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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dsyevd (f08fc)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dsyevd (f08fc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric matrix. If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the QL or QR algorithm.

Syntax

[a, w, info] = f08fc(job, uplo, a, 'n', n)
[a, w, info] = nag_lapack_dsyevd(job, uplo, a, 'n', n)

Description

nag_lapack_dsyevd (f08fc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric matrix A. In other words, it can compute the spectral factorization of A as
A=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
Azi=λizi,  i=1,2,,n.  

References

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

Parameters

Compulsory Input Parameters

1:     job – string (length ≥ 1)
Indicates whether eigenvectors are computed.
job='N'
Only eigenvalues are computed.
job='V'
Eigenvalues and eigenvectors are computed.
Constraint: job='N' or 'V'.
2:     uplo – string (length ≥ 1)
Indicates whether the upper or lower triangular part of A is stored.
uplo='U'
The upper triangular part of A is stored.
uplo='L'
The lower triangular part of A is stored.
Constraint: uplo='U' or 'L'.
3:     alda: – double array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
The n by n symmetric matrix A.
  • If uplo='U', the upper triangular part of a must be stored and the elements of the array below the diagonal are not referenced.
  • If uplo='L', the lower triangular part of a must be stored and the elements of the array above the diagonal are not referenced.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a.
n, the order of the matrix A.
Constraint: n0.

Output Parameters

1:     alda: – double array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,n.
If job='V', a stores the orthogonal matrix Z which contains the eigenvectors of A.
2:     w: – double array
The dimension of the array w will be max1,n
The eigenvalues of the matrix A in ascending order.
3:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   info=-i
If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: job, 2: uplo, 3: n, 4: a, 5: lda, 6: w, 7: work, 8: lwork, 9: iwork, 10: liwork, 11: 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.
   info>0
if info=i and job='N', the algorithm failed to converge; i elements of an intermediate tridiagonal form did not converge to zero; if info=i and job='V', then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column i/n+1 through i mod n+1.

Accuracy

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 complex analogue of this function is nag_lapack_zheevd (f08fq).

Example

This example computes all the eigenvalues and eigenvectors of the symmetric matrix A, where
A = 1.0 2.0 3.0 4.0 2.0 2.0 3.0 4.0 3.0 3.0 3.0 4.0 4.0 4.0 4.0 4.0 .  
function f08fc_example


fprintf('f08fc example results\n\n');

job = 'V';
uplo = 'L';
a = [1, 0, 0, 0;
     2, 2, 0, 0;
     3, 3, 3, 0;
     4, 4, 4, 4];

[z, w, info] = f08fc( ...
		      job, uplo, a);

disp('Eigenvalues');
disp(w');

[ifail] = x04ca( ...
		 'General', ' ', z, 'Eigenvectors');


f08fc example results

Eigenvalues
   -2.0531   -0.5146   -0.2943   12.8621

 Eigenvectors
          1       2       3       4
 1  -0.7003 -0.5144 -0.2767 -0.4103
 2  -0.3592  0.4851  0.6634 -0.4422
 3   0.1569  0.5420 -0.6504 -0.5085
 4   0.5965 -0.4543  0.2457 -0.6144

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

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