of the $n\times n$ symmetric band matrix $A$, with the diagonal of the matrix stored in the $(m+1)$th row of the array, and the $m$ superdiagonals within the band stored in the first $m$ rows of the array. Each column of the matrix is stored in the corresponding column of the array. For example, if $n=6$ and $m=2$, the storage scheme is

$$\begin{array}{c}\begin{array}{cccccc}\text{*}& \text{*}& a_{13}& a_{24}& a_{35}& a_{46}\\ \text{*}& a_{12}& a_{23}& a_{34}& a_{45}& a_{56}\\ a_{11}& a_{22}& a_{33}& a_{44}& a_{55}& a_{66}\end{array}\end{array}$$

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 $(\text{m1},n)$

$A$ is overwritten by the corresponding elements of $L$, $D$ and $U$.

Raises

NagValueError

(errno $1$)

On entry, $k=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $k\le n$.

(errno $1$)

On entry, $k=⟨value⟩$ and $\text{m1}-1=⟨value⟩$.

Constraint: $k\ge \text{m1}-1$.

(errno $2$)

The matrix $A$ is not positive definite.

(errno $3$)

The matrix $A$ is not positive definite.

Notes

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

The symmetric positive definite matrix $A$, of order $n$ and bandwidth $2m+1$, is divided into the leading principal sub-matrix of order $k$ and its complement, where $m\le k\le n$. A $UD{U}^T$ decomposition of the latter and an $LD{L}^T$ decomposition of the former are obtained by means of a sequence of elementary transformations, where $U$ is unit upper triangular, $L$ is unit lower triangular and $D$ is diagonal. Thus if $k=n$, an $LD{L}^T$ decomposition of $A$ is obtained.

This function is specifically designed to precede real_symm_posdef_geneig() for the transformation of the symmetric-definite eigenproblem $Ax=\lambda Bx$ by the method of Crawford where $A$ and $B$ are of band form. In this context, $k$ is chosen to be close to $n/2$ and the decomposition is applied to the matrix $B$.

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