NAG Library Routine Document

F08JYF (ZSTEGR)

INTEGER	N, IL, IU, M, LDZ, ISUPPZ(*), LWORK, IWORK(max(1,LIWORK)), LIWORK, INFO
REAL (KIND=nag_wp)	D(), E(), VL, VU, ABSTOL, W(*), WORK(max(1,LWORK))
COMPLEX (KIND=nag_wp)	Z(LDZ,*)
CHARACTER(1)	JOBZ, RANGE

The routine may be called by its LAPACK name zstegr.

3 Description

F08JYF (ZSTEGR) computes all the eigenvalues and, optionally, the eigenvectors, of a real symmetric tridiagonal matrix

T

. That is, the routine computes the spectral factorization of

T

given by

T = Z Λ Z^{T},

where

Λ

is a diagonal matrix whose diagonal elements are the eigenvalues,

λ_{i}

, of

T

and

Z

is an orthogonal matrix whose columns are the eigenvectors,

z_{i}

, of

T

. Thus

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

The routine stores the real orthogonal matrix

Z

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

A

which has been reduced to tridiagonal form

T

\begin{array}{l} A & = Q T Q^{H}, where ​ Q ​ is unitary \\ = (Q Z) Λ {(Q Z)}^{H} . \end{array}

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 (ZHETRD) and F08FUF (ZUNMTR)
full matrix, packed storage	F08GSF (ZHPTRD) and F08GUF (ZUPMTR)
band matrix	F08HSF (ZHBTRD) with $VECT ='V'$ and F06ZAF (ZGEMM).

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 (ZSTEGR) 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 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

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 Parameters

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 – INTEGERInput

On entry:

n

, the order of the matrix

T

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: D is overwritten.

5: E( $*$ ) – REAL (KIND=nag_wp) arrayInput/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 – INTEGERInput

9: IU – INTEGERInput

On entry: if

RANGE ='I'

, IL and IU contains 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 – 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 eigenvalues in ascending order.

13: Z(LDZ, $*$ ) – COMPLEX (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'

, 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 – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08JYF (ZSTEGR) 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)

16: WORK( $\max (1, LWORK)$ ) – REAL (KIND=nag_wp) arrayWorkspace

On exit: if

INFO = 0

WORK (1)

returns the minimum LWORK.

17: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08JYF (ZSTEGR) 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 arrayWorkspace

On exit: if

INFO = 0

WORK (1)

returns the minimum LIWORK.

19: LIWORK – INTEGERInput

On entry: the dimension of the array IWORK as declared in the (sub)program from which F08JYF (ZSTEGR) is called.

LIWORK = - 1

Constraint:

LIWORK \geq \max (1, 10 \times N)

LIWORK = - 1

20: INFO – INTEGEROutput

On exit:

INFO = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

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

$INFO > 0$: If $INFO = 1$ , the $dqds$ algorithm failed to converge, if $INFO = 2$ , inverse iteration 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 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 (DSTEGR).

9 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.

NAG Library Routine DocumentF08JYF (ZSTEGR)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F08JYF (ZSTEGR)