Arguments

AP

(input/output) REAL or COMPLEX array, shape $(:)$ with $size$(AP) $= n(n+1)/2$, where $n$ is the order of $A$ and $B$.
On entry, the upper or lower triangle of matrix $A$ in packed storage. The elements are stored columnwise as follows:

$$\begin{array}{c\vert c\vert c} A_{i,j} & i,j & {\bf UPLO} ... ... \leq i \leq n \end{array} & \mbox{ 'L'} \\ \hline \end{array}$$

On exit, the contents of AP are destroyed.

BP

(input/output) REAL or COMPLEX array, shape $(:)$ with $size$(BP) $= size$(AP).
On entry, the upper or lower triangle of matrix $B$ in packed storage. The elements are stored columnwise as follows:

$$\begin{array}{c\vert c\vert c} B_{i,j} & i,j & {\bf UPLO} ... ...j \leq i \leq n \end{array} & \mbox{'L'} \\ \hline \end{array}$$

On exit, the triangular factor $U$ or $L$ of the Cholesky factorization $B = U^T\,U$ or $B = L\,L^T$, in the same storage format as $B$.

W

(output) REAL array, shape $(:)$ with $size({\bf W}) = n$.
The eigenvalues in ascending order.

ITYPE

Optional (input) INTEGER.
Specifies the problem type to be solved:


Default value: 1.

UPLO

Optional (input) CHARACTER(LEN=1).


Default value: 'U'.

Z

Optional (output) REAL or COMPLEX rectangular array, shape $(:,:)$ with $size({\bf Z},1) = n$ and $size$(Z,2) = M.
The first M columns of Z contain the orthonormal eigenvectors corresponding to the selected eigenvalues, with the $i^{th}$ column of Z holding the eigenvector associated with the eigenvalue in W$_i$. The eigenvectors are normalized as follows:


If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector and the index of the eigenvector is returned in IFAIL.

VL,VU

Optional (input) REAL.
The lower and upper bounds of the interval to be searched for eigenvalues. VL $<$ VU.
Default values: VL $=$ -HUGE(wp) and VU $=$ HUGE(wp), where wp ::= KIND(1.0) $\mid$ KIND(1.0D0).
Note: Neither VL nor VU may be present if IL and/or IU is present.

IL,IU

Optional (input) INTEGER.
The indices of the smallest and largest eigenvalues to be returned. The ${\bf IL}^{th}$ through ${\bf IU}^{th}$ eigenvalues will be found. $1 \leq {\bf IL} \leq {\bf IU} \leq size({\bf A},1)$.
Default values: IL $= 1$ and IU $=$ $size$(A,1).
Note: Neither IL nor IU may be present if VL and/or VU is present.
Note: All eigenvalues are calculated if none of the arguments VL, VU, IL and IU are present.

M

Optional (output) INTEGER.
The total number of eigenvalues found. $0 \leq {\bf M} \leq size({\bf A},1)$.
Note: If ${\bf IL}$ and ${\bf IU}$ are present then ${\bf M} = {\bf IU}-{\bf IL}+1$.

IFAIL

Optional (output) INTEGER array, shape $(:)$ with $size$(IFAIL) $=$ $size$(A,1).
If INFO $= 0$, the first M elements of IFAIL are zero.
If INFO $> 0$, then IFAIL contains the indices of the eigenvectors that failed to converge.
Note: If Z is present then IFAIL should also be present.

ABSTOL

Optional (input) REAL.
The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval $[a,b]$ of width less than or equal to

$${\bf ABSTOL} + {\bf EPSILON}(1.0\_{\it wp})\times \max(\mid a\mid,\mid b\mid),$$

where wp is the working precision. If ABSTOL $\leq 0$, then ${\bf EPSILON}(1.0\_{\it wp})\times \Vert T \Vert _1$ will be used in its place, where $\Vert T \Vert _1$ is the $l_1$ norm of the tridiagonal matrix obtained by reducing the generalized eigenvalue problem to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold $2\times {\bf LA\_LAMCH}(1.0\_{\it wp},\mbox{'S'})$, not zero.
Default value: $0.0\_{\it wp}$.
Note: If this routine returns with $0 < {\bf INFO} \leq n$, then some eigenvectors did not converge. Try setting ABSTOL to $2\times {\bf LA\_LAMCH}(1.0\_{\it wp},\mbox{'S'})$.

INFO

Optional (output) INTEGER.


If INFO is not present and an error occurs, then the program is terminated with an error message.

References: [1] and [17,9,20,21].