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

naginterfaces.library.matop.real_symm_posdef_geneig(k, a, b)

real_symm_posdef_geneig transforms the generalized symmetric-definite eigenproblem $Ax=\lambda bx$ to the equivalent standard eigenproblem $Cy=\lambda y$, where $A$, $b$ and $C$ are symmetric band matrices and $b$ is positive definite. $b$ must have been decomposed by real_symm_posdef_fac().

For full information please refer to the NAG Library document for f01bv

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f01/f01bvf.html

Parameters

kint

$k$, the change-over point in the transformations. It must be the same as the value used by real_symm_posdef_fac() in the decomposition of $B$.

Suggested value: the optimum value is the multiple of $m_A$ nearest to $n/2$.

afloat, array-like, shape $(\text{ma1},n)$

The upper triangle of the $n\times n$ symmetric band matrix $A$, with the diagonal of the matrix stored in the $(m_A+1)$th row of the array, and the $m_A$ superdiagonals within the band stored in the first $m_A$ rows of the array. Each column of the matrix is stored in the corresponding column of the array. For example, if $n=6$ and $m_A=2$, 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 need not be set. 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-ma1), j+1):
        a[i-j+ma1-1, j] = A[i, j]

bfloat, array-like, shape $(\text{mb1},n)$

The elements of the decomposition of matrix $B$ as returned by real_symm_posdef_fac().

Returns

afloat, ndarray, shape $(\text{ma1},n)$

Is overwritten by the corresponding elements of $C$.

bfloat, ndarray, shape $(\text{mb1},n)$

The elements of $b$ will have been permuted.

Raises

NagValueError

(errno $1$)

On entry, $\text{mb1}=⟨value⟩$ and $\text{ma1}=⟨value⟩$.

Constraint: $\text{mb1}\le \text{ma1}$.

Notes

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

$A$ is a symmetric band matrix of order $n$ and bandwidth $2m_A+1$. The positive definite symmetric band matrix $B$, of order $n$ and bandwidth $2m_B+1$, must have been previously decomposed by real_symm_posdef_fac() as $ULD{L}^T{U}^T$. real_symm_posdef_geneig applies $U$, $L$ and $D$ to $A$, $m_A$ rows at a time, restoring the band form of $A$ at each stage by plane rotations. The argument $k$ defines the change-over point in the decomposition of $B$ as used by real_symm_posdef_fac() and is also used as a change-over point in the transformations applied by this function. For maximum efficiency, $k$ should be chosen to be the multiple of $m_A$ nearest to $n/2$. The resulting symmetric band matrix $C$ is overwritten on $a$. The eigenvalues of $C$, and thus of the original problem, may be found using lapackeig.dsbtrd and lapackeig.dsterf. For selected eigenvalues, use lapackeig.dsbtrd and lapackeig.dstebz.

References

Crawford, C R, 1973, Reduction of a band-symmetric generalized eigenvalue problem, Comm. ACM (16), 41–44