NAG Library Function Document
nag_zgeqrt (f08apc)
1
Purpose
nag_zgeqrt (f08apc) recursively computes, with explicit blocking, the factorization of a complex by matrix.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_zgeqrt (Nag_OrderType order,
Integer m,
Integer n,
Integer nb,
Complex a[],
Integer pda,
Complex t[],
Integer pdt,
NagError *fail) |
|
3
Description
nag_zgeqrt (f08apc) forms the factorization of an arbitrary rectangular complex by matrix. No pivoting is performed.
It differs from
nag_zgeqrf (f08asc) in that it: requires an explicit block size; stores reflector factors that are upper triangular matrices of the chosen block size (rather than scalars); and recursively computes the
factorization based on the algorithm of
Elmroth and Gustavson (2000).
If
, the factorization is given by:
where
is an
by
upper triangular matrix (with real diagonal elements) and
is an
by
unitary matrix. It is sometimes more convenient to write the factorization as
which reduces to
where
consists of the first
columns of
, and
the remaining
columns.
If
,
is upper trapezoidal, and the factorization can be written
where
is upper triangular and
is rectangular.
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).
Note also that for any , the information returned represents a factorization of the first columns of the original matrix .
4
References
Elmroth E and Gustavson F (2000) Applying Recursion to Serial and Parallel Factorization Leads to Better Performance IBM Journal of Research and Development. (Volume 44) 4 605–624
Golub G H and Van Loan C F (2012) Matrix Computations (4th 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:
– IntegerInput
-
On entry: the explicitly chosen block size to be used in computing the
factorization. See
Section 9 for details.
Constraint:
if , .
- 5:
– 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 elements below the diagonal are overwritten by details of the unitary matrix
and the upper triangle is overwritten by the corresponding elements of the
by
upper triangular matrix
.
If , the strictly lower triangular part is overwritten by details of the unitary matrix and the remaining elements are overwritten by the corresponding elements of the by upper trapezoidal matrix .
The diagonal 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.
Constraints:
- if ,
;
- if , .
- 7:
– ComplexOutput
-
Note: the dimension,
dim, of the array
t
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On exit: further details of the unitary matrix
. The number of blocks is
, where
and each block is of order
nb except for the last block, which is of order
. For each of the blocks, an upper triangular block reflector factor is computed:
. These are stored in the
by
matrix
as
.
- 8:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
t.
Constraints:
- if ,
;
- if , .
- 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: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INT_3
-
On entry, , and .
Constraint: if , .
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 factorization is the exact factorization of a nearby matrix
, where
and
is the
machine precision.
8
Parallelism and Performance
nag_zgeqrt (f08apc) 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 .
To apply
to an arbitrary complex rectangular matrix
,
nag_zgeqrt (f08apc) may be followed by a call to
nag_zgemqrt (f08aqc). For example,
nag_zgemqrt(order,Nag_LeftSide,Nag_ConjTrans,m,p,MIN(m,n),nb,a,pda,
t,pdt,c,pdc,&fail)
forms
, where
is
by
.
To form the unitary matrix
explicitly, simply initialize the
by
matrix
to the identity matrix and form
using
nag_zgemqrt (f08aqc) as above.
The block size,
nb, used by
nag_zgeqrt (f08apc) is supplied explicitly through the interface. For moderate and large sizes of matrix, the block size can have a marked effect on the efficiency of the algorithm with the optimal value being dependent on problem size and platform. A value of
is likely to achieve good efficiency and it is unlikely that an optimal value would exceed
.
To compute a
factorization with column pivoting, use
nag_ztpqrt (f08bpc) or
nag_zgeqpf (f08bsc).
The real analogue of this function is
nag_dgeqrt (f08abc).
10
Example
This example solves the linear least squares problems
where
and
are the columns of the matrix
,
and
10.1
Program Text
Program Text (f08apce.c)
10.2
Program Data
Program Data (f08apce.d)
10.3
Program Results
Program Results (f08apce.r)