NAG CL Interface
f08gsc (zhptrd)
1
Purpose
f08gsc reduces a complex Hermitian matrix to tridiagonal form, using packed storage.
2
Specification
void |
f08gsc (Nag_OrderType order,
Nag_UploType uplo,
Integer n,
Complex ap[],
double d[],
double e[],
Complex tau[],
NagError *fail) |
|
The function may be called by the names: f08gsc, nag_lapackeig_zhptrd or nag_zhptrd.
3
Description
f08gsc reduces a complex Hermitian matrix , held in packed storage, to real symmetric tridiagonal form by a unitary similarity transformation: .
The matrix
is not formed explicitly but is represented as a product of
elementary reflectors (see the
F08 Chapter Introduction for details). Functions are provided to work with
in this representation (see
Section 9).
4
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments
-
1:
– Nag_OrderType
Input
-
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
-
2:
– Nag_UploType
Input
-
On entry: indicates whether the upper or lower triangular part of
is stored.
- The upper triangular part of is stored.
- The lower triangular part of is stored.
Constraint:
or .
-
3:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
4:
– Complex
Input/Output
-
Note: the dimension,
dim, of the array
ap
must be at least
.
On entry: the upper or lower triangle of the
by
Hermitian matrix
, packed by rows or columns.
The storage of elements
depends on the
order and
uplo arguments as follows:
- if and ,
- is stored in , for ;
- if and ,
- is stored in , for ;
- if and ,
- is stored in , for ;
- if and ,
- is stored in , for .
On exit:
ap is overwritten by the tridiagonal matrix
and details of the unitary matrix
.
-
5:
– double
Output
-
On exit: the diagonal elements of the tridiagonal matrix .
-
6:
– double
Output
-
On exit: the off-diagonal elements of the tridiagonal matrix .
-
7:
– Complex
Output
-
On exit: further details of the unitary matrix .
-
8:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
7
Accuracy
The computed tridiagonal matrix
is exactly similar to a nearby matrix
, where
is a modestly increasing function of
, and
is the
machine precision.
The elements of themselves may be sensitive to small perturbations in or to rounding errors in the computation, but this does not affect the stability of the eigenvalues and eigenvectors.
8
Parallelism and Performance
f08gsc 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 real floating-point operations is approximately .
To form the unitary matrix
f08gsc may be followed by a call to
f08gtc:
nag_lapackeig_zupgtr(order,uplo,n,ap,tau,&q,pdq,&fail)
To apply
to an
by
complex matrix
f08gsc may be followed by a call to
f08guc. For example,
nag_lapackeig_zupmtr(order,Nag_LeftSide,uplo,Nag_NoTrans,n,p,ap,tau,&c,
pdc,&fail)
forms the matrix product
.
The real analogue of this function is
f08gec.
10
Example
This example reduces the matrix
to tridiagonal form, where
using packed storage.
10.1
Program Text
10.2
Program Data
10.3
Program Results