NAG Library Routine Document

Integer, Intent (In)	::	n, il, iu, ldz, lwork, liwork
Integer, Intent (Inout)	::	isuppz(*)
Integer, Intent (Out)	::	m, iwork(max(1,liwork)), info
Real (Kind=nag_wp), Intent (In)	::	vl, vu, abstol
Real (Kind=nag_wp), Intent (Inout)	::	d(), e(), w(), z(ldz,)
Real (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))
Character (1), Intent (In)	::	jobz, range

C Header Interface

#include <nagmk26.h>

void

f08jdf_ (const char *jobz, const char *range, const Integer *n, double d[], double e[], const double *vl, const double *vu, const Integer *il, const Integer *iu, const double *abstol, Integer *m, double w[], double z[], const Integer *ldz, Integer isuppz[], double work[], const Integer *lwork, Integer iwork[], const Integer *liwork, Integer *info, const Charlen length_jobz, const Charlen length_range)

The routine may be called by its LAPACK name dstevr.

3

Description

Whenever possible f08jdf (dstevr) computes the eigenspectrum using Relatively Robust Representations. f08jdf (dstevr) computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various ‘good’

L D L^{T}

representations (also known as Relatively Robust Representations). Gram–Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the

i

th unreduced block of

T

(a)	compute $T - σ_{i} I = L_{i} D_{i} L_{i}^{T}$ , such that $L_{i} D_{i} L_{i}^{T}$ is a relatively robust representation,
(b)	compute the eigenvalues, $λ_{j}$ , of $L_{i} D_{i} L_{i}^{T}$ to high relative accuracy by the dqds algorithm,
(c)	if there is a cluster of close eigenvalues, ‘choose’ $σ_{i}$ close to the cluster, and go to (a),
(d)	given the approximate eigenvalue $λ_{j}$ of $L_{i} D_{i} L_{i}^{T}$ , compute the corresponding eigenvector by forming a rank-revealing twisted factorization.

The desired accuracy of the output can be specified by the argument abstol. For more details, see Dhillon (1997) and Parlett and Dhillon (2000).

4

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791

Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912

Dhillon I (1997) A new

O (n^{2})

algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem Computer Science Division Technical Report No. UCB//CSD-97-971 UC Berkeley

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

Parlett B N and Dhillon I S (2000) Relatively robust representations of symmetric tridiagonals Linear Algebra Appl. 309 121–151

5

Arguments

1: $jobz$ – Character(1)Input

On entry: indicates whether eigenvectors are computed.

$jobz ='N'$: Only eigenvalues are computed.
$jobz ='V'$: Eigenvalues and eigenvectors are computed.

Constraint:

jobz ='N'

'V'

2: $range$ – Character(1)Input

On entry: if

range ='A'

, all eigenvalues will be found.

range ='V'

, all eigenvalues in the half-open interval

(vl, vu]

will be found.

range ='I'

, the ilth to iuth eigenvalues will be found.

Constraint:

range ='A'

'V'

'I'

3: $n$ – IntegerInput

On entry:

n

, the order of the matrix.

Constraint:

n \geq 0

4: $d (*)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the dimension of the array d must be at least

\max (1, n)

On entry: the

n

diagonal elements of the tridiagonal matrix

T

On exit: may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.

5: $e (*)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the dimension of the array e must be at least

\max (1, n - 1)

On entry: the

(n - 1)

subdiagonal elements of the tridiagonal matrix

T

On exit: may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.

6: $vl$ – Real (Kind=nag_wp)Input

7: $vu$ – Real (Kind=nag_wp)Input

On entry: if

range ='V'

, the lower and upper bounds of the interval to be searched for eigenvalues.

range ='A'

'I'

, vl and vu are not referenced.

Constraint: if

range ='V'

vl < vu

8: $il$ – IntegerInput

9: $iu$ – IntegerInput

On entry: if

range ='I'

, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.

range ='A'

'V'

, il and iu are not referenced.

Constraints:

if $range ='I'$ and $n = 0$ , $il = 1$ and $iu = 0$ ;
if $range ='I'$ and $n > 0$ , $1 \leq il \leq iu \leq n$ .

10: $abstol$ – Real (Kind=nag_wp)Input

On entry: 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

abstol + ε \max (|a|, |b|),

where

ε

is the machine precision. If abstol is less than or equal to zero, then

ε {‖T‖}_{1}

will be used in its place. See Demmel and Kahan (1990).

If high relative accuracy is important, set abstol to

x02amf ()

, although doing so does not currently guarantee that eigenvalues are computed to high relative accuracy. See Barlow and Demmel (1990) for a discussion of which matrices can define their eigenvalues to high relative accuracy.

11: $m$ – IntegerOutput

On exit: the total number of eigenvalues found.

0 \leq m \leq n

range ='A'

m = n

range ='I'

m = iu - il + 1

12: $w (*)$ – Real (Kind=nag_wp) arrayOutput

Note: the dimension of the array w must be at least

\max (1, n)

On exit: the first m elements contain the selected eigenvalues in ascending order.

13: $z (ldz *)$ – Real (Kind=nag_wp) arrayOutput

Note: the second dimension of the array z must be at least

\max (1, m)

jobz ='V'

, and at least

1

otherwise.

On exit: if

jobz ='V'

, the first m columns of

Z

contain the orthonormal eigenvectors of the matrix

A

corresponding to the selected eigenvalues, with the

i

th column of

Z

holding the eigenvector associated with

w (i)

jobz ='N'

, z is not referenced.

Note: you must ensure that at least

\max (1, m)

columns are supplied in the array z; if

range ='V'

, the exact value of m is not known in advance and an upper bound of at least n must be used.

14: $ldz$ – IntegerInput

On entry: the first dimension of the array z as declared in the (sub)program from which f08jdf (dstevr) is called.

Constraints:

if $jobz ='V'$ , $ldz \geq \max (1, n)$ ;
otherwise $ldz \geq 1$ .

15: $isuppz (*)$ – Integer arrayOutput

Note: the dimension of the array isuppz must be at least

\max (1, 2 \times m)

On exit: the support of the eigenvectors in z, i.e., the indices indicating the nonzero elements in z. The

i

th eigenvector is nonzero only in elements

isuppz (2 \times i - 1)

through

isuppz (2 \times i)

. Implemented only for

range ='A'

'I'

and

iu - il = n - 1

16: $work (\max (1, lwork))$ – Real (Kind=nag_wp) arrayWorkspace

On exit: if

info = 0

work (1)

contains the minimum value of lwork required for optimal performance.

17: $lwork$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f08jdf (dstevr) is called.

lwork = - 1

, a workspace query is assumed; the routine only calculates the minimum sizes of the work and iwork arrays, returns these values as the first entries of the work and iwork arrays, and no error message related to lwork or liwork is issued.

Constraint:

lwork \geq \max (1, 20 \times n)

18: $iwork (\max (1, liwork))$ – Integer arrayWorkspace

On exit: if

info = 0

iwork (1)

returns the minimum liwork.

19: $liwork$ – IntegerInput

On entry: the dimension of the array iwork as declared in the (sub)program from which f08jdf (dstevr) is called.

liwork = - 1

Constraint:

liwork \geq \max (1, 120 \times n)

20: $info$ – IntegerOutput

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.

$info > 0$: An internal error has occurred in this routine. Please refer to info in f08jjf (dstebz).

7

Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix

(A + E)

, where

{‖E‖}_{2} = O (ε) {‖A‖}_{2},

and

ε

is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

8

Parallelism and Performance

f08jdf (dstevr) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08jdf (dstevr) 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.

9

Further Comments

The total number of floating-point operations is proportional to

n^{2}

jobz ='N'

and is proportional to

n^{3}

jobz ='V'

and

range ='A'

, otherwise the number of floating-point operations will depend upon the number of computed eigenvectors.

10

Example

This example finds the eigenvalues with indices in the range

[2, 3]

, and the corresponding eigenvectors, of the symmetric tridiagonal matrix

T = (\begin{array}{r} 1 & 1 & 0 & 0 \\ 1 & 4 & 2 & 0 \\ 0 & 2 & 9 & 3 \\ 0 & 0 & 3 & 16 \end{array}) .

NAG Library Routine Document

f08jdf (dstevr)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results