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NAG Toolbox: nag_lapack_dsygvx (f08sb)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dsygvx (f08sb) computes selected eigenvalues and, optionally, 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. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Syntax

[a, b, m, w, z, jfail, info] = f08sb(itype, jobz, range, uplo, a, b, vl, vu, il, iu, abstol, 'n', n)
[a, b, m, w, z, jfail, info] = nag_lapack_dsygvx(itype, jobz, range, uplo, a, b, vl, vu, il, iu, abstol, 'n', n)

Description

nag_lapack_dsygvx (f08sb) 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 desired eigenvalues and 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
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:     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:     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'.
4:     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'.
5:     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.
6:     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 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.
7:     vl – double scalar
8:     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.
9:     il int64int32nag_int scalar
10:   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 .
11:   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 T1  will be used in its place, where T is the tridiagonal matrix obtained by reducing C to tridiagonal form. 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=1ton, 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 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.
The lower triangle (if uplo='L') or the upper triangle (if uplo='U') 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.
The triangular factor U or L from the Cholesky factorization B=UTU or B=LLT.
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. The eigenvectors are normalized as follows:
    • if itype=1 or 2, ZTBZ=I;
    • if itype=3, ZTB-1Z=I;
  • if an eigenvector fails to converge (info=1ton), 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=1ton, 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

   info=-i
If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: itype, 2: jobz, 3: range, 4: uplo, 5: n, 6: a, 7: lda, 8: b, 9: ldb, 10: vl, 11: vu, 12: il, 13: iu, 14: abstol, 15: m, 16: w, 17: z, 18: ldz, 19: work, 20: lwork, 21: iwork, 22: jfail, 23: 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_dsyevx (f08fb) failed to converge; i eigenvectors failed to converge. Their indices are stored in array jfail.
   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.

Further Comments

The total number of floating-point operations is proportional to n3.
The complex analogue of this function is nag_lapack_zhegvx (f08sp).

Example

This example finds the eigenvalues in the half-open interval -1.0,1.0, and corresponding 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 .  
The example program for nag_lapack_dsygvd (f08sc) illustrates solving a generalized symmetric eigenproblem of the form ABz=λz.
function f08sb_example


fprintf('f08sb 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
% Selecting eigenvalues in the range [-1,1]
itype = int64(1);
jobz  = 'Vectors';
range = 'Values in range';
[vl,vu] = deal(-1, 1);
[il,iu] = deal(int64(0), int64(0));
abstol = 0;
[~, ~, m, w, z, jfail, info] = ...
  f08sb( ...
	 itype, jobz, range, uplo, a, b, 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('Selected Eigenvalues');
disp(w(1:m)');
disp('Corresponding eigenvectors');
disp(z);


f08sb example results

Selected Eigenvalues
   -0.4548    0.1001

Corresponding eigenvectors
   -0.3080   -0.4469
   -0.5329   -0.0371
    0.3496    0.0505
    0.6211    0.4743


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