# NAG FL Interfacef08ahf (dgelqf)

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## 1Purpose

f08ahf computes the $LQ$ factorization of a real $m×n$ matrix.

## 2Specification

Fortran Interface
 Subroutine f08ahf ( 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 f08ahf_ (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 f08ahf, nagf_lapackeig_dgelqf or its LAPACK name dgelqf.

## 3Description

f08ahf forms the $LQ$ factorization of an arbitrary rectangular real $m×n$ matrix. No pivoting is performed.
If $m\le n$, the factorization is given by:
 $A = ( L 0 ) Q$
where $L$ is an $m×m$ lower triangular matrix and $Q$ is an $n×n$ orthogonal matrix. It is sometimes more convenient to write the factorization as
 $A = ( L 0 ) ( Q1 Q2 )$
which reduces to
 $A = LQ1 ,$
where ${Q}_{1}$ consists of the first $m$ rows of $Q$, and ${Q}_{2}$ the remaining $n-m$ rows.
If $m>n$, $L$ is trapezoidal, and the factorization can be written
 $A = ( L1 L2 ) Q$
where ${L}_{1}$ is lower triangular and ${L}_{2}$ is rectangular.
The $LQ$ factorization of $A$ is essentially the same as the $QR$ factorization of ${A}^{\mathrm{T}}$, since
 $A = ( L 0 ) Q⇔AT= QT ( LT 0 ) .$
The matrix $Q$ is not formed explicitly but is represented as a product of $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ elementary reflectors (see the F08 Chapter Introduction for details). Routines are provided to work with $Q$ in this representation (see Section 9).
Note also that for any $k, the information returned in the first $k$ rows of the array a represents an $LQ$ factorization of the first $k$ rows of the original matrix $A$.

None.

## 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: ${\mathbf{n}}\ge 0$.
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 $m×n$ matrix $A$.
On exit: if $m\le n$, the elements above the diagonal are overwritten by details of the orthogonal matrix $Q$ and the lower triangle is overwritten by the corresponding elements of the $m×m$ lower triangular matrix $L$.
If $m>n$, the strictly upper triangular part is overwritten by details of the orthogonal matrix $Q$ and the remaining elements are overwritten by the corresponding elements of the $m×n$ lower trapezoidal matrix $L$.
4: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08ahf 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)\right)$.
On exit: further details of the orthogonal matrix $Q$.
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 f08ahf 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 $\left(A+E\right)$, where
 $‖E‖2 = O(ε) ‖A‖2 ,$
and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f08ahf 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 $\frac{2}{3}{m}^{2}\left(3n-m\right)$ if $m\le n$ or $\frac{2}{3}{n}^{2}\left(3m-n\right)$ if $m>n$.
To form the orthogonal matrix $Q$ f08ahf may be followed by a call to f08ajf :
`Call dorglq(n,n,min(m,n),a,lda,tau,work,lwork,info)`
but note that the first dimension of the array a must be at least n, which may be larger than was required by f08ahf.
When $m\le n$, it is often only the first $m$ rows of $Q$ that are required, and they may be formed by the call:
`Call dorglq(m,n,m,a,lda,tau,work,lwork,info)`
To apply $Q$ to an arbitrary $m×p$ real rectangular matrix $C$, f08ahf may be followed by a call to f08akf . For example,
```Call dormlq('Left','Transpose',m,p,min(m,n),a,lda,tau,c,ldc, &
work,lwork,info)```
forms the matrix product $C={Q}^{\mathrm{T}}C$.
The complex analogue of this routine is f08avf.

## 10Example

This example finds the minimum norm solutions of the underdetermined systems of linear equations
 $Ax1= b1 and Ax2= b2$
where ${b}_{1}$ and ${b}_{2}$ are the columns of the matrix $B$,
 $A = ( -5.42 3.28 -3.68 0.27 2.06 0.46 -1.65 -3.40 -3.20 -1.03 -4.06 -0.01 -0.37 2.35 1.90 4.31 -1.76 1.13 -3.15 -0.11 1.99 -2.70 0.26 4.50 ) and B= ( -2.87 -5.23 1.63 0.29 -3.52 4.76 0.45 -8.41 ) .$

### 10.1Program Text

Program Text (f08ahfe.f90)

### 10.2Program Data

Program Data (f08ahfe.d)

### 10.3Program Results

Program Results (f08ahfe.r)