nag_dsterf (f08jfc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dsterf (f08jfc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dsterf (f08jfc) computes all the eigenvalues of a real symmetric tridiagonal matrix.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dsterf (Integer n, double d[], double e[], NagError *fail)

3  Description

nag_dsterf (f08jfc) computes all the eigenvalues of a real symmetric tridiagonal matrix, using a square-root-free variant of the QR algorithm.
The function uses an explicit shift, and, like nag_dsteqr (f08jec), 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  Arguments

1:     nIntegerInput
On entry: n, the order of the matrix T.
Constraint: n0.
2:     d[dim]doubleInput/Output
Note: the dimension, dim, 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 fail.code= NE_CONVERGENCE (in which case see Section 6).
3:     e[dim]doubleInput/Output
Note: the dimension, dim, 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:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_CONVERGENCE
The algorithm has failed to find all the eigenvalues after a total of 30×n iterations; value elements of e have not converged to zero.
NE_INT
On entry, n=value.
Constraint: n0.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

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 function.

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 (f08jfce.c)

9.2  Program Data

Program Data (f08jfce.d)

9.3  Program Results

Program Results (f08jfce.r)


nag_dsterf (f08jfc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012