NAG Library Function Document
nag_zhetrd (f08fsc)
1
Purpose
nag_zhetrd (f08fsc) reduces a complex Hermitian matrix to tridiagonal form.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_zhetrd (Nag_OrderType order,
Nag_UploType uplo,
Integer n,
Complex a[],
Integer pda,
double d[],
double e[],
Complex tau[],
NagError *fail) |
|
3
Description
nag_zhetrd (f08fsc) reduces a complex Hermitian matrix 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_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_UploTypeInput
-
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:
– IntegerInput
-
On entry: , the order of the matrix .
Constraint:
.
- 4:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
a
must be at least
.
On entry: the
by
Hermitian matrix
.
If , is stored in .
If , is stored in .
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.
On exit:
a is overwritten by the tridiagonal matrix
and details of the unitary matrix
as specified by
uplo.
- 5:
– IntegerInput
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix
in the array
a.
Constraint:
.
- 6:
– doubleOutput
-
Note: the dimension,
dim, of the array
d
must be at least
.
On exit: the diagonal elements of the tridiagonal matrix .
- 7:
– doubleOutput
-
Note: the dimension,
dim, of the array
e
must be at least
.
On exit: the off-diagonal elements of the tridiagonal matrix .
- 8:
– ComplexOutput
-
Note: the dimension,
dim, of the array
tau
must be at least
.
On exit: further details of the unitary matrix .
- 9:
– 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_INT
-
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 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
nag_zhetrd (f08fsc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zhetrd (f08fsc) 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
nag_zhetrd (f08fsc) may be followed by a call to
nag_zungtr (f08ftc):
nag_zungtr(order,uplo,n,&a,pda,tau,&fail)
To apply
to an
by
complex matrix
nag_zhetrd (f08fsc) may be followed by a call to
nag_zunmtr (f08fuc). For example,
nag_zunmtr(order,Nag_LeftSide,uplo,Nag_NoTrans,n,p,&a,pda,
tau,&c,pdc,&fail)
forms the matrix product
.
The real analogue of this function is
nag_dsytrd (f08fec).
10
Example
This example reduces the matrix
to tridiagonal form, where
10.1
Program Text
Program Text (f08fsce.c)
10.2
Program Data
Program Data (f08fsce.d)
10.3
Program Results
Program Results (f08fsce.r)