NAG CL Interface
f08bfc (dgeqp3)
1
Purpose
f08bfc computes the $QR$ factorization, with column pivoting, of a real $m$ by $n$ matrix.
2
Specification
void 
f08bfc (Nag_OrderType order,
Integer m,
Integer n,
double a[],
Integer pda,
Integer jpvt[],
double tau[],
NagError *fail) 

The function may be called by the names: f08bfc, nag_lapackeig_dgeqp3 or nag_dgeqp3.
3
Description
f08bfc forms the $QR$ factorization, with column pivoting, of an arbitrary rectangular real $m$ by $n$ matrix.
If
$m\ge n$, the factorization is given by:
where
$R$ is an
$n$ by
$n$ upper triangular matrix,
$Q$ is an
$m$ by
$m$ orthogonal matrix and
$P$ is an
$n$ by
$n$ permutation 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 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). Functions are provided to work with
$Q$ in this representation (see
Section 9).
Note also that for any
$k<n$, the information returned in the first
$k$ columns of the array
a represents a
$QR$ factorization of the first
$k$ columns of the permuted matrix
$AP$.
The function allows specified columns of $A$ to be moved to the leading columns of $AP$ at the start of the factorization and fixed there. The remaining columns are free to be interchanged so that at the $i$th stage the pivot column is chosen to be the column which maximizes the $2$norm of elements $i$ to $m$ over columns $i$ to $n$.
4
References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments

1:
$\mathbf{order}$ – Nag_OrderType
Input

On entry: the
order argument specifies the twodimensional storage scheme being used, i.e., rowmajor ordering or columnmajor ordering. C language defined storage is specified by
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$. 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:
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$ or $\mathrm{Nag\_ColMajor}$.

2:
$\mathbf{m}$ – Integer
Input

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

3:
$\mathbf{n}$ – Integer
Input

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

4:
$\mathbf{a}\left[\mathit{dim}\right]$ – double
Input/Output

Note: the dimension,
dim, of the array
a
must be at least
 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}\times {\mathbf{n}}\right)$ when
${\mathbf{order}}=\mathrm{Nag\_ColMajor}$;
 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\times {\mathbf{pda}}\right)$ when
${\mathbf{order}}=\mathrm{Nag\_RowMajor}$.
The
$\left(i,j\right)$th element of the matrix
$A$ is stored in
 ${\mathbf{a}}\left[\left(j1\right)\times {\mathbf{pda}}+i1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_ColMajor}$;
 ${\mathbf{a}}\left[\left(i1\right)\times {\mathbf{pda}}+j1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_RowMajor}$.
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:
$\mathbf{pda}$ – Integer
Input

On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraints:
 if ${\mathbf{order}}=\mathrm{Nag\_ColMajor}$,
${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
 if ${\mathbf{order}}=\mathrm{Nag\_RowMajor}$, ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

6:
$\mathbf{jpvt}\left[\mathit{dim}\right]$ – Integer
Input/Output

Note: the dimension,
dim, of the array
jpvt
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: if ${\mathbf{jpvt}}\left[j1\right]\ne 0$, the $j$ th column of $A$ is moved to the beginning of $AP$ before the decomposition is computed and is fixed in place during the computation. Otherwise, the $j$ th column of $A$ is a free column (i.e., one which may be interchanged during the computation with any other free column).
On exit: details of the permutation matrix $P$. More precisely, if ${\mathbf{jpvt}}\left[j1\right]=k$, the $k$th column of $A$ is moved to become the $j$ th column of $AP$; in other words, the columns of $AP$ are the columns of $A$ in the order ${\mathbf{jpvt}}\left[0\right],{\mathbf{jpvt}}\left[1\right],\dots ,{\mathbf{jpvt}}\left[n1\right]$.

7:
$\mathbf{tau}\left[\mathit{dim}\right]$ – double
Output

Note: the dimension,
dim, of the array
tau
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: the scalar factors of the elementary reflectors.

8:
$\mathbf{fail}$ – 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 $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{pda}}>0$.
 NE_INT_2

On entry, ${\mathbf{pda}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pda}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
 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
$\left(A+E\right)$, where
and
$\epsilon $ is the
machine precision.
8
Parallelism and Performance
f08bfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08bfc 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 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 form the orthogonal matrix
$Q$ f08bfc may be followed by a call to
f08afc:
nag_lapackeig_dorgqr(order,m,m,MIN(m,n),&a,pda,tau,&fail)
but note that the second dimension of the array
a must be at least
m, which may be larger than was required by
f08bfc.
When
$m\ge n$, it is often only the first
$n$ columns of
$Q$ that are required, and they may be formed by the call:
nag_lapackeig_dorgqr(order,m,n,n,&a,pda,tau,&fail)
To apply
$Q$ to an arbitrary real rectangular matrix
$C$,
f08bfc may be followed by a call to
f08agc. For example,
nag_lapackeig_dormqr(order,Nag_LeftSide,Nag_Trans,m,p,MIN(m,n),&a,pda,tau,
&c,pdc,&fail)
forms
$C={Q}^{\mathrm{T}}C$, where
$C$ is
$m$ by
$p$.
To compute a
$QR$ factorization without column pivoting, use
f08aec.
The complex analogue of this function is
f08btc.
10
Example
This example solves the linear least squares problems
for the basic solutions
${x}_{1}$ and
${x}_{2}$, where
and
${b}_{j}$ is the
$j$th column of the matrix
$B$. The solution is obtained by first obtaining a
$QR$ factorization with column pivoting of the matrix
$A$. A tolerance of
$0.01$ is used to estimate the rank of
$A$ from the upper triangular factor,
$R$.
10.1
Program Text
10.2
Program Data
10.3
Program Results