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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dsteqr (f08je)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dsteqr (f08je) 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.

Syntax

[d, e, z, info] = f08je(compz, d, e, 'n', n, 'z', z)
[d, e, z, info] = nag_lapack_dsteqr(compz, d, e, 'n', n, 'z', z)

Description

nag_lapack_dsteqr (f08je) 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
T=ZΛZT,  
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_lapack_dsteqr (f08je), which must be called with compz='V'. The functions which must be called to perform the reduction to tridiagonal form and form Q are:
full matrix nag_lapack_dsytrd (f08fe) and nag_lapack_dorgtr (f08ff)
full matrix, packed storage nag_lapack_dsptrd (f08ge) and nag_lapack_dopgtr (f08gf)
band matrix nag_lapack_dsbtrd (f08he) with vect='V'.
nag_lapack_dsteqr (f08je) 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_lapack_dsterf (f08jf) instead. If T is positive definite, small eigenvalues can be computed more accurately by nag_lapack_dpteqr (f08jg).

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 http://www.netlib.org/lapack/lawnspdf/lawn17.pdf
Parlett B N (1998) The Symmetric Eigenvalue Problem SIAM, Philadelphia

Parameters

Compulsory Input Parameters

1:     compz – string (length ≥ 1)
Indicates whether the eigenvectors are to be computed.
compz='N'
Only the eigenvalues are computed (and the array z is not referenced).
compz='V'
The eigenvalues and eigenvectors of A are computed (and the array z must contain the matrix Q on entry).
compz='I'
The eigenvalues and eigenvectors of T are computed (and the array z is initialized by the function).
Constraint: compz='N', 'V' or 'I'.
2:     d: – double array
The dimension of the array d must be at least max1,n
The diagonal elements of the tridiagonal matrix T.
3:     e: – double array
The dimension of the array e must be at least max1,n-1
The off-diagonal elements of the tridiagonal matrix T.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array d and the second dimension of the array d. (An error is raised if these dimensions are not equal.)
n, the order of the matrix T.
Constraint: n0.
2:     zldz: – double array
The first dimension, ldz, of the array z must satisfy
  • if compz='V' or 'I', ldz max1,n ;
  • if compz='N', ldz1.
The second dimension of the array z must be at least max1,n if compz='V' or 'I' and at least 1 if compz='N'.
If compz='V', z must contain the orthogonal matrix Q from the reduction to tridiagonal form.
If compz='I', z need not be set.

Output Parameters

1:     d: – double array
The dimension of the array d will be max1,n
The n eigenvalues in ascending order, unless info>0 (in which case see Error Indicators and Warnings).
2:     e: – double array
The dimension of the array e will be max1,n-1
3:     zldz: – double array
The first dimension, ldz, of the array z will be
  • if compz='V' or 'I', ldz= max1,n ;
  • if compz='N', ldz=1.
The second dimension of the array z will be max1,n if compz='V' or 'I' and at least 1 if compz='N'.
If compz='V' or 'I', the n required orthonormal eigenvectors stored as columns of Z; the ith column corresponds to the ith eigenvalue, where i=1,2,,n, unless info>0.
If compz='N', z is not referenced.
4:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   info=-i
If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: compz, 2: n, 3: d, 4: e, 5: z, 6: ldz, 7: work, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
W  info>0
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. If info=i, then i off-diagonal elements have not converged to zero.

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.

Further Comments

The total number of floating-point operations is typically about 24n2 if compz='N' and about 7n3 if compz='V' or 'I', but depends on how rapidly the algorithm converges. When compz='N', the operations are all performed in scalar mode; the additional operations to compute the eigenvectors when compz='V' or 'I' can be vectorized and on some machines may be performed much faster.
The complex analogue of this function is nag_lapack_zsteqr (f08js).

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_lapack_dorgtr (f08ff), nag_lapack_dopgtr (f08gf) or nag_lapack_dsbtrd (f08he), which illustrate the use of this function to compute the eigenvalues and eigenvectors of a full or band symmetric matrix.
function f08je_example


fprintf('f08je example results\n\n');

% Symmetric tridiagonal A stored as diagonal and off-diagonal
n = 4;
d = [-6.99;     7.92;     2.34;     0.32];
e = [-0.44;    -2.63;    -1.18];

% All eigenvalues and eigenvectors of A
compz = 'I';
z = zeros(n, n);
[w, ~, z, info] = f08je( ...
                         compz, d, e, 'z', z);

% Normalize eigenvectors: largest element positive
for j = 1:n
  [~,k] = max(abs(z(:,j)));
  if z(k,j) < 0;
    z(:,j) = -z(:,j);
  end
end                            

disp('Eigenvalues');
disp(w');
disp('Eigenvectors');
disp(z);


f08je example results

Eigenvalues
   -7.0037   -0.4059    2.0028    8.9968

Eigenvectors
    0.9995   -0.0109   -0.0167   -0.0255
    0.0310    0.1627    0.3408    0.9254
    0.0089    0.5170    0.7696   -0.3746
    0.0014    0.8403   -0.5397    0.0509


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NAG Toolbox

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