# NAG FL Interfacef08bhf (dtzrzf)

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## 1Purpose

f08bhf reduces the $m×n$ ($m\le n$) real upper trapezoidal matrix $A$ to upper triangular form by means of orthogonal transformations.

## 2Specification

Fortran Interface
 Subroutine f08bhf ( m, n, a, lda, tau, work, info)
 Integer, Intent (In) :: m, n, lda, lwork Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), tau(*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
#include <nag.h>
 void f08bhf_ (const Integer *m, const Integer *n, double a[], const Integer *lda, double tau[], double work[], const Integer *lwork, Integer *info)
The routine may be called by the names f08bhf, nagf_lapackeig_dtzrzf or its LAPACK name dtzrzf.

## 3Description

The $m×n$ ($m\le n$) real upper trapezoidal matrix $A$ given by
 $A = ( R1 R2 ) ,$
where ${R}_{1}$ is an $m×m$ upper triangular matrix and ${R}_{2}$ is an $m×\left(n-m\right)$ matrix, is factorized as
 $A = ( R 0 ) Z ,$
where $R$ is also an $m×m$ upper triangular matrix and $Z$ is an $n×n$ 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 https://www.netlib.org/lapack/lug

## 5Arguments

1: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge {\mathbf{m}}$.
3: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (Kind=nag_wp) array Input/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 $m×n$ upper trapezoidal part of the array a must contain the matrix to be factorized.
On exit: the leading $m×m$ 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).
4: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08bhf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
5: $\mathbf{tau}\left(*\right)$Real (Kind=nag_wp) array Output
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: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Real (Kind=nag_wp) array Workspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
7: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08bhf 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: $\mathbf{info}$Integer Output
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 $A+E$, where
 $‖E‖2 = O⁡ε ‖A‖2$
and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f08bhf 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 $4{m}^{2}\left(n-m\right)$.
The complex analogue of this routine is f08bvf.

## 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 $k×k$ 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.

### 10.1Program Text

Program Text (f08bhfe.f90)

### 10.2Program Data

Program Data (f08bhfe.d)

### 10.3Program Results

Program Results (f08bhfe.r)