NAG FL Interface
f08abf (dgeqrt)
1
Purpose
f08abf recursively computes, with explicit blocking, the $QR$ factorization of a real $m$ by $n$ matrix.
2
Specification
Fortran Interface
Integer, Intent (In) 
:: 
m, n, nb, lda, ldt 
Integer, Intent (Out) 
:: 
info 
Real (Kind=nag_wp), Intent (Inout) 
:: 
a(lda,*), t(ldt,*) 
Real (Kind=nag_wp), Intent (Out) 
:: 
work(nb*n) 

C Header Interface
#include <nag.h>
void 
f08abf_ (const Integer *m, const Integer *n, const Integer *nb, double a[], const Integer *lda, double t[], const Integer *ldt, double work[], Integer *info) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
f08abf_ (const Integer &m, const Integer &n, const Integer &nb, double a[], const Integer &lda, double t[], const Integer &ldt, double work[], Integer &info) 
}

The routine may be called by the names f08abf, nagf_lapackeig_dgeqrt or its LAPACK name dgeqrt.
3
Description
f08abf forms the $QR$ factorization of an arbitrary rectangular real $m$ by $n$ matrix. No pivoting is performed.
It differs from
f08aef 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
$QR$ factorization based on the algorithm of
Elmroth and Gustavson (2000).
If
$m\ge n$, the factorization is given by:
where
$R$ is an
$n$ by
$n$ upper triangular matrix and
$Q$ is an
$m$ by
$m$ orthogonal matrix. It is sometimes more convenient to write the factorization as
which reduces to
where
${Q}_{1}$ consists of the first
$n$ columns of
$Q$, and
${Q}_{2}$ the remaining
$mn$ columns.
If
$m<n$,
$R$ is upper trapezoidal, and the factorization can be written
where
${R}_{1}$ is upper triangular and
${R}_{2}$ is rectangular.
The matrix
$Q$ is not formed explicitly but is represented as a product of
$\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ elementary reflectors (see the
F08 Chapter Introduction for details). Routines are provided to work with
$Q$ in this representation (see
Section 9).
Note also that for any $k<n$, the information returned represents a $QR$ factorization of the first $k$ columns of the original matrix $A$.
4
References
Elmroth E and Gustavson F (2000) Applying Recursion to Serial and Parallel $QR$ 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:
$\mathbf{m}$ – Integer
Input

On entry: $m$, the number of rows of the matrix $A$.
Constraint:
${\mathbf{m}}\ge 0$.

2:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of columns of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

3:
$\mathbf{nb}$ – Integer
Input

On entry: the explicitly chosen block size to be used in computing the
$QR$ factorization. See
Section 9 for details.
Constraint:
if $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)>0$, $1\le {\mathbf{nb}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.

4:
$\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) array
Input/Output

Note: the second dimension of the array
a
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $m$ by $n$ matrix $A$.
On exit: if
$m\ge n$, the elements below the diagonal are overwritten by details of the orthogonal matrix
$Q$ and the upper triangle is overwritten by the corresponding elements of the
$n$ by
$n$ upper triangular matrix
$R$.
If $m<n$, the strictly lower triangular part is overwritten by details of the orthogonal matrix $Q$ and the remaining elements are overwritten by the corresponding elements of the $m$ by $n$ upper trapezoidal matrix $R$.

5:
$\mathbf{lda}$ – Integer
Input

On entry: the first dimension of the array
a as declared in the (sub)program from which
f08abf is called.
Constraint:
${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.

6:
$\mathbf{t}\left({\mathbf{ldt}},*\right)$ – Real (Kind=nag_wp) array
Output

Note: the second dimension of the array
t
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
On exit: further details of the orthogonal matrix
$Q$. The number of blocks is
$b=\u2308\frac{k}{{\mathbf{nb}}}\u2309$, where
$k=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ and each block is of order
nb except for the last block, which is of order
$k\left(b1\right)\times {\mathbf{nb}}$. For each of the blocks, an upper triangular block reflector factor is computed:
${\mathit{T}}_{1},{\mathit{T}}_{2},\dots ,{\mathit{T}}_{b}$. These are stored in the
${\mathbf{nb}}$ by
$n$ matrix
$T$ as
$\mathit{T}=\left[{\mathit{T}}_{1}\left{\mathit{T}}_{2}\right\cdots {\mathit{T}}_{b}\right]$.

7:
$\mathbf{ldt}$ – Integer
Input

On entry: the first dimension of the array
t as declared in the (sub)program from which
f08abf is called.
Constraint:
${\mathbf{ldt}}\ge {\mathbf{nb}}$.

8:
$\mathbf{work}\left({\mathbf{nb}}\times {\mathbf{n}}\right)$ – Real (Kind=nag_wp) array
Workspace


9:
$\mathbf{info}$ – Integer
Output
On exit:
${\mathbf{info}}=0$ unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
 ${\mathbf{info}}<0$
If ${\mathbf{info}}=i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
7
Accuracy
The computed factorization is the exact factorization of a nearby matrix
$\left(A+E\right)$, where
and
$\epsilon $ is the
machine precision.
8
Parallelism and Performance
f08abf 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 routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The total number of floatingpoint operations is approximately $\frac{2}{3}{n}^{2}\left(3mn\right)$ if $m\ge n$ or $\frac{2}{3}{m}^{2}\left(3nm\right)$ if $m<n$.
To apply
$Q$ to an arbitrary real rectangular matrix
$C$,
f08abf may be followed by a call to
f08acf. For example,
Call dgemqrt('Left','Transpose',m,p,min(m,n),nb,a,lda,t,ldt,c,ldc, &
work,info)
forms
$C={Q}^{\mathrm{T}}C$, where
$C$ is
$m$ by
$p$.
To form the orthogonal matrix
$Q$ explicitly, simply initialize the
$m$ by
$m$ matrix
$C$ to the identity matrix and form
$C=QC$ using
f08acf as above.
The block size,
nb, used by
f08abf 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
${\mathbf{nb}}=64\ll \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ is likely to achieve good efficiency and it is unlikely that an optimal value would exceed
$340$.
To compute a
$QR$ factorization with column pivoting, use
f08bbf or
f08bef.
The complex analogue of this routine is
f08apf.
10
Example
This example solves the linear least squares problems
where
${b}_{1}$ and
${b}_{2}$ are the columns of the matrix
$B$,
10.1
Program Text
10.2
Program Data
10.3
Program Results