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NAG Toolbox: nag_lapack_zhegst (f08ss)
Purpose
nag_lapack_zhegst (f08ss) reduces a complex Hermitian-definite generalized eigenproblem
,
or
to the standard form
, where
is a complex Hermitian matrix and
has been factorized by
nag_lapack_zpotrf (f07fr).
Syntax
Description
To reduce the complex Hermitian-definite generalized eigenproblem
,
or
to the standard form
,
nag_lapack_zhegst (f08ss) must be preceded by a call to
nag_lapack_zpotrf (f07fr) which computes the Cholesky factorization of
;
must be positive definite.
The different problem types are specified by the argument
itype, as indicated in the table below. The table shows how
is computed by the function, and also how the eigenvectors
of the original problem can be recovered from the eigenvectors of the standard form.
itype |
Problem |
uplo |
|
|
|
|
|
'U'
'L' |
|
|
|
|
|
'U'
'L' |
|
|
|
|
|
'U'
'L' |
|
|
|
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Parameters
Compulsory Input Parameters
- 1:
– int64int32nag_int scalar
-
Indicates how the standard form is computed.
-
- if , ;
- if , .
- or
-
- if , ;
- if , .
Constraint:
, or .
- 2:
– string (length ≥ 1)
-
Indicates whether the upper or lower triangular part of
is stored and how
has been factorized.
- The upper triangular part of is stored and .
- The lower triangular part of is stored and .
Constraint:
or .
- 3:
– complex array
-
The first dimension of the array
a must be at least
.
The second dimension of the array
a must be at least
.
The
by
Hermitian matrix
.
- If , the upper triangular part of must be stored and the elements of the array below the diagonal are not referenced.
- If , the lower triangular part of must be stored and the elements of the array above the diagonal are not referenced.
- 4:
– complex array
-
The first dimension of the array
b must be at least
.
The second dimension of the array
b must be at least
.
The Cholesky factor of
as specified by
uplo and returned by
nag_lapack_zpotrf (f07fr).
Optional Input Parameters
- 1:
– int64int32nag_int scalar
-
Default:
the first dimension of the arrays
a,
b and the second dimension of the arrays
a,
b.
, the order of the matrices and .
Constraint:
.
Output Parameters
- 1:
– complex array
-
The first dimension of the array
a will be
.
The second dimension of the array
a will be
.
The upper or lower triangle of
a stores the corresponding upper or lower triangle of
as specified by
itype and
uplo.
- 2:
– int64int32nag_int scalar
unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
-
If , parameter had an illegal value on entry. The parameters are numbered as follows:
1:
itype, 2:
uplo, 3:
n, 4:
a, 5:
lda, 6:
b, 7:
ldb, 8:
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.
Accuracy
Forming the reduced matrix
is a stable procedure. However it involves implicit multiplication by
(if
) or
(if
or
). When
nag_lapack_zhegst (f08ss) 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
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 .
The real analogue of this function is
nag_lapack_dsygst (f08se).
Example
This example computes all the eigenvalues of
, where
and
Here
is Hermitian positive definite and must first be factorized by
nag_lapack_zpotrf (f07fr). The program calls
nag_lapack_zhegst (f08ss) to reduce the problem to the standard form
; then
nag_lapack_zhetrd (f08fs) to reduce
to tridiagonal form, and
nag_lapack_dsterf (f08jf) to compute the eigenvalues.
Open in the MATLAB editor:
f08ss_example
function f08ss_example
fprintf('f08ss example results\n\n');
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];
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];
uplo = 'L';
[bfac, info] = f07fr(uplo, b);
itype = int64(1);
[c info] = f08ss(itype, uplo, a, bfac);
jobz = 'No Vectors';
[~, w, info] = f08fn(jobz, uplo, c);
disp('Eigenvalues:');
disp(w');
f08ss example results
Eigenvalues:
-5.9990 -2.9936 0.5047 3.9990
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