F08JAF (DSTEV) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08JAF (DSTEV)

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

F08JAF (DSTEV) computes all the eigenvalues and, optionally, all the eigenvectors of a real n by n symmetric tridiagonal matrix A.

2  Specification

SUBROUTINE F08JAF ( JOBZ, N, D, E, Z, LDZ, WORK, INFO)
INTEGER  N, LDZ, INFO
REAL (KIND=nag_wp)  D(*), E(*), Z(LDZ,*), WORK(*)
CHARACTER(1)  JOBZ
The routine may be called by its LAPACK name dstev.

3  Description

F08JAF (DSTEV) computes all the eigenvalues and, optionally, all the eigenvectors of A using a combination of the QR and QL algorithms, with an implicit shift.

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

5  Parameters

1:     JOBZ – CHARACTER(1)Input
On entry: if JOBZ='N', compute eigenvalues only.
If JOBZ='V', compute eigenvalues and eigenvectors.
Constraint: JOBZ='N' or 'V'.
2:     N – INTEGERInput
On entry: n, the order of the matrix.
Constraint: N0.
3:     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 A.
On exit: if INFO=0, the eigenvalues in ascending order.
4:     E(*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array E must be at least max1,N-1.
On entry: the n-1 subdiagonal elements of the tridiagonal matrix A.
On exit: the contents of E are destroyed.
5:     Z(LDZ,*) – REAL (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, Z contains the orthonormal eigenvectors of the matrix A, with the ith column of Z holding the eigenvector associated with Di.
If JOBZ='N', Z is not referenced.
6:     LDZ – INTEGERInput
On entry: the first dimension of the array Z as declared in the (sub)program from which F08JAF (DSTEV) is called.
Constraints:
  • if JOBZ='V', LDZmax1,N;
  • otherwise LDZ1.
7:     WORK(*) – REAL (KIND=nag_wp) arrayWorkspace
Note: the dimension of the array WORK must be at least max1,2×N-2.
On exit: if JOBZ='N', WORK is not referenced.
8:     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

7  Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix A+E, where
E2 = Oε A2 ,
and ε is the machine precision. See Section 4.7 of
Anderson et al. (1999) for further details.

8  Further Comments

The total number of floating point operations is proportional to n2 if JOBZ='N' and is proportional to n3 if JOBZ='V'.

9  Example

This example finds all the eigenvalues and eigenvectors of the symmetric tridiagonal matrix
A = 1 1 0 0 1 4 2 0 0 2 9 3 0 0 3 16 ,
together with approximate error bounds for the computed eigenvalues and eigenvectors.

9.1  Program Text

Program Text (f08jafe.f90)

9.2  Program Data

Program Data (f08jafe.d)

9.3  Program Results

Program Results (f08jafe.r)


F08JAF (DSTEV) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

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