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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dspev (f08ga)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_dspev (f08ga) computes all the eigenvalues and, optionally, all the eigenvectors of a real n by n symmetric matrix A in packed storage.


[ap, w, z, info] = f08ga(jobz, uplo, n, ap)
[ap, w, z, info] = nag_lapack_dspev(jobz, uplo, n, ap)


The symmetric matrix A is first reduced to tridiagonal form, using orthogonal similarity transformations, and then the QR algorithm is applied to the tridiagonal matrix to compute the eigenvalues and (optionally) the eigenvectors.


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


Compulsory Input Parameters

1:     jobz – string (length ≥ 1)
Indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
Constraint: jobz='N' or 'V'.
2:     uplo – string (length ≥ 1)
If uplo='U', the upper triangular part of A is stored.
If uplo='L', the lower triangular part of A is stored.
Constraint: uplo='U' or 'L'.
3:     n int64int32nag_int scalar
n, the order of the matrix A.
Constraint: n0.
4:     ap: – double array
The dimension of the array ap must be at least max1,n×n+1/2
The upper or lower triangle of the n by n symmetric matrix A, packed by columns.
More precisely,
  • if uplo='U', the upper triangle of A must be stored with element Aij in api+jj-1/2 for ij;
  • if uplo='L', the lower triangle of A must be stored with element Aij in api+2n-jj-1/2 for ij.

Optional Input Parameters


Output Parameters

1:     ap: – double array
The dimension of the array ap will be max1,n×n+1/2
ap stores the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements of A.
2:     wn – double array
The eigenvalues in ascending order.
3:     zldz: – double array
The first dimension, ldz, of the array z will be
  • if jobz='V', ldz= max1,n ;
  • otherwise ldz=1.
The second dimension of the array z will be max1,n if jobz='V' and 1 otherwise.
If jobz='V', z contains the orthonormal eigenvectors of the matrix A, with the ith column of Z holding the eigenvector associated with wi.
If jobz='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

If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: jobz, 2: uplo, 3: n, 4: ap, 5: w, 6: z, 7: ldz, 8: work, 9: 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.
If info=i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.


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.

Further Comments

The total number of floating-point operations is proportional to n3.
The complex analogue of this function is nag_lapack_zhpev (f08gn).


This example finds all the eigenvalues of the symmetric matrix
A = 1 2 3 4 2 2 3 4 3 3 3 4 4 4 4 4 ,  
together with approximate error bounds for the computed eigenvalues.
function f08ga_example

fprintf('f08ga example results\n\n');

% A is symmetric matrix stored in symmetric (Upper) packed format
uplo = 'U';
n = int64(4);
ap = [1;
      2;   2;
      3;   3;   3;
      4;   4;   4;   4];

% Eigenvalues of A only
jobz = 'No vectors';
[apf, w, ~, info] = f08ga( ...
                           jobz, uplo, n, ap);


% Eigenvalue error bound
errbnd = x02aj*max(abs(w(1)),abs(w(end)));
disp('Error estimate for the eigenvalues');

f08ga example results

   -2.0531   -0.5146   -0.2943   12.8621

Error estimate for the eigenvalues

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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