nag_dsteqr (f08jec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_dsteqr (f08jec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dsteqr (f08jec) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix, or of a real symmetric matrix which has been reduced to tridiagonal form.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dsteqr (Nag_OrderType order, Nag_ComputeZType compz, Integer n, double d[], double e[], double z[], Integer pdz, NagError *fail)

3  Description

nag_dsteqr (f08jec) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric tridiagonal matrix T. In other words, it can compute the spectral factorization of T as
where Λ is a diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the orthogonal matrix whose columns are the eigenvectors zi. Thus
Tzi=λizi,  i=1,2,,n.
The function may also be used to compute all the eigenvalues and eigenvectors of a real symmetric matrix A which has been reduced to tridiagonal form T:
A =QTQT, where ​Q​ is orthogonal =QZΛQZT.
In this case, the matrix Q must be formed explicitly and passed to nag_dsteqr (f08jec), which must be called with compz=Nag_UpdateZ. The functions which must be called to perform the reduction to tridiagonal form and form Q are:
full matrix nag_dsytrd (f08fec) and nag_dorgtr (f08ffc)
full matrix, packed storage nag_dsptrd (f08gec) and nag_dopgtr (f08gfc)
band matrix nag_dsbtrd (f08hec) with vect=Nag_FormQ.
nag_dsteqr (f08jec) uses the implicitly shifted QR algorithm, switching between the QR and QL variants in order to handle graded matrices effectively (see Greenbaum and Dongarra (1980)). The eigenvectors are normalized so that zi2=1, but are determined only to within a factor ±1.
If only the eigenvalues of T are required, it is more efficient to call nag_dsterf (f08jfc) instead. If T is positive definite, small eigenvalues can be computed more accurately by nag_dpteqr (f08jgc).

4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
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:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     compzNag_ComputeZTypeInput
On entry: indicates whether the eigenvectors are to be computed.
Only the eigenvalues are computed (and the array z is not referenced).
The eigenvalues and eigenvectors of T are computed (and the array z is initialized by the function).
The eigenvalues and eigenvectors of A are computed (and the array z must contain the matrix Q on entry).
Constraint: compz=Nag_NotZ, Nag_UpdateZ or Nag_InitZ.
3:     nIntegerInput
On entry: n, the order of the matrix T.
Constraint: n0.
4:     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).
5:     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.
6:     z[dim]doubleInput/Output
Note: the dimension, dim, of the array z must be at least pdz×n when compz=Nag_UpdateZ or Nag_InitZ.
The i,jth element of the matrix Z is stored in
  • z[j-1×pdz+i-1] when order=Nag_ColMajor;
  • z[i-1×pdz+j-1] when order=Nag_RowMajor.
On entry: if compz=Nag_UpdateZ, z must contain the orthogonal matrix Q from the reduction to tridiagonal form.
If compz=Nag_InitZ, z must be allocated, but its contents need not be set.
If compz=Nag_NotZ, z is not referenced and may be NULL.
On exit: if compz=Nag_InitZ or Nag_UpdateZ, the n required orthonormal eigenvectors stored as columns of Z; the ith column corresponds to the ith eigenvalue, where i=1,2,,n, unless fail.errnum>0.
z is not changed if compz=Nag_NotZ.
7:     pdzIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array z.
  • if compz=Nag_InitZ or Nag_UpdateZ, pdz n ;
  • if compz=Nag_NotZ, z may be NULL.
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
The algorithm has failed to find all the eigenvalues after a total of 30×n iterations. In this case, d and e contain on exit the diagonal and off-diagonal elements, respectively, of a tridiagonal matrix orthogonally similar to T. value off-diagonal elements have not converged to zero.
On entry, compz=value, pdz=value and n=value.
Constraint: if compz=Nag_InitZ or Nag_UpdateZ, pdz n .
On entry, n=value.
Constraint: n0.
On entry, pdz=value.
Constraint: pdz>0.
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 and eigenvectors 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.
If zi is the corresponding exact eigenvector, and z~i is the corresponding computed eigenvector, then the angle θz~i,zi between them is bounded as follows:
θ z~i,zi cnεT2 minijλi-λj .
Thus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other eigenvalues.

8  Parallelism and Performance

nag_dsteqr (f08jec) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dsteqr (f08jec) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The total number of floating-point operations is typically about 24n2 if compz=Nag_NotZ and about 7n3 if compz=Nag_UpdateZ or Nag_InitZ, but depends on how rapidly the algorithm converges. When compz=Nag_NotZ, the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when compz=Nag_UpdateZ or Nag_InitZ can be vectorized and on some machines may be performed much faster.
The complex analogue of this function is nag_zsteqr (f08jsc).

10  Example

This example computes all the eigenvalues and eigenvectors 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 .
See also the examples for nag_dorgtr (f08ffc)nag_dopgtr (f08gfc) or nag_dsbtrd (f08hec), which illustrate the use of this function to compute the eigenvalues and eigenvectors of a full or band symmetric matrix.

10.1  Program Text

Program Text (f08jece.c)

10.2  Program Data

Program Data (f08jece.d)

10.3  Program Results

Program Results (f08jece.r)

nag_dsteqr (f08jec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014