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NAG Toolbox

NAG Toolbox: nag_lapack_dspgvd (f08tc)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dspgvd (f08tc) computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form
Az=λBz ,   ABz=λz   or   BAz=λz ,  
where A and B are symmetric, stored in packed format, and B is also positive definite. If eigenvectors are desired, it uses a divide-and-conquer algorithm.

Syntax

[ap, bp, w, z, info] = f08tc(itype, jobz, uplo, n, ap, bp)
[ap, bp, w, z, info] = nag_lapack_dspgvd(itype, jobz, uplo, n, ap, bp)

Description

nag_lapack_dspgvd (f08tc) first performs a Cholesky factorization of the matrix B as B=UTU , when uplo='U' or B=LLT , when uplo='L'. The generalized problem is then reduced to a standard symmetric eigenvalue problem
Cx=λx ,  
which is solved for the eigenvalues and, optionally, the eigenvectors; the eigenvectors are then backtransformed to give the eigenvectors of the original problem.
For the problem Az=λBz , the eigenvectors are normalized so that the matrix of eigenvectors, z, satisfies
ZT A Z = Λ   and   ZT B Z = I ,  
where Λ  is the diagonal matrix whose diagonal elements are the eigenvalues. For the problem A B z = λ z  we correspondingly have
Z-1 A Z-T = Λ   and   ZT B Z = I ,  
and for B A z = λ z  we have
ZT A Z = Λ   and   ZT B-1 Z = I .  

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:     itype int64int32nag_int scalar
Specifies the problem type to be solved.
itype=1
Az=λBz.
itype=2
ABz=λz.
itype=3
BAz=λz.
Constraint: itype=1, 2 or 3.
2:     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'.
3:     uplo – string (length ≥ 1)
If uplo='U', the upper triangles of A and B are stored.
If uplo='L', the lower triangles of A and B are stored.
Constraint: uplo='U' or 'L'.
4:     n int64int32nag_int scalar
n, the order of the matrices A and B.
Constraint: n0.
5:     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.
6:     bp: – double array
The dimension of the array bp must be at least max1,n×n+1/2
The upper or lower triangle of the n by n symmetric matrix B, packed by columns.
More precisely,
  • if uplo='U', the upper triangle of B must be stored with element Bij in bpi+jj-1/2 for ij;
  • if uplo='L', the lower triangle of B must be stored with element Bij in bpi+2n-jj-1/2 for ij.

Optional Input Parameters

None.

Output Parameters

1:     ap: – double array
The dimension of the array ap will be max1,n×n+1/2
The contents of ap are destroyed.
2:     bp: – double array
The dimension of the array bp will be max1,n×n+1/2
The triangular factor U or L from the Cholesky factorization B=UTU or B=LLT, in the same storage format as B.
3:     wn – double array
The eigenvalues in ascending order.
4:     zldz: – double array
The first dimension of the array z will be max1,n.
The second dimension of the array z will be max1,n if jobz='V' and 1 otherwise.
If jobz='V', z contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows:
  • if itype=1 or 2, ZTBZ=I;
  • if itype=3, ZTB-1Z=I.
If jobz='N', z is not referenced.
5:     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: itype, 2: jobz, 3: uplo, 4: n, 5: ap, 6: bp, 7: w, 8: z, 9: ldz, 10: work, 11: lwork, 12: iwork, 13: liwork, 14: 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
nag_lapack_dpptrf (f07gd) or nag_lapack_dspevd (f08gc) returned an error code:
n if info=i, nag_lapack_dspevd (f08gc) failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero;
>n if info=n+i, for 1in, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.

Accuracy

If B is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of B differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of B would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds.
The example program below illustrates the computation of approximate error bounds.

Further Comments

The total number of floating-point operations is proportional to n3 .
The complex analogue of this function is nag_lapack_zhpgvd (f08tq).

Example

This example finds all the eigenvalues and eigenvectors of the generalized symmetric eigenproblem ABz=λz , where
A = 0.24 0.39 0.42 -0.16 0.39 -0.11 0.79 0.63 0.42 0.79 -0.25 0.48 -0.16 0.63 0.48 -0.03   and   B = 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.09 0.56 -0.83 0.76 0.34 -0.10 1.09 0.34 1.18 ,  
together with an estimate of the condition number of B, and approximate error bounds for the computed eigenvalues and eigenvectors.
The example program for nag_lapack_dspgv (f08ta) illustrates solving a generalized symmetric eigenproblem of the form Az = λ Bz .
function f08tc_example


fprintf('f08tc example results\n\n');

% Symmetric matrices A and B stored in packed (Upper) format
n = int64(4);
uplo = 'U';
ap = [0.24;
      0.39; -0.11;
      0.42;  0.79; -0.25;
     -0.16;  0.63;  0.48; -0.03];
bp = [4.16;
     -3.12;  5.03;
      0.56; -0.83;  0.76;
      -0.10;  1.09;  0.34;  1.18];

% Generalized eigenvalues for ABz = lambda z
itype = int64(2);
jobz = 'No vectors';
[~, L, w, z, info] = f08tc( ...
                            itype, jobz, uplo, n, ap, bp);

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

% Calculate condition number and error estimates
% Unpack ap and bp to calculate 1 norms of A and B
[a, ap, ifail] = f01za( ...
                        'Unpack', uplo, 'N', zeros(n,n), ap);
[b, bp, ifail] = f01za( ...
                        'Unpack', uplo, 'N', zeros(n,n), bp);
anorm = norm(a + a' - diag(diag(a)),1);
bnorm = norm(b + b' - diag(diag(b)),1);

% Estimate condition number
[rcond, info] = f07ug( ...
                       '1', uplo, 'N', n, L);

rcondb = rcond^2;
fprintf('Estimate of reciprocal condition number for B\n%12.1e\n', rcondb);

% Error bounds
t1 = x02aj/rcondb;
t2 = anorm*bnorm;
errbnd(1:n) = x02aj*t2 + t1.*abs(w);

fprintf('\n');
disp('Error estimate for the eigenvalues');
fprintf('%12.1e',errbnd);
fprintf('\n');


f08tc example results

Eigenvalues
   -3.5411   -0.3347    0.2983    2.2544

Estimate of reciprocal condition number for B
     5.8e-03

Error estimate for the eigenvalues
     7.0e-14     8.6e-15     7.9e-15     4.6e-14

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

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