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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dsbev (f08ha)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_dsbev (f08ha) computes all the eigenvalues and, optionally, all the eigenvectors of a real n by n symmetric band matrix A of bandwidth 2kd+1 .


[ab, w, z, info] = f08ha(jobz, uplo, kd, ab, 'n', n)
[ab, w, z, info] = nag_lapack_dsbev(jobz, uplo, kd, ab, 'n', n)


The symmetric band 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:     kd int64int32nag_int scalar
If uplo='U', the number of superdiagonals, kd, of the matrix A.
If uplo='L', the number of subdiagonals, kd, of the matrix A.
Constraint: kd0.
4:     abldab: – double array
The first dimension of the array ab must be at least kd+1.
The second dimension of the array ab must be at least max1,n.
The upper or lower triangle of the n by n symmetric band matrix A.
The matrix is stored in rows 1 to kd+1, more precisely,
  • if uplo='U', the elements of the upper triangle of A within the band must be stored with element Aij in abkd+1+i-jj​ for ​max1,j-kdij;
  • if uplo='L', the elements of the lower triangle of A within the band must be stored with element Aij in ab1+i-jj​ for ​jiminn,j+kd.

Optional Input Parameters

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

Output Parameters

1:     abldab: – double array
The first dimension of the array ab will be kd+1.
The second dimension of the array ab will be max1,n.
ab stores values generated during the reduction to tridiagonal form.
The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix T are returned in ab using the same storage format as described above.
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: kd, 5: ab, 6: ldab, 7: w, 8: z, 9: ldz, 10: work, 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.
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 if jobz='V' and is proportional to kd n2  otherwise.
The complex analogue of this function is nag_lapack_zhbev (f08hn).


This example finds all the eigenvalues and eigenvectors of the symmetric band matrix
A = 1 2 3 0 0 2 2 3 4 0 3 3 3 4 5 0 4 4 4 5 0 0 5 5 5 ,  
together with approximate error bounds for the computed eigenvalues and eigenvectors.
function f08ha_example

fprintf('f08ha example results\n\n');

% Symmetric band matrix A, stored on symmetric banded format
uplo = 'U';
kd = int64(2);
n  = int64(5);
ab = [0, 0, 3, 4, 5;
      0, 2, 3, 4, 5;
      1, 2, 3, 4, 5];

% Calculate all eigenvalues and eigenvectors
jobz = 'Vectors';
[abf, w, z, info] = f08ha( ...
                           jobz, uplo, kd, ab);

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


% Eigenvalue error bound
errbnd = x02aj*max(abs(w(1)),abs(w(end)));
% Eigenvector condition numbers
[rcondz, info] = f08fl( ...
		        'Eigenvectors', n, n, w);

% Eigenvector error bounds
zerrbd = errbnd./rcondz;

disp('Error estimate for the eigenvalues');
disp('Error estimates for the eigenvectors');

f08ha example results


    0.0394    0.6238    0.5635   -0.5165    0.1582
    0.5721   -0.2575   -0.3896   -0.5955    0.3161
   -0.4372   -0.5900    0.4008   -0.1470    0.5277
   -0.4424    0.4308   -0.5581    0.0470    0.5523
    0.5332    0.1039    0.2421    0.5956    0.5400

Error estimate for the eigenvalues
Error estimates for the eigenvectors
     2.9e-15     2.9e-15     6.9e-16     6.9e-16     1.5e-16

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