NAG Library Function Document
nag_zgebrd (f08ksc)
1
Purpose
nag_zgebrd (f08ksc) reduces a complex by matrix to bidiagonal form.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_zgebrd (Nag_OrderType order,
Integer m,
Integer n,
Complex a[],
Integer pda,
double d[],
double e[],
Complex tauq[],
Complex taup[],
NagError *fail) |
|
3
Description
nag_zgebrd (f08ksc) reduces a complex by matrix to real bidiagonal form by a unitary transformation: , where and are unitary matrices of order and respectively.
If
, the reduction is given by:
where
is a real
by
upper bidiagonal matrix and
consists of the first
columns of
.
If
, the reduction is given by
where
is a real
by
lower bidiagonal matrix and
consists of the first
rows of
.
The unitary matrices
and
are not formed explicitly but are represented as products of elementary reflectors (see the
f08 Chapter Introduction for details). Functions are provided to work with
and
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:
– IntegerInput
-
On entry: , the number of rows of the matrix .
Constraint:
.
- 3:
– IntegerInput
-
On entry: , the number of columns of the matrix .
Constraint:
.
- 4:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
a
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by matrix .
On exit: if
, the diagonal and first superdiagonal are overwritten by the upper bidiagonal matrix
, elements below the diagonal are overwritten by details of the unitary matrix
and elements above the first superdiagonal are overwritten by details of the unitary matrix
.
If , the diagonal and first subdiagonal are overwritten by the lower bidiagonal matrix , elements below the first subdiagonal are overwritten by details of the unitary matrix and elements above the diagonal are overwritten by details of the unitary matrix .
- 5:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraints:
- if ,
;
- if , .
- 6:
– doubleOutput
-
Note: the dimension,
dim, of the array
d
must be at least
.
On exit: the diagonal elements of the bidiagonal matrix .
- 7:
– doubleOutput
-
Note: the dimension,
dim, of the array
e
must be at least
.
On exit: the off-diagonal elements of the bidiagonal matrix .
- 8:
– ComplexOutput
-
Note: the dimension,
dim, of the array
tauq
must be at least
.
On exit: further details of the unitary matrix .
- 9:
– ComplexOutput
-
Note: the dimension,
dim, of the array
taup
must be at least
.
On exit: further details of the unitary matrix .
- 10:
– 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: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
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 bidiagonal form
satisfies
, 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 singular values and vectors.
8
Parallelism and Performance
nag_zgebrd (f08ksc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zgebrd (f08ksc) 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 if or if .
If
, it can be more efficient to first call
nag_zgeqrf (f08asc) to perform a
factorization of
, and then to call
nag_zgebrd (f08ksc) to reduce the factor
to bidiagonal form. This requires approximately
floating-point operations.
If
, it can be more efficient to first call
nag_zgelqf (f08avc) to perform an
factorization of
, and then to call
nag_zgebrd (f08ksc) to reduce the factor
to bidiagonal form. This requires approximately
operations.
To form the unitary matrices
and/or
nag_zgebrd (f08ksc) may be followed by calls to
nag_zungbr (f08ktc):
to form the
by
unitary matrix
nag_zungbr(order,Nag_FormQ,m,m,n,&a,pda,tauq,&fail)
but note that the second dimension of the array
a must be at least
m, which may be larger than was required by
nag_zgebrd (f08ksc);
to form the
by
unitary matrix
nag_zungbr(order,Nag_FormP,n,n,m,&a,pda,taup,&fail)
but note that the first dimension of the array
a, specified by the argument
pda, must be at least
n, which may be larger than was required by
nag_zgebrd (f08ksc).
To apply
or
to a complex rectangular matrix
,
nag_zgebrd (f08ksc) may be followed by a call to
nag_zunmbr (f08kuc).
The real analogue of this function is
nag_dgebrd (f08kec).
10
Example
This example reduces the matrix
to bidiagonal form, where
10.1
Program Text
Program Text (f08ksce.c)
10.2
Program Data
Program Data (f08ksce.d)
10.3
Program Results
Program Results (f08ksce.r)