F08JFF (DSTERF) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08JFF (DSTERF)

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

F08JFF (DSTERF) computes all the eigenvalues of a real symmetric tridiagonal matrix.

2  Specification

SUBROUTINE F08JFF ( N, D, E, INFO)
INTEGER  N, INFO
REAL (KIND=nag_wp)  D(*), E(*)
The routine may be called by its LAPACK name dsterf.

3  Description

F08JFF (DSTERF) computes all the eigenvalues of a real symmetric tridiagonal matrix, using a square-root-free variant of the QR algorithm.
The routine uses an explicit shift, and, like F08JEF (DSTEQR), switches between the QR and QL variants in order to handle graded matrices effectively (see Greenbaum and Dongarra (1980)).

4  References

Greenbaum A and Dongarra J J (1980) Experiments with QR/QL methods for the symmetric triangular eigenproblem LAPACK Working Note No. 17 (Technical Report CS-89-92) University of Tennessee, Knoxville
Parlett B N (1998) The Symmetric Eigenvalue Problem SIAM, Philadelphia

5  Parameters

1:     N – INTEGERInput
On entry: n, the order of the matrix T.
Constraint: N0.
2:     D(*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array D must be at least max1,N.
On entry: the diagonal elements of the tridiagonal matrix T.
On exit: the n eigenvalues in ascending order, unless INFO>0 (in which case see Section 6).
3:     E(*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array E must be at least max1,N-1.
On entry: the off-diagonal elements of the tridiagonal matrix T.
On exit: E is overwritten.
4:     INFO – INTEGEROutput

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>0

7  Accuracy

The computed eigenvalues are exact for a nearby matrix T+E, where
E2 = Oε T2 ,
and ε is the machine precision.
If λi is an exact eigenvalue and λ~i is the corresponding computed value, then
λ~i - λi c n ε T2 ,
where cn is a modestly increasing function of n.

8  Further Comments

The total number of floating point operations is typically about 14n2, but depends on how rapidly the algorithm converges. The operations are all performed in scalar mode.
There is no complex analogue of this routine.

9  Example

This example computes all the eigenvalues of the symmetric tridiagonal matrix T, where
T = -6.99 -0.44 0.00 0.00 -0.44 7.92 -2.63 0.00 0.00 -2.63 2.34 -1.18 0.00 0.00 -1.18 0.32 .

9.1  Program Text

Program Text (f08jffe.f90)

9.2  Program Data

Program Data (f08jffe.d)

9.3  Program Results

Program Results (f08jffe.r)


F08JFF (DSTERF) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

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