NAG CL Interface
f08bbc (dtpqrt)
1
Purpose
f08bbc computes the factorization of a real by triangular-pentagonal matrix.
2
Specification
void |
f08bbc (Nag_OrderType order,
Integer m,
Integer n,
Integer l,
Integer nb,
double a[],
Integer pda,
double b[],
Integer pdb,
double t[],
Integer pdt,
NagError *fail) |
|
The function may be called by the names: f08bbc, nag_lapackeig_dtpqrt or nag_dtpqrt.
3
Description
f08bbc forms the
factorization of a real
by
triangular-pentagonal matrix
,
where
is an upper triangular
by
matrix and
is an
by
pentagonal matrix consisting of an
by
rectangular matrix
on top of an
by
upper trapezoidal matrix
:
The upper trapezoidal matrix consists of the first rows of an by upper triangular matrix, where . If , is by rectangular; if and , is upper triangular.
A recursive, explicitly blocked,
factorization (see
f08abc) is performed on the matrix
. The upper triangular matrix
, details of the orthogonal matrix
, and further details (the block reflector factors) of
are returned.
Typically the matrix or contains the matrix from the factorization of a subproblem and f08bbc performs the update operation from the inclusion of matrix .
For example, consider the
factorization of an
by
matrix
with
:
,
, where
is
by
upper triangular and
is
by
rectangular (this can be performed by
f08abc). Given an initial least squares problem
where
and
are
by
matrices, we have
.
Now, adding an additional
rows to the original system gives the augmented least squares problem
where
is an
by
matrix formed by adding
rows on top of
and
is an
by
matrix formed by adding
rows on top of
.
f08bbc can then be used to perform the factorization of the pentagonal matrix ; the by matrix will be zero on input and contain on output.
In the case where is by , , is by upper triangular (forming ) on top of rows of zeros (forming first rows of ). Augmentation is then performed by adding rows to the bottom of with .
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_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:
– Integer
Input
-
On entry: , the number of rows of the matrix .
Constraint:
.
-
3:
– Integer
Input
-
On entry: , the number of columns of the matrix and the order of the upper triangular matrix .
Constraint:
.
-
4:
– Integer
Input
-
On entry: , the number of rows of the trapezoidal part of (i.e., ).
Constraint:
.
-
5:
– Integer
Input
-
On entry: the explicitly chosen block-size to be used in the algorithm for computing the
factorization. See
Section 9 for details.
-
6:
– double
Input/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 upper triangular matrix .
On exit: the upper triangle is overwritten by the corresponding elements of the by upper triangular matrix .
-
7:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
.
-
8:
– double
Input/Output
-
Note: the dimension,
dim, of the array
b
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by pentagonal matrix composed of an by rectangular matrix above an by upper trapezoidal matrix .
On exit: details of the orthogonal matrix .
-
9:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
-
10:
– double
Output
-
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 orthogonal 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
.
-
11:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
t.
Constraints:
- if ,
;
- if , .
-
12:
– 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: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: and
if , .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INT_3
-
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 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 factorization is the exact factorization of a nearby matrix
, where
and
is the
machine precision.
8
Parallelism and Performance
f08bbc 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 floating-point operations is approximately if or if .
The block size,
nb, used by
f08bbc 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 apply
to an arbitrary real rectangular matrix
,
f08bbc may be followed by a call to
f08bcc. For example,
nag_lapackeig_dtpmqrt(Nag_ColMajor,Nag_LeftSide,Nag_Trans,m,p,n,l,nb,b,pdb,
t,pdt,c,pdc,&c(n+1,1),ldc,&fail)
forms
, where
is
by
.
To form the orthogonal matrix
explicitly set
, initialize
to the identity matrix and make a call to
f08bcc as above.
10
Example
This example finds the basic solutions for the linear least squares problems
where
and
are the columns of the matrix
,
A
factorization is performed on the first
rows of
using
f08abc after which the first
rows of
are updated by applying
using
f08acc. The remaining row is added by performing a
update using
f08bbc;
is updated by applying the new
using
f08bcc; the solution is finally obtained by triangular solve using
from the updated
.
10.1
Program Text
10.2
Program Data
10.3
Program Results