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(*)
Real (Kind=nag_wp), Intent (Out)	::	work(max(1,lwork))
Complex (Kind=nag_wp), Intent (Inout)	::	z(ldz,*)
Character (1), Intent (In)	::	jobz, range

C Header Interface

#include <nag.h>

void

f08jyf_ (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[], Complex 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 the names f08jyf, nagf_lapackeig_zstegr or its LAPACK name zstegr.

3 Description

f08jyf computes selected eigenvalues and, optionally, the corresponding eigenvectors, of a real symmetric tridiagonal matrix

T

. That is, the routine computes the (partial) spectral factorization of

T

given by

Z Λ Z^{T},

where

Λ

is a diagonal matrix whose diagonal elements are the selected eigenvalues,

λ_{i}

, of

T

and

Z

is an orthogonal matrix whose columns are the corresponding eigenvectors,

z_{i}

, of

T

. Thus

T z_{i} = λ_{i} z_{i}, i = 1, 2, \dots, m

where

m

is the number of selected eigenvectors computed.

The routine stores the real orthogonal matrix

Z

in a complex array, so that it may also be used to compute selected eigenvalues and the corresponding eigenvectors of a complex Hermitian matrix

A

which has been reduced to tridiagonal form

T

(Q Z) Λ {(Q Z)}^{H}, where ​ Q ​ is unitary.

In this case, the matrix

Q

must be explicitly applied to the output matrix

Z

. The routines which must be called to perform the reduction to tridiagonal form and apply

Q

are:

full matrix	f08fsf and f08fuf
full matrix, packed storage	f08gsf and f08guf
band matrix	f08hsf with $vect ='V'$ and f06zaf.

This routine uses the dqds and the Relatively Robust Representation algorithms to compute the eigenvalues and eigenvectors respectively; see for example Parlett and Dhillon (2000) and Dhillon and Parlett (2004) for further details. f08jyf can usually compute all the eigenvalues and eigenvectors in

O (n^{2})

floating-point operations and so, for large matrices, is often considerably faster than the other symmetric tridiagonal routines in this chapter when all the eigenvectors are required, particularly so compared to those routines that are based on the

Q R

algorithm.

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 https://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

Dhillon I S and Parlett B N (2004) Orthogonal eigenvectors and relative gaps SIAM J. Appl. Math. 25 858–899

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: indicates which eigenvalues should be returned.

$range ='A'$: All eigenvalues will be found.
$range ='V'$: All eigenvalues in the half-open interval $(vl, vu]$ will be found.
$range ='I'$: The ilth through iuth eigenvectors will be found.

Constraint:

range ='A'

'V'

'I'

3: $n$ – Integer Input

On entry:

n

, the order of the matrix

T

Constraint:

n \geq 0

4: $d (*)$ – Real (Kind=nag_wp) array Input/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: d is overwritten.

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

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

\max (1, n)

On entry:

e (1 : n - 1)

contains the subdiagonal elements of the tridiagonal matrix

T

e (n)

need not be set.

On exit: e is overwritten.

6: $vl$ – Real (Kind=nag_wp) Input

7: $vu$ – Real (Kind=nag_wp) Input

On entry: if

range ='V'

, vl and vu contain the lower and upper bounds respectively of the interval to be searched for eigenvalues.

range ='A'

'I'

, vl and vu are not referenced.

Constraint: if

range ='V'

vl < vu

8: $il$ – Integer Input

9: $iu$ – Integer Input

On entry: if

range ='I'

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

range ='A'

'V'

, il and iu are not referenced.

Constraints:

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

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

On entry: in earlier versions, this argument was the absolute error tolerance for the eigenvalues/eigenvectors. It is now deprecated, and only included for backwards-compatibility.

11: $m$ – Integer Output

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) array Output

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

\max (1, n)

On exit: the eigenvalues in ascending order.

13: $z (ldz, *)$ – Complex (Kind=nag_wp) array Output

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'

, then if

info = 0

, the columns of z contain the orthonormal eigenvectors of the matrix

T

, 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: $ldz$ – Integer Input

On entry: the first dimension of the array z as declared in the (sub)program from which f08jyf is called.

Constraints:

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

15: $isuppz (*)$ – Integer array Output

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)

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

On exit: if

info = 0

work (1)

returns the minimum lwork.

17: $lwork$ – Integer Input

On entry: the dimension of the array work as declared in the (sub)program from which f08jyf 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, 18 \times n)

lwork = - 1

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

On exit: if

info = 0

work (1)

returns the minimum liwork.

19: $liwork$ – Integer Input

On entry: the dimension of the array iwork as declared in the (sub)program from which f08jyf is called.

liwork = - 1

Constraint:

liwork \geq \max (1, 10 \times n)

liwork = - 1

20: $info$ – 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.

$info > 0expr$: Inverse iteration failed to converge.

The $dqds$ algorithm failed to converge.

7 Accuracy

See the description for abstol. See also Section 4.7 of Anderson et al. (1999) and Barlow and Demmel (1990) for further details.

8 Parallelism and Performance

f08jyf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08jyf 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 required to compute all the eigenvalues and eigenvectors is approximately proportional to

n^{2}

The real analogue of this routine is f08jlf.

10 Example

This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix

T = (\begin{array}{r} 1.0 & 1.0 & 0 & 0 \\ 1.0 & 4.0 & 2.0 & 0 \\ 0 & 2.0 & 9.0 & 3.0 \\ 0 & 0 & 3.0 & 16.0 \end{array}) .

abstol is set to zero so that the default tolerance of

n ε {‖ T ‖}_{1}

is used.

f08jy: FL CL CPP AD

NAG FL Interfacef08jyf (zstegr)

▸▿ 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

NAG FL Interface
f08jyf (zstegr)