NAG Library Function Document
nag_zhbevx (f08hpc)
1
Purpose
nag_zhbevx (f08hpc) computes selected eigenvalues and, optionally, eigenvectors of a complex by Hermitian band matrix of bandwidth . Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_zhbevx (Nag_OrderType order,
Nag_JobType job,
Nag_RangeType range,
Nag_UploType uplo,
Integer n,
Integer kd,
Complex ab[],
Integer pdab,
Complex q[],
Integer pdq,
double vl,
double vu,
Integer il,
Integer iu,
double abstol,
Integer *m,
double w[],
Complex z[],
Integer pdz,
Integer jfail[],
NagError *fail) |
|
3
Description
The Hermitian band 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.
4
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
5
Arguments
- 1:
– Nag_OrderTypeInput
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_JobTypeInput
-
On entry: indicates whether eigenvectors are computed.
- Only eigenvalues are computed.
- Eigenvalues and eigenvectors are computed.
Constraint:
or .
- 3:
– Nag_RangeTypeInput
-
On entry: 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 .
- 4:
– Nag_UploTypeInput
-
On entry: if
, the upper triangular part of
is stored.
If , the lower triangular part of is stored.
Constraint:
or .
- 5:
– IntegerInput
-
On entry: , the order of the matrix .
Constraint:
.
- 6:
– IntegerInput
-
On entry: if
, the number of superdiagonals,
, of the matrix
.
If , the number of subdiagonals, , of the matrix .
Constraint:
.
- 7:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
ab
must be at least
.
On entry: the upper or lower triangle of the
by
Hermitian band matrix
.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of
, depends on the
order and
uplo arguments as follows:
- if and ,
is stored in , for and ; - if and ,
is stored in , for and ; - if and ,
is stored in , for and ; - if and ,
is stored in , for and .
On exit:
ab is overwritten by values generated during the reduction to tridiagonal form.
The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix
are returned in
ab using the same storage format as described above.
- 8:
– IntegerInput
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix
in the array
ab.
Constraint:
.
- 9:
– ComplexOutput
-
Note: the dimension,
dim, of the array
q
must be at least
- when
;
- otherwise.
The
th element of the matrix
is stored in
- when ;
- when .
On exit: if
, the
by
unitary matrix used in the reduction to tridiagonal form.
If
,
q is not referenced.
- 10:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
q.
Constraints:
- if , ;
- otherwise .
- 11:
– doubleInput
- 12:
– doubleInput
-
On entry: if
, the lower and upper bounds of the interval to be searched for eigenvalues.
If
or
,
vl and
vu are not referenced.
Constraint:
if , .
- 13:
– IntegerInput
- 14:
– IntegerInput
-
On entry: 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 , .
- 15:
– doubleInput
-
On entry: 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
NE_CONVERGENCE, indicating that some eigenvectors did not converge, try setting
abstol to
. See
Demmel and Kahan (1990).
- 16:
– Integer *Output
-
On exit: the total number of eigenvalues found.
.
If , .
If , .
- 17:
– doubleOutput
-
On exit: the first
m elements contain the selected eigenvalues in ascending order.
- 18:
– ComplexOutput
-
Note: the dimension,
dim, of the array
z
must be at least
- when
;
- otherwise.
The
th element of the matrix
is stored in
- when ;
- when .
On exit: if
, then
- if NE_NOERROR, 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 ( NE_CONVERGENCE), 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.
- 19:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
z.
Constraints:
- if , ;
- otherwise .
- 20:
– IntegerOutput
-
Note: the dimension,
dim, of the array
jfail
must be at least
.
On exit: if
, then
- if NE_NOERROR, the first m elements of jfail are zero;
- if NE_CONVERGENCE, jfail contains the indices of the eigenvectors that failed to converge.
If
,
jfail is not referenced.
- 21:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_CONVERGENCE
-
The algorithm failed to converge;
eigenvectors did not converge. Their indices are stored in array
jfail.
- NE_ENUM_INT_2
-
On entry, , and .
Constraint: if , ;
otherwise .
On entry, , and .
Constraint: if , ;
otherwise .
- NE_ENUM_INT_3
-
On entry, , , and .
Constraint: if and , and ;
if and , .
- NE_ENUM_REAL_2
-
On entry, , and .
Constraint: if , .
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
7
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.
8
Parallelism and Performance
nag_zhbevx (f08hpc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zhbevx (f08hpc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The total number of floating-point operations is proportional to if
, and is proportional to if and , otherwise the number of floating-point operations will depend upon the number of computed eigenvectors.
The real analogue of this function is
nag_dsbevx (f08hbc).
10
Example
This example finds the eigenvalues in the half-open interval
, and the corresponding eigenvectors, of the Hermitian band matrix
10.1
Program Text
Program Text (f08hpce.c)
10.2
Program Data
Program Data (f08hpce.d)
10.3
Program Results
Program Results (f08hpce.r)