Arguments

D

(input/output) REAL array, shape $(:)$ with $size$(D) $= n$, where $n$ is the order of $A$.
On entry, the diagonal elements of the matrix $A$.
On exit, the original contents of D possibly multiplied by a constant factor to avoid over/underflow in computing the eigenvalues.

E

(input/output) REAL array, shape $(:)$ with $size$(E) $= n$.
On entry, the $n - 1$ subdiagonal elements of $A$ in E$_1$ to E$_{n-1}\,$. E$_n$ need not be set.
On exit, the original contents of E possibly multiplied by a constant factor to avoid over/underflow in computing the eigenvalues.

W

(output) REAL array with $size({\bf W}) = n$.
The first M elements contain the selected eigenvalues in ascending order.

Z

Optional (output) REAL or COMPLEX array, shape $(:,:)$ with $size$(Z,1) $= n$ and $size$(Z,2) $=$ M.
The first M columns of Z contain the orthonormal eigenvectors of $A$ corresponding to the selected eigenvalues, with the $i^{th}$ column of Z containing the eigenvector associated with the eigenvalue in W$_i$.
Note: The user must ensure that at least M columns are supplied in the array Z. When the exact value of M is not known in advance, an upper bound must be used. In all cases M $\leq n$.

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. $1 \leq {\bf IL} \leq {\bf IU} \leq n$.
Default values: IL $= 1$ and IU $= n$.
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. $0 \leq {\bf M} \leq n$.
Note: If ${\bf IL}$ and ${\bf IU}$ are present then ${\bf M} = {\bf IU}-{\bf IL}+1$.

ISUPPZ

Optional (output) INTEGER array, shape $(:)$ with size(ISUPPZ) $= 2\max(1$,M).
The support of the eigenvectors in A, i.e., the indices indicating the nonzero elements. The $i^{th}$ eigenvector is nonzero only in elements ISUPPZ$_{2i-1}$ through ISUPPZ$_{2i}$.

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 ${\bf EPSILON}(1.0\_{\it wp}) \times \Vert A \Vert _1$ will be used in its place. Eigenvalues will be computed most accurately if ABSTOL is set to LA_LAMCH( 1.0_wp, 'Safe minimum'), not zero.
Default value: $0.0\_{\it wp}$.

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].