# NAG Library Routine Document

## 1Purpose

f08bbf (dtpqrt) computes the $QR$ factorization of a real $\left(m+n\right)$ by $n$ triangular-pentagonal matrix.

## 2Specification

Fortran Interface
 Subroutine f08bbf ( m, n, l, nb, a, lda, b, ldb, t, ldt, work, info)
 Integer, Intent (In) :: m, n, l, nb, lda, ldb, ldt Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*), t(ldt,*), work(*)
#include <nagmk26.h>
 void f08bbf_ (const Integer *m, const Integer *n, const Integer *l, const Integer *nb, double a[], const Integer *lda, double b[], const Integer *ldb, double t[], const Integer *ldt, double work[], Integer *info)
The routine may be called by its LAPACK name dtpqrt.

## 3Description

f08bbf (dtpqrt) forms the $QR$ factorization of a real $\left(m+n\right)$ by $n$ triangular-pentagonal matrix $C$,
 $C= A B$
where $A$ is an upper triangular $n$ by $n$ matrix and $B$ is an $m$ by $n$ pentagonal matrix consisting of an $\left(m-l\right)$ by $n$ rectangular matrix ${B}_{1}$ on top of an $l$ by $n$ upper trapezoidal matrix ${B}_{2}$:
 $B= B1 B2 .$
The upper trapezoidal matrix ${B}_{2}$ consists of the first $l$ rows of an $n$ by $n$ upper triangular matrix, where $0\le l\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$. If $l=0$, $B$ is $m$ by $n$ rectangular; if $l=n$ and $m=n$, $B$ is upper triangular.
A recursive, explicitly blocked, $QR$ factorization (see f08abf (dgeqrt)) is performed on the matrix $C$. The upper triangular matrix $R$, details of the orthogonal matrix $Q$, and further details (the block reflector factors) of $Q$ are returned.
Typically the matrix $A$ or ${B}_{2}$ contains the matrix $R$ from the $QR$ factorization of a subproblem and f08bbf (dtpqrt) performs the $QR$ update operation from the inclusion of matrix ${B}_{1}$.
For example, consider the $QR$ factorization of an $l$ by $n$ matrix $\stackrel{^}{B}$ with $l: $\stackrel{^}{B}=\stackrel{^}{Q}\stackrel{^}{R}$, $\stackrel{^}{R}=\left(\begin{array}{cc}\stackrel{^}{{R}_{1}}& \stackrel{^}{{R}_{2}}\end{array}\right)$, where $\stackrel{^}{{R}_{1}}$ is $l$ by $l$ upper triangular and $\stackrel{^}{{R}_{2}}$ is $\left(n-l\right)$ by $n$ rectangular (this can be performed by f08abf (dgeqrt)). Given an initial least squares problem $\stackrel{^}{B}\stackrel{^}{X}=\stackrel{^}{Y}$ where $X$ and $Y$ are $l$ by $\mathit{nrhs}$ matrices, we have $\stackrel{^}{R}\stackrel{^}{X}={\stackrel{^}{Q}}^{\mathrm{T}}\stackrel{^}{Y}$.
Now, adding an additional $m-l$ rows to the original system gives the augmented least squares problem
 $BX=Y$
where $B$ is an $m$ by $n$ matrix formed by adding $m-l$ rows on top of $\stackrel{^}{R}$ and $Y$ is an $m$ by $\mathit{nrhs}$ matrix formed by adding $m-l$ rows on top of ${\stackrel{^}{Q}}^{\mathrm{T}}\stackrel{^}{Y}$.
f08bbf (dtpqrt) can then be used to perform the $QR$ factorization of the pentagonal matrix $B$; the $n$ by $n$ matrix $A$ will be zero on input and contain $R$ on output.
In the case where $\stackrel{^}{B}$ is $r$ by $n$, $r\ge n$, $\stackrel{^}{R}$ is $n$ by $n$ upper triangular (forming $A$) on top of $r-n$ rows of zeros (forming first $r-n$ rows of $B$). Augmentation is then performed by adding rows to the bottom of $B$ with $l=0$.

## 4References

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

## 5Arguments

1:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $B$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrix $B$ and the order of the upper triangular matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{l}$ – IntegerInput
On entry: $l$, the number of rows of the trapezoidal part of $B$ (i.e., ${B}_{2}$).
Constraint: $0\le {\mathbf{l}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
4:     $\mathbf{nb}$ – IntegerInput
On entry: the explicitly chosen block-size to be used in the algorithm for computing the $QR$ factorization. See Section 9 for details.
Constraints:
• ${\mathbf{nb}}\ge 1$;
• if ${\mathbf{n}}>0$, ${\mathbf{nb}}\le {\mathbf{n}}$.
5:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput/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 $n$ by $n$ upper triangular matrix $A$.
On exit: the upper triangle is overwritten by the corresponding elements of the $n$ by $n$ upper triangular matrix $R$.
6:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08bbf (dtpqrt) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $m$ by $n$ pentagonal matrix $B$ composed of an $\left(m-l\right)$ by $n$ rectangular matrix ${B}_{1}$ above an $l$ by $n$ upper trapezoidal matrix ${B}_{2}$.
On exit: details of the orthogonal matrix $Q$.
8:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08bbf (dtpqrt) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
9:     $\mathbf{t}\left({\mathbf{ldt}},*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array t must be at least ${\mathbf{n}}$.
On exit: further details of the orthogonal matrix $Q$. The number of blocks is $b=⌈\frac{k}{{\mathbf{nb}}}⌉$, 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 $k-\left(b-1\right)×{\mathbf{nb}}$. For each of the blocks, an upper triangular block reflector factor is computed: ${\mathbit{T}}_{1},{\mathbit{T}}_{2},\dots ,{\mathbit{T}}_{b}$. These are stored in the ${\mathbf{nb}}$ by $n$ matrix $T$ as $\mathbit{T}=\left[{\mathbit{T}}_{1}|{\mathbit{T}}_{2}|\dots |{\mathbit{T}}_{b}\right]$.
10:   $\mathbf{ldt}$ – IntegerInput
On entry: the first dimension of the array t as declared in the (sub)program from which f08bbf (dtpqrt) is called.
Constraint: ${\mathbf{ldt}}\ge {\mathbf{nb}}$.
11:   $\mathbf{work}\left(*\right)$ – Real (Kind=nag_wp) arrayWorkspace
Note: the dimension of the array work must be at least ${\mathbf{nb}}×{\mathbf{n}}$.
12:   $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error 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.

## 7Accuracy

The computed factorization is the exact factorization of a nearby matrix $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f08bbf (dtpqrt) 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 implementation-specific information.

The total number of floating-point operations is approximately $\frac{2}{3}{n}^{2}\left(3m-n\right)$ if $m\ge n$ or $\frac{2}{3}{m}^{2}\left(3n-m\right)$ if $m.
The block size, nb, used by f08bbf (dtpqrt) 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 apply $Q$ to an arbitrary real rectangular matrix $C$, f08bbf (dtpqrt) may be followed by a call to f08bcf (dtpmqrt). For example,
```Call dtpmqrt('Left','Transpose',m,p,n,l,nb,b,ldb, &
t,ldt,c,ldc,c(n+1,1),ldc,work,info)```
forms $C={Q}^{\mathrm{T}}C$, where $C$ is $\left(m+n\right)$ by $p$.
To form the orthogonal matrix $Q$ explicitly set $p=m+n$, initialize $C$ to the identity matrix and make a call to f08bcf (dtpmqrt) as above.

## 10Example

This example finds the basic solutions for the linear least squares problems
 $minimize⁡ Axi - bi 2 , i=1,2$
where ${b}_{1}$ and ${b}_{2}$ are the columns of the matrix $B$,
 $A = -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 and B= -2.67 0.41 -0.55 -3.10 3.34 -4.01 -0.77 2.76 0.48 -6.17 4.10 0.21 .$
A $QR$ factorization is performed on the first $4$ rows of $A$ using f08abf (dgeqrt) after which the first $4$ rows of $B$ are updated by applying ${Q}^{T}$ using f08acf (dgemqrt). The remaining row is added by performing a $QR$ update using f08bbf (dtpqrt); $B$ is updated by applying the new ${Q}^{T}$ using f08bcf (dtpmqrt); the solution is finally obtained by triangular solve using $R$ from the updated $QR$.

### 10.1Program Text

Program Text (f08bbfe.f90)

### 10.2Program Data

Program Data (f08bbfe.d)

### 10.3Program Results

Program Results (f08bbfe.r)