F08JYF (ZSTEGR) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08JYF (ZSTEGR)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

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
Constraint: RANGE='A', 'V' or 'I'.
3:     N – INTEGERInput
On entry: n, the order of the matrix T.
Constraint: N0.
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, 1ILIUN;
  • 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. 0MN.
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', LDZmax1,N;
  • otherwise LDZ1.
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
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
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.
Constraint: LIWORKmax1,10×N or LIWORK=-1.
20:   INFO – INTEGEROutput

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 n2.
The real analogue of this routine is F08JLF (DSTEGR). The specification of F08JYF (ZSTEGR) differs from that of F08JLF (DSTEGR) only in that the array Z is declared as complex in F08JYF (ZSTEGR).

9  Example

This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
T = 1.0 1.0 0.0 0.0 1.0 4.0 2.0 0.0 0.0 2.0 9.0 3.0 0.0 0.0 3.0 16.0 .
ABSTOL is set to zero so that the default tolerance of nε T1 is used.

9.1  Program Text

Program Text (f08jyfe.f90)

9.2  Program Data

Program Data (f08jyfe.d)

9.3  Program Results

Program Results (f08jyfe.r)


F08JYF (ZSTEGR) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2011