F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08BVF (ZTZRZF)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08BVF (ZTZRZF) reduces the $m$ by $n$ ($m\le n$) complex upper trapezoidal matrix $A$ to upper triangular form by means of unitary transformations.

## 2  Specification

 SUBROUTINE F08BVF ( M, N, A, LDA, TAU, WORK, LWORK, INFO)
 INTEGER M, N, LDA, LWORK, INFO COMPLEX (KIND=nag_wp) A(LDA,*), TAU(*), WORK(max(1,LWORK))
The routine may be called by its LAPACK name ztzrzf.

## 3  Description

The $m$ by $n$ ($m\le n$) complex 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 $n$ unitary matrix.

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

## 5  Arguments

1:     $\mathrm{M}$ – INTEGERInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{M}}\ge 0$.
2:     $\mathrm{N}$ – INTEGERInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     $\mathrm{A}\left({\mathbf{LDA}},*\right)$ – COMPLEX (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 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 unitary matrix $Z$ as a product of $m$ elementary reflectors (see Section 3.3.6 in the F08 Chapter Introduction).
4:     $\mathrm{LDA}$ – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08BVF (ZTZRZF) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$.
5:     $\mathrm{TAU}\left(*\right)$ – COMPLEX (KIND=nag_wp) arrayOutput
Note: the dimension 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.
6:     $\mathrm{WORK}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)\right)$ – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, the real part of ${\mathbf{WORK}}\left(1\right)$ contains the minimum value of LWORK required for optimal performance.
7:     $\mathrm{LWORK}$ – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08BVF (ZTZRZF) is called.
If ${\mathbf{LWORK}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued.
Suggested value: for optimal performance, ${\mathbf{LWORK}}\ge {\mathbf{M}}×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{LWORK}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{M}}\right)$ or ${\mathbf{LWORK}}=-1$.
8:     $\mathrm{INFO}$ – INTEGEROutput
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 $A+E$, where
 $E2 = O⁡ε A2$
and $\epsilon$ is the machine precision.

## 8  Parallelism and Performance

F08BVF (ZTZRZF) 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 $16{m}^{2}\left(n-m\right)$.
The real analogue of this routine is F08BHF (DTZRZF).

## 10  Example

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 $B$,
 $A = 0.47-0.34i -0.40+0.54i 0.60+0.01i 0.80-1.02i -0.32-0.23i -0.05+0.20i -0.26-0.44i -0.43+0.17i 0.35-0.60i -0.52-0.34i 0.87-0.11i -0.34-0.09i 0.89+0.71i -0.45-0.45i -0.02-0.57i 1.14-0.78i -0.19+0.06i 0.11-0.85i 1.44+0.80i 0.07+1.14i$
and
 $B = -1.08-2.59i 2.22+2.35i -2.61-1.49i 1.62-1.48i 3.13-3.61i 1.65+3.43i 7.33-8.01i -0.98+3.08i 9.12+7.63i -2.84+2.78i .$
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, $R$.
Note that the block size (NB) of $64$ assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

### 10.1  Program Text

Program Text (f08bvfe.f90)

### 10.2  Program Data

Program Data (f08bvfe.d)

### 10.3  Program Results

Program Results (f08bvfe.r)