f08jxf computes the eigenvectors of a real symmetric tridiagonal matrix corresponding to specified eigenvalues, by inverse iteration, storing the eigenvectors in a complex array.

2 Specification

Fortran Interface

Subroutine f08jxf (

n, d, e, m, w, iblock, isplit, z, ldz, work, iwork, ifailv, info)

Integer, Intent (In)	::	n, m, iblock(), isplit(), ldz
Integer, Intent (Out)	::	iwork(n), ifailv(m), info
Real (Kind=nag_wp), Intent (In)	::	d(), e(), w(*)
Real (Kind=nag_wp), Intent (Out)	::	work(5*n)
Complex (Kind=nag_wp), Intent (Inout)	::	z(ldz,*)

C Header Interface

#include <nag.h>

void	f08jxf_ (const Integer n, const double d[], const double e[], const Integer m, const double w[], const Integer iblock[], const Integer isplit[], Complex z[], const Integer ldz, double work[], Integer iwork[], Integer ifailv[], Integer info)

The routine may be called by the names f08jxf, nagf_lapackeig_zstein or its LAPACK name zstein.

3 Description

f08jxf computes the eigenvectors of a real symmetric tridiagonal matrix

T

corresponding to specified eigenvalues, by inverse iteration (see Jessup and Ipsen (1992)). It is designed to be used in particular after the specified eigenvalues have been computed by f08jjf with

order ='B'

, but may also be used when the eigenvalues have been computed by other routines in Chapters F02 or F08.

The eigenvectors of

T

are real, but are stored by this routine in a complex array. If

T

has been formed by reduction of a full complex Hermitian matrix

A

to tridiagonal form, then eigenvectors of

T

may be transformed to (complex) eigenvectors of

A

by a call to f08fuf or f08guf.

f08jjf determines whether the matrix

T

splits into block diagonal form:

T = (\begin{array}{r} T_{1} \\ T_{2} \\ . \\ . \\ . \\ T_{p} \end{array})

and passes details of the block structure to this routine in the arrays iblock and isplit. This routine can then take advantage of the block structure by performing inverse iteration on each block

T_{i}

separately, which is more efficient than using the whole matrix.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Jessup E and Ipsen I C F (1992) Improving the accuracy of inverse iteration SIAM J. Sci. Statist. Comput. 13 550–572

5 Arguments

1: $n$ – Integer Input: On entry: $n$ , the order of the matrix $T$ .

Constraint: $n \geq 0$ .
2: $d (*)$ – Real (Kind=nag_wp) array Input: Note: the dimension of the array d must be at least $\max (1, n)$ .

On entry: the diagonal elements of the tridiagonal matrix $T$ .
3: $e (*)$ – Real (Kind=nag_wp) array Input: Note: the dimension of the array e must be at least $\max (1, n - 1)$ .

On entry: the off-diagonal elements of the tridiagonal matrix $T$ .
4: $m$ – Integer Input: On entry: $m$ , the number of eigenvectors to be returned.

Constraint: $0 \leq m \leq n$ .
5: $w (*)$ – Real (Kind=nag_wp) array Input: Note: the dimension of the array w must be at least $\max (1, n)$ .

On entry: the eigenvalues of the tridiagonal matrix $T$ stored in $w (1)$ to $w (m)$ , as returned by f08jjf with $order ='B'$ . Eigenvalues associated with the first sub-matrix must be supplied first, in nondecreasing order; then those associated with the second sub-matrix, again in nondecreasing order; and so on.

Constraint: if $iblock (i) = iblock (i + 1)$ , $w (i) \leq w (i + 1)$ , for $i = 1, 2, \dots, m - 1$ .
6: $iblock (*)$ – Integer array Input: Note: the dimension of the array iblock must be at least $\max (1, n)$ .

On entry: the first $m$ elements must contain the sub-matrix indices associated with the specified eigenvalues, as returned by f08jjf with $order ='B'$ . If the eigenvalues were not computed by f08jjf with $order ='B'$ , set $iblock (i)$ to $1$ , for $i = 1, 2, \dots, m$ .

Constraint: $iblock (i) \leq iblock (i + 1)$ , for $i = 1, 2, \dots, m - 1$ .
7: $isplit (*)$ – Integer array Input: Note: the dimension of the array isplit must be at least $\max (1, n)$ .

On entry: the points at which $T$ breaks up into sub-matrices, as returned by f08jjf with $order ='B'$ . If the eigenvalues were not computed by f08jjf with $order ='B'$ , set $isplit (1)$ to n.
8: $z (ldz, *)$ – Complex (Kind=nag_wp) array Output: Note: the second dimension of the array z must be at least $\max (1, m)$ .

On exit: the $m$ eigenvectors, stored as columns of $Z$ ; the $i$ th column corresponds to the $i$ th specified eigenvalue, unless $info > 0$ (in which case see Section 6).
9: $ldz$ – Integer Input: On entry: the first dimension of the array z as declared in the (sub)program from which f08jxf is called.

Constraint: $ldz \geq \max (1, n)$ .
10: $work (5 \times n)$ – Real (Kind=nag_wp) array Workspace
11: $iwork (n)$ – Integer array Workspace
12: $ifailv (m)$ – Integer array Output: On exit: if $info = i > 0$ , the first $i$ elements of ifailv contain the indices of any eigenvectors which have failed to converge. The rest of the first m elements of ifailv are set to $0$ .
13: $info$ – Integer Output: On exit: $info = 0$ unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

$info > 0$: $⟨ value ⟩$ eigenvectors (as indicated by argument ifailv) each failed to converge in five iterations. The current iterate after five iterations is stored in the corresponding column of z.

7 Accuracy

Each computed eigenvector

z_{i}

is the exact eigenvector of a nearby matrix

A + E_{i}

, such that

‖ E_{i} ‖ = O (ε) ‖ A ‖,

where

ε

is the machine precision. Hence the residual is small:

‖ A z_{i} - λ_{i} z_{i} ‖ = O (ε) ‖ A ‖ .

However, a set of eigenvectors computed by this routine may not be orthogonal to so high a degree of accuracy as those computed by f08jsf.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

f08jxf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08jxf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.