# NAG C Library Function Document

## 1Purpose

nag_dtzrzf (f08bhc) reduces the $m$ by $n$ ($m\le n$) real upper trapezoidal matrix $A$ to upper triangular form by means of orthogonal transformations.

## 2Specification

 #include #include
 void nag_dtzrzf (Nag_OrderType order, Integer m, Integer n, double a[], Integer pda, double tau[], NagError *fail)

## 3Description

The $m$ by $n$ ($m\le n$) real upper trapezoidal matrix $A$ given by
 $A = R1 R2 ,$
where ${R}_{1}$ is an $m$ by $m$ upper triangular matrix and ${R}_{2}$ is an $m$ by $\left(n-m\right)$ matrix, is factorized as
 $A = R 0 Z ,$
where $R$ is also an $m$ by $m$ upper triangular matrix and $Z$ is an $n$ by $n$ orthogonal matrix.

## 4References

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 http://www.netlib.org/lapack/lug

## 5Arguments

1:    $\mathbf{order}$Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge {\mathbf{m}}$.
4:    $\mathbf{a}\left[\mathit{dim}\right]$doubleInput/Output
Note: the dimension, dim, of the array a must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}×{\mathbf{pda}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ 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 leading $m$ by $n$ upper trapezoidal part of the array a must contain the matrix to be factorized.
On exit: the leading $m$ by $m$ upper triangular part of a contains the upper triangular matrix $R$, and elements ${\mathbf{m}}+1$ to n of the first $m$ rows of a, with the array tau, represent the orthogonal matrix $Z$ as a product of $m$ elementary reflectors (see Section 3.3.6 in the f08 Chapter Introduction).
5:    $\mathbf{pda}$IntegerInput
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:    $\mathbf{tau}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array tau must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On exit: the scalar factors of the elementary reflectors.
7:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}>0$.
NE_INT_2
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
On entry, ${\mathbf{pda}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The computed factorization is the exact factorization of a nearby matrix $A+E$, where
 $E2 = O⁡ε A2$
and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

nag_dtzrzf (f08bhc) 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 $4{m}^{2}\left(n-m\right)$.
The complex analogue of this function is nag_ztzrzf (f08bvc).

## 10Example

This example solves the linear least squares problems
 $minx bj - Axj 2 , j=1,2$
for the minimum norm solutions ${x}_{1}$ and ${x}_{2}$, where ${b}_{j}$ is the $j$th column of the matrix $B$,
 $A = -0.09 0.14 -0.46 0.68 1.29 -1.56 0.20 0.29 1.09 0.51 -1.48 -0.43 0.89 -0.71 -0.96 -1.09 0.84 0.77 2.11 -1.27 0.08 0.55 -1.13 0.14 1.74 -1.59 -0.72 1.06 1.24 0.34 and B= 7.4 2.7 4.2 -3.0 -8.3 -9.6 1.8 1.1 8.6 4.0 2.1 -5.7 .$
The solution is obtained by first obtaining a $QR$ factorization with column pivoting of the matrix $A$, and then the $RZ$ factorization of the leading $k$ by $k$ part of $R$ is computed, where $k$ is the estimated rank of $A$. A tolerance of $0.01$ is used to estimate the rank of $A$ from the upper triangular factor, $R$.

### 10.1Program Text

Program Text (f08bhce.c)

### 10.2Program Data

Program Data (f08bhce.d)

### 10.3Program Results

Program Results (f08bhce.r)