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NAG Toolbox: nag_lapack_zheevx (f08fp)
Purpose
nag_lapack_zheevx (f08fp) computes selected eigenvalues and, optionally, eigenvectors of a complex by Hermitian matrix . Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
Syntax
[
a,
m,
w,
z,
jfail,
info] = f08fp(
jobz,
range,
uplo,
a,
vl,
vu,
il,
iu,
abstol, 'n',
n)
[
a,
m,
w,
z,
jfail,
info] = nag_lapack_zheevx(
jobz,
range,
uplo,
a,
vl,
vu,
il,
iu,
abstol, 'n',
n)
Description
The Hermitian matrix is first reduced to real tridiagonal form, using unitary similarity transformations. The required eigenvalues and eigenvectors are then computed from the tridiagonal matrix; the method used depends upon whether all, or selected, eigenvalues and eigenvectors are required.
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:
– string (length ≥ 1)
-
Indicates whether eigenvectors are computed.
- Only eigenvalues are computed.
- Eigenvalues and eigenvectors are computed.
Constraint:
or .
- 2:
– string (length ≥ 1)
-
If
, all eigenvalues will be found.
If , all eigenvalues in the half-open interval will be found.
If
, the
ilth to
iuth eigenvalues will be found.
Constraint:
, or .
- 3:
– string (length ≥ 1)
-
If
, the upper triangular part of
is stored.
If , the lower triangular part of is stored.
Constraint:
or .
- 4:
– 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.
- 5:
– double scalar
- 6:
– double scalar
-
If
, the lower and upper bounds of the interval to be searched for eigenvalues.
If
or
,
vl and
vu are not referenced.
Constraint:
if , .
- 7:
– int64int32nag_int scalar
- 8:
– int64int32nag_int scalar
-
If
, the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If
or
,
il and
iu are not referenced.
Constraints:
- if and , and ;
- if and , .
- 9:
– 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
of width less than or equal to
where
is the
machine precision. If
abstol is less than or equal to zero, then
will be used in its place, where
is the tridiagonal matrix obtained by reducing
to tridiagonal form. Eigenvalues will be computed most accurately when
abstol is set to twice the underflow threshold
, not zero. If this function returns with
, indicating that some eigenvectors did not converge, try setting
abstol to
. See
Demmel and Kahan (1990).
Optional Input Parameters
- 1:
– int64int32nag_int scalar
-
Default:
the first dimension of the array
a and the second dimension of the array
a. (An error is raised if these dimensions are not equal.)
, the order of the matrix .
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 lower triangle (if
) or the upper triangle (if
) of
a, including the diagonal, is overwritten.
- 2:
– int64int32nag_int scalar
-
The total number of eigenvalues found.
.
If , .
If , .
- 3:
– double array
-
The dimension of the array
w will be
The first
m elements contain the selected eigenvalues in ascending order.
- 4:
– complex array
-
The first dimension,
, of the array
z will be
- if , ;
- otherwise .
The second dimension of the array
z will be
if
and
otherwise.
If
, then
- if , the first m columns of contain the orthonormal eigenvectors of the matrix corresponding to the selected eigenvalues, with the th column of holding the eigenvector associated with ;
- if an eigenvector fails to converge (), then that column of contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
If
,
z is not referenced.
- 5:
– int64int32nag_int array
-
The dimension of the array
jfail will be
If
, then
- if , the first m elements of jfail are zero;
- if , jfail contains the indices of the eigenvectors that failed to converge.
If
,
jfail is not referenced.
- 6:
– int64int32nag_int scalar
unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
Cases prefixed with W are classified as warnings and
do not generate an error of type NAG:error_n. See nag_issue_warnings.
-
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
- W
-
The algorithm failed to converge;
eigenvectors did not converge. Their indices are stored in array
jfail.
Accuracy
The computed eigenvalues and eigenvectors are exact for a nearby matrix
, where
and
is the
machine precision. See Section 4.7 of
Anderson et al. (1999) for further details.
Further Comments
The total number of floating-point operations is proportional to .
The real analogue of this function is
nag_lapack_dsyevx (f08fb).
Example
This example finds the eigenvalues in the half-open interval
, and the corresponding eigenvectors, of the Hermitian matrix
Open in the MATLAB editor:
f08fp_example
function f08fp_example
fprintf('f08fp example results\n\n');
a = [ 1 + 0i, 2 - 1i, 3 - 1i, 4 - 1i;
0 + 0i, 2 + 0i, 3 - 2i, 4 - 2i;
0 + 0i, 0 + 0i, 3 + 0i, 4 - 3i;
0 + 0i, 0 + 0i, 0 + 0i, 4 + 0i];
jobz = 'Vectors';
range = 'Values in range';
uplo = 'Upper';
vl = -2;
vu = 2;
il = int64(0);
iu = int64(0);
abstol = 0;
[~, m, w, z, jfail, info] = ...
f08fp(...
jobz, range, uplo, a, vl, vu, il, iu, abstol);
for i = 1:m
[~,k] = max(abs(real(z(:,i)))+abs(imag(z(:,i))));
z(:,i) = z(:,i)*conj(z(k,i))/abs(z(k,i));
end
fprintf('Number of eigenvalues in [-2,2] is %2d\n',m);
fprintf('\n Eigenvalues are:\n');
disp(w(1:m));
ncols = int64(80);
indent = int64(0);
[ifail] = x04db( ...
'General', ' ', z, 'Bracketed', 'F7.4', ...
'Corresponding eigenvectors', 'Integer', 'Integer', ...
ncols, indent);
f08fp example results
Number of eigenvalues in [-2,2] is 2
Eigenvalues are:
-0.6886
1.1412
Corresponding eigenvectors
1 2
1 ( 0.6470, 0.0000) ( 0.0179,-0.4453)
2 (-0.4984,-0.1130) ( 0.5706, 0.0000)
3 ( 0.2949, 0.3165) (-0.1530, 0.5273)
4 (-0.2241,-0.2878) (-0.2118,-0.3598)
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