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NAG Toolbox

NAG Toolbox: nag_lapack_dsygv (f08sa)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dsygv (f08sa) 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 and B is also positive definite.

Syntax

[a, b, w, info] = f08sa(itype, jobz, uplo, a, b, 'n', n)
[a, b, w, info] = nag_lapack_dsygv(itype, jobz, uplo, a, b, 'n', n)

Description

nag_lapack_dsygv (f08sa) 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:     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.
5:     bldb: – double array
The first dimension of the array b must be at least max1,n.
The second dimension of the array b must be at least max1,n.
The n by n symmetric positive definite matrix B.
  • If uplo='U', the upper triangular part of b must be stored and the elements of the array below the diagonal are not referenced.
  • If uplo='L', the lower triangular part of b 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 arrays a, b and the second dimension of the arrays a, b. (An error is raised if these dimensions are not equal.)
n, the order of the matrices A and B.
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 jobz='V', a 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', the upper triangle (if uplo='U') or the lower triangle (if uplo='L') of a, including the diagonal, is overwritten.
2:     bldb: – double array
The first dimension of the array b will be max1,n.
The second dimension of the array b will be max1,n.
If 0infon, the part of b containing the matrix stores the triangular factor U or L from the Cholesky factorization B=UTU or B=LLT.
3:     wn – double array
The eigenvalues in ascending order.
4:     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: a, 6: lda, 7: b, 8: ldb, 9: w, 10: work, 11: lwork, 12: 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=1ton
If info=i, nag_lapack_dsyev (f08fa) failed to converge; i i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
   info>n
nag_lapack_dpotrf (f07fd) returned an error code; i.e., 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_zhegv (f08sn).

Example

This example finds all the eigenvalues and eigenvectors of the generalized symmetric eigenproblem Az=λBz, 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 and estimate of the condition number of B, and approximate error bounds for the computed eigenvalues and eigenvectors.
The example program for nag_lapack_dsygvd (f08sc) illustrates solving a generalized symmetric eigenproblem of the form ABz=λz.
function f08sa_example


fprintf('f08sa example results\n\n');

% Upper triangular parts of symmetric matrix A and symmetric definite matrix B
uplo = 'Upper';
n = 4;
a = [0.24,  0.39,  0.42, -0.16;
     0,    -0.11,  0.79,  0.63;
     0,     0,    -0.25,  0.48;
     0,     0,     0,    -0.03];
b = [4.16, -3.12,  0.56, -0.10;
     0,     5.03, -0.83,  1.09;
     0,     0,     0.76,  0.34;
     0,     0,     0,     1.18];

% Generalized eigenvalues and eigenvectors for problem Az = lambda Bz
itype = int64(1);
jobz = 'Vectors';
[Z, U, w, info] = f08sa( ...
			 itype, jobz, uplo, a, b);

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

disp('Eigenvalues');
disp(w');
disp('Eigenvectors');
disp(Z);


f08sa example results

Eigenvalues
   -2.2254   -0.4548    0.1001    1.1270

Eigenvectors
    0.0690   -0.3080   -0.4469    0.5528
    0.5740   -0.5329   -0.0371    0.6766
    1.5428    0.3496    0.0505    0.9276
   -1.4004    0.6211    0.4743   -0.2510


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