NAG Library Routine Document

F08JYF (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

1 Purpose

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

2 Specification

SUBROUTINE F08JYF (

JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)

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 = ZΛZT ,

where Λ is a diagonal matrix whose diagonal elements are the eigenvalues, λi, of T and Z is an orthogonal matrix whose columns are the eigenvectors, zi, of T. Thus

Tzi= λi zi , i = 1,2,…,n .

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 On2 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 QR algorithm.

In the future this routine may be extended to allow for the computation of selected eigenvalues and eigenvectors.

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' or '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'

Constraint: RANGE='A', 'V' or 'I'.

3: N – INTEGERInput

On entry: n, the order of the matrix T.

Constraint: N≥0.

4: D(*) – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array D must be at least max1,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 max1,N.

On entry: E1:N-1 contains the subdiagonal elements of the tridiagonal matrix T. EN 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. If RANGE='A' or 'I', VL and VL are not referenced.

Constraint: 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. If RANGE='A' or 'V', IL and IU are not referenced.

Constraints:

if RANGE='I' and N>0, 1≤IL≤IU≤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≤M≤N.

If RANGE='A', M=N.

If RANGE='I', M=IU-IL+1.

12: W(*) – REAL (KIND=nag_wp) arrayOutput

Note: the dimension of the array W must be at least max1,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 max1,N if 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 ith column of Z holding the eigenvector associated with Wi.

If JOBZ='N', Z is not referenced.

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≥max1,N;
otherwise LDZ≥1.

15: ISUPPZ(*) – INTEGER arrayOutput

Note: the dimension of the array ISUPPZ must be at least max1,2×M.

On exit: the support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The ith eigenvector is nonzero only in elements ISUPPZ2×i-1 through ISUPPZ2×i.

16: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace

On exit: if INFO=0, WORK1 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.

If 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 ≥ max1,18×N or LWORK=-1 .

18: IWORK(max1,LIWORK) – INTEGER arrayWorkspace

On exit: if INFO=0, WORK1 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.

If LIWORK=-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.