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NAG Toolbox

NAG Toolbox: nag_lapack_zhpgst (f08ts)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zhpgst (f08ts) reduces a complex Hermitian-definite generalized eigenproblem Az=λBz, ABz=λz or BAz=λz to the standard form Cy=λy, where A is a complex Hermitian matrix and B has been factorized by nag_lapack_zpptrf (f07gr), using packed storage.

Syntax

[ap, info] = f08ts(itype, uplo, n, ap, bp)
[ap, info] = nag_lapack_zhpgst(itype, uplo, n, ap, bp)

Description

To reduce the complex Hermitian-definite generalized eigenproblem Az=λBz, ABz=λz or BAz=λz to the standard form Cy=λy using packed storage, nag_lapack_zhpgst (f08ts) must be preceded by a call to nag_lapack_zpptrf (f07gr) which computes the Cholesky factorization of B; B must be positive definite.
The different problem types are specified by the argument itype, as indicated in the table below. The table shows how C is computed by the function, and also how the eigenvectors z of the original problem can be recovered from the eigenvectors of the standard form.
itype Problem uplo B C z
1 Az=λBz 'U'
'L'
UHU 
LLH
U-HAU-1 
L-1AL-H
U-1y 
L-Hy
2 ABz=λz 'U'
'L'
UHU 
LLH
UAUH 
LHAL
U-1y 
L-Hy
3 BAz=λz 'U'
'L'
UHU 
LLH
UAUH 
LHAL
UHy 
Ly

References

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
Indicates how the standard form is computed.
itype=1
  • if uplo='U', C=U-HAU-1;
  • if uplo='L', C=L-1AL-H.
itype=2 or 3
  • if uplo='U', C=UAUH;
  • if uplo='L', C=LHAL.
Constraint: itype=1, 2 or 3.
2:     uplo – string (length ≥ 1)
Indicates whether the upper or lower triangular part of A is stored and how B has been factorized.
uplo='U'
The upper triangular part of A is stored and B=UHU.
uplo='L'
The lower triangular part of A is stored and B=LLH.
Constraint: uplo='U' or 'L'.
3:     n int64int32nag_int scalar
n, the order of the matrices A and B.
Constraint: n0.
4:     ap: – complex 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 Hermitian 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.
5:     bp: – complex array
The dimension of the array bp must be at least max1,n×n+1/2
The Cholesky factor of B as specified by uplo and returned by nag_lapack_zpptrf (f07gr).

Optional Input Parameters

None.

Output Parameters

1:     ap: – complex array
The dimension of the array ap will be max1,n×n+1/2
The upper or lower triangle of ap stores the corresponding upper or lower triangle of C as specified by itype and uplo, using the same packed storage format as described above.
2:     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: uplo, 3: n, 4: ap, 5: bp, 6: info.

Accuracy

Forming the reduced matrix C is a stable procedure. However it involves implicit multiplication by B-1 if (itype=1) or B (if itype=2 or 3). When nag_lapack_zhpgst (f08ts) is used as a step in the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of accuracy if B is ill-conditioned with respect to inversion. See the document for nag_lapack_zhegv (f08sn) for further details.

Further Comments

The total number of real floating-point operations is approximately 4n3.
The real analogue of this function is nag_lapack_dspgst (f08te).

Example

This example computes all the eigenvalues of Az=λBz, where
A = -7.36+0.00i 0.77-0.43i -0.64-0.92i 3.01-6.97i 0.77+0.43i 3.49+0.00i 2.19+4.45i 1.90+3.73i -0.64+0.92i 2.19-4.45i 0.12+0.00i 2.88-3.17i 3.01+6.97i 1.90-3.73i 2.88+3.17i -2.54+0.00i  
and
B = 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i ,  
using packed storage. Here B is Hermitian positive definite and must first be factorized by nag_lapack_zpptrf (f07gr). The program calls nag_lapack_zhpgst (f08ts) to reduce the problem to the standard form Cy=λy; then nag_lapack_zhptrd (f08gs) to reduce C to tridiagonal form, and nag_lapack_dsterf (f08jf) to compute the eigenvalues.
function f08ts_example


fprintf('f08ts example results\n\n');

% Hermitian matrices A and B stored in packed (Lower) format
uplo = 'L';
n = int64(4);
ap = [-7.36;  0.77 + 0.43i; -0.64 + 0.92i;  3.01 + 6.97i;
              3.49 + 0i;     2.19 - 4.45i;  1.90 - 3.73i;
                             0.12 + 0i;     2.88 + 3.17i;
                                           -2.54 + 0i];
bp = [ 3.23;  1.51 + 1.92i;  1.90 - 0.84i;  0.42 - 2.50i; 
              3.58 + 0i;    -0.23 - 1.11i; -1.18 - 1.37i;
                             4.09 - 0i;     2.33 + 0.14i;
                                            4.29 - 0i];

% Compute Cholesky factorization of B
[lp, info] = f07gr( ...
		    uplo, n, bp);

% Reduce problem to standard form Cy = lambda y,
% where y = L^H x, C = L^-1 A L^-H
itype = int64(1);
[cp, info] = f08ts( ...
		   itype, uplo, n, ap, lp);

% Reduce C to tridiagonal form T = Q^H C Q
[qp, d, e, tau, info] = f08gs( ...
			       uplo, n, cp);

% Calculate eigenvalues of T (same as C)
[w, ~, info] = f08jf(d, e);

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


f08ts example results

Eigenvalues
   -5.9990   -2.9936    0.5047    3.9990


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