f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_ztpqrt (f08bpc)

## 1  Purpose

nag_ztpqrt (f08bpc) computes the $QR$ factorization of a complex $\left(m+n\right)$ by $n$ triangular-pentagonal matrix.

## 2  Specification

 #include #include
 void nag_ztpqrt (Nag_OrderType order, Integer m, Integer n, Integer l, Integer nb, Complex a[], Integer pda, Complex b[], Integer pdb, Complex t[], Integer pdt, NagError *fail)

## 3  Description

nag_ztpqrt (f08bpc) forms the $QR$ factorization of a complex $\left(m+n\right)$ 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 nag_zgeqrt (f08apc)) is performed on the matrix $C$. The upper triangular matrix $R$, details of the unitary 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 nag_ztpqrt (f08bpc) 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 nag_zgeqrt (f08apc)). Given an initial least-squares problem $\stackrel{^}{B}\stackrel{^}{X}=\stackrel{^}{Y}$ where $X$ and $Y$ are $l$ by $\mathit{nrhs}$ matrices, we have $\stackrel{^}{R}\stackrel{^}{X}={\stackrel{^}{Q}}^{\mathrm{H}}\stackrel{^}{Y}$.
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 ${\stackrel{^}{Q}}^{\mathrm{H}}\stackrel{^}{Y}$.
nag_ztpqrt (f08bpc) 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 $l=0$.

## 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:     orderNag_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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:     mIntegerInput
On entry: $m$, the number of rows of the matrix $B$.
Constraint: ${\mathbf{m}}\ge 0$.
3:     nIntegerInput
On entry: $n$, the number of columns of the matrix $B$ and the order of the upper triangular matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     lIntegerInput
On entry: $l$, the number of rows of the trapezoidal part of $B$ (i.e., ${B}_{2}$).
Constraint: $0\le {\mathbf{l}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
5:     nbIntegerInput
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 ${\mathbf{n}}>0$, ${\mathbf{nb}}\le {\mathbf{n}}$.
6:     a[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
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 $R$.
7:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8:     b[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array b must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pdb}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
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 unitary matrix $Q$.
9:     pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10:   t[$\mathit{dim}$]ComplexOutput
Note: the dimension, dim, of the array t must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdt}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nb}}×{\mathbf{pdt}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $T$ is stored in
• ${\mathbf{t}}\left[\left(j-1\right)×{\mathbf{pdt}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{t}}\left[\left(i-1\right)×{\mathbf{pdt}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: further details of the unitary 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 $\mathbit{T}=\left[{\mathbit{T}}_{1}|{\mathbit{T}}_{2}|\dots |{\mathbit{T}}_{b}\right]$.
11:   pdtIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array t.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdt}}\ge {\mathbf{nb}}$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdt}}\ge {\mathbf{n}}$.
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{nb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nb}}\ge 1$ and
if ${\mathbf{n}}>0$, ${\mathbf{nb}}\le {\mathbf{n}}$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdt}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdt}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{pdt}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nb}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdt}}\ge {\mathbf{nb}}$.
NE_INT_3
On entry, ${\mathbf{l}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: $0\le {\mathbf{l}}\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$.
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.

## 7  Accuracy

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.

## 8  Parallelism and Performance

nag_ztpqrt (f08bpc) is not threaded by NAG in any implementation.
nag_ztpqrt (f08bpc) 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.

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 nag_ztpqrt (f08bpc) 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 complex rectangular matrix $C$, nag_ztpqrt (f08bpc) may be followed by a call to nag_ztpmqrt (f08bqc). For example,
```nag_ztpmqrt(Nag_ColMajor,Nag_LeftSide,Nag_Trans,m,p,n,l,nb,b,pdb,
t,pdt,c,pdc,&c(n+1,1),ldc,&fail)
```
forms $C={Q}^{\mathrm{H}}C$, where $C$ is $\left(m+n\right)$ by $p$.
To form the unitary matrix $Q$ explicitly set $p=m+n$, initialize $C$ to the identity matrix and make a call to nag_ztpmqrt (f08bqc) as above.

## 10  Example

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 $B$,
 $A = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i and$
 $B= -2.09+1.93i 3.26-2.70i 3.34-3.53i -6.22+1.16i -4.94-2.04i 7.94-3.13i 0.17+4.23i 1.04-4.26i -5.19+3.63i -2.31-2.12i 0.98+2.53i -1.39-4.05i .$
A $QR$ factorization is performed on the first $4$ rows of $A$ using nag_zgeqrt (f08apc) after which the first $4$ rows of $B$ are updated by applying ${Q}^{T}$ using nag_zgemqrt (f08aqc). The remaining row is added by performing a $QR$ update using nag_ztpqrt (f08bpc); $B$ is updated by applying the new ${Q}^{T}$ using nag_ztpmqrt (f08bqc); the solution is finally obtained by triangular solve using $R$ from the updated $QR$.

### 10.1  Program Text

Program Text (f08bpce.c)

### 10.2  Program Data

Program Data (f08bpce.d)

### 10.3  Program Results

Program Results (f08bpce.r)