NAG Library Function Document
nag_zgehrd (f08nsc)
1
Purpose
nag_zgehrd (f08nsc) reduces a complex general matrix to Hessenberg form.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_zgehrd (Nag_OrderType order,
Integer n,
Integer ilo,
Integer ihi,
Complex a[],
Integer pda,
Complex tau[],
NagError *fail) |
|
3
Description
nag_zgehrd (f08nsc) reduces a complex general matrix to upper Hessenberg form by a unitary similarity transformation: . has real subdiagonal elements.
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).
The function can take advantage of a previous call to
nag_zgebal (f08nvc), which may produce a matrix with the structure:
where
and
are upper triangular. If so, only the central diagonal block
, in rows and columns
to
, needs to be reduced to Hessenberg form (the blocks
and
will also be affected by the reduction). Therefore the values of
and
determined by
nag_zgebal (f08nvc) can be supplied to the function directly. If
nag_zgebal (f08nvc) has not previously been called however, then
must be set to
and
to
.
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:
– IntegerInput
-
On entry: , the order of the matrix .
Constraint:
.
- 3:
– IntegerInput
- 4:
– IntegerInput
-
On entry: if
has been output by
nag_zgebal (f08nvc),
ilo and
ihi must contain the values returned by that function. Otherwise,
ilo must be set to
and
ihi to
n.
Constraints:
- if , ;
- if , and .
- 5:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
a
must be at least
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by general matrix .
On exit:
a is overwritten by the upper Hessenberg matrix
and details of the unitary matrix
. The subdiagonal elements of
are real.
- 6:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
.
- 7:
– ComplexOutput
-
Note: the dimension,
dim, of the array
tau
must be at least
.
On exit: further details of the unitary matrix .
- 8:
– 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_INT_3
-
On entry, , and .
Constraint: if , ;
if , and .
- 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 Hessenberg 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, eigenvectors or Schur factorization.
8
Parallelism and Performance
nag_zgehrd (f08nsc) 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 , where ; if and , the number is approximately .
To form the unitary matrix
nag_zgehrd (f08nsc) may be followed by a call to
nag_zunghr (f08ntc):
nag_zunghr(order,n,ilo,ihi,&a,pda,tau,&fail)
To apply
to an
by
complex matrix
nag_zgehrd (f08nsc) may be followed by a call to
nag_zunmhr (f08nuc). For example,
nag_zunmhr(order,Nag_LeftSide,Nag_NoTrans,m,n,ilo,ihi,&a,pda,
tau,&c,pdc,&fail)
forms the matrix product .
The real analogue of this function is
nag_dgehrd (f08nec).
10
Example
This example computes the upper Hessenberg form of the matrix
, where
10.1
Program Text
Program Text (f08nsce.c)
10.2
Program Data
Program Data (f08nsce.d)
10.3
Program Results
Program Results (f08nsce.r)