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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dstevx (f08jb)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dstevx (f08jb) computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Syntax

[d, e, m, w, z, jfail, info] = f08jb(jobz, range, d, e, vl, vu, il, iu, abstol, 'n', n)
[d, e, m, w, z, jfail, info] = nag_lapack_dstevx(jobz, range, d, e, vl, vu, il, iu, abstol, 'n', n)

Description

nag_lapack_dstevx (f08jb) computes the required eigenvalues and eigenvectors of A by reducing the tridiagonal matrix to diagonal form using the QR algorithm. Bisection is used to determine selected eigenvalues.

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
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1:     jobz – string (length ≥ 1)
Indicates whether eigenvectors are computed.
jobz='N'
Only eigenvalues are computed.
jobz='V'
Eigenvalues and eigenvectors are computed.
Constraint: jobz='N' or 'V'.
2:     range – string (length ≥ 1)
If range='A', all eigenvalues will be found.
If range='V', all eigenvalues in the half-open interval vl,vu will be found.
If range='I', the ilth to iuth eigenvalues will be found.
Constraint: range='A', 'V' or 'I'.
3:     d: – double array
The dimension of the array d must be at least max1,n
The n diagonal elements of the tridiagonal matrix A.
4:     e: – double array
The dimension of the array e must be at least max1,n-1
The n-1 subdiagonal elements of the tridiagonal matrix A.
5:     vl – double scalar
6:     vu – double scalar
If range='V', the lower and upper bounds of the interval to be searched for eigenvalues.
If range='A' or 'I', vl and vu are not referenced.
Constraint: if range='V', vl<vu.
7:     il int64int32nag_int scalar
8:     iu int64int32nag_int scalar
If range='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If range='A' or 'V', il and iu are not referenced.
Constraints:
  • if range='I' and n=0, il=1 and iu=0;
  • if range='I' and n>0, 1 il iu n .
9:     abstol – double scalar
The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval a,b  of width less than or equal to
abstol+ε maxa,b ,  
where ε  is the machine precision. If abstol is less than or equal to zero, then ε A1  will be used in its place. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold 2 × x02am   , not zero. If this function returns with info>0, indicating that some eigenvectors did not converge, try setting abstol to 2 × x02am   . See Demmel and Kahan (1990).

Optional Input Parameters

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

Output Parameters

1:     d: – double array
The dimension of the array d will be max1,n
May be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
2:     e: – double array
The dimension of the array e will be max1,n-1
May be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.
3:     m int64int32nag_int scalar
The total number of eigenvalues found. 0mn.
If range='A', m=n.
If range='I', m=iu-il+1.
4:     wn – double array
The first m elements contain the selected eigenvalues in ascending order.
5:     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,m if jobz='V' and 1 otherwise.
If jobz='V', then
  • if info=0, the first m columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the ith column of Z holding the eigenvector associated with wi;
  • if an eigenvector fails to converge (info>0), then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
If jobz='N', z is not referenced.
6:     jfail: int64int32nag_int array
The dimension of the array jfail will be max1,n
If jobz='V', then
  • if info=0, the first m elements of jfail are zero;
  • if info>0, jfail contains the indices of the eigenvectors that failed to converge.
If jobz='N', jfail is not referenced.
7:     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<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  info>0
The algorithm failed to converge; _ eigenvectors did not converge. Their indices are stored in array jfail.

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 total number of floating-point operations is proportional to n2 if jobz='N' and is proportional to n3 if jobz='V' and range='A', otherwise the number of floating-point operations will depend upon the number of computed eigenvectors.

Example

This example finds the eigenvalues in the half-open interval 0,5 , and the corresponding eigenvectors, of the symmetric tridiagonal matrix
A = 1 1 0 0 1 4 2 0 0 2 9 3 0 0 3 16 .  
function f08jb_example


fprintf('f08jb example results\n\n');

% Symmetric tridiagonal A stored as diagonal and off-diagonal
d = [1;     4;     9;     16];
e = [1;     2;     3];

% Eigenvalues in range [0,5] and corresponding eigenvectors
jobz  = 'Vectors';
range = 'Values in range';
vl = 0;
vu = 5;
il = int64(0);
iu = int64(0);
abstol = 0;
[~, ~, m, w, z, jfail, info] = ...
f08jb( ...
       jobz, range, d, e, vl, vu, il, iu, abstol);

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

disp(' Eigenvalues of A in range:');
disp(w(1:m));
disp(' Corresponding eigenvectors:');
disp(z);


f08jb example results

 Eigenvalues of A in range:
    0.6476
    3.5470

 Corresponding eigenvectors:
    0.9396    0.3388
   -0.3311    0.8628
    0.0853   -0.3648
   -0.0167    0.0879


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