F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08AVF (ZGELQF)

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

F08AVF (ZGELQF) computes the $LQ$ factorization of a complex $m$ by $n$ matrix.

## 2  Specification

 SUBROUTINE F08AVF ( 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 zgelqf.

## 3  Description

F08AVF (ZGELQF) forms the $LQ$ factorization of an arbitrary rectangular complex $m$ by $n$ matrix. No pivoting is performed.
If $m\le n$, the factorization is given by:
 $A = L 0 Q$
where $L$ is an $m$ by $m$ lower triangular matrix (with real diagonal elements) and $Q$ is an $n$ by $n$ unitary 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 ${A}^{\mathrm{H}}$, since
 $A = L 0 Q⇔AH= QH LH 0 .$
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.

## 5  Parameters

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 $m$ by $n$ matrix $A$.
On exit: if $m\le n$, the elements above the diagonal are overwritten by details of the unitary matrix $Q$ and the lower triangle is overwritten by the corresponding elements of the $m$ by $m$ lower triangular matrix $L$.
If $m>n$, the strictly upper triangular part is overwritten by details of the unitary matrix $Q$ and the remaining elements are overwritten by the corresponding elements of the $m$ by $n$ lower trapezoidal matrix $L$.
The diagonal elements of $L$ are real.
4:     $\mathrm{LDA}$ – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08AVF (ZGELQF) 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{M}},{\mathbf{N}}\right)\right)$.
On exit: further details of the unitary matrix $Q$.
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 F08AVF (ZGELQF) 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 $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision.

## 8  Parallelism and Performance

F08AVF (ZGELQF) is not threaded by NAG in any implementation.
F08AVF (ZGELQF) 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 real floating-point operations is approximately $\frac{8}{3}{m}^{2}\left(3n-m\right)$ if $m\le n$ or $\frac{8}{3}{n}^{2}\left(3m-n\right)$ if $m>n$.
To form the unitary matrix $Q$ F08AVF (ZGELQF) may be followed by a call to F08AWF (ZUNGLQ):
```CALL ZUNGLQ(N,N,MIN(M,N),A,LDA,TAU,WORK,LWORK,INFO)
```
but note that the first dimension of the array A, specified by the parameter LDA, must be at least N, which may be larger than was required by F08AVF (ZGELQF).
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 ZUNGLQ(M,N,M,A,LDA,TAU,WORK,LWORK,INFO)
```
To apply $Q$ to an arbitrary complex rectangular matrix $C$, F08AVF (ZGELQF) may be followed by a call to F08AXF (ZUNMLQ). For example,
```CALL ZUNMLQ('Left','Conjugate Transpose',M,P,MIN(M,N),A,LDA,TAU, &
C,LDC,WORK,LWORK,INFO)```
forms the matrix product $C={Q}^{\mathrm{H}}C$, where $C$ is $m$ by $p$.
The real analogue of this routine is F08AHF (DGELQF).

## 10  Example

This example finds the minimum norm solutions of the under-determined systems of linear equations
 $Ax1= b1 and Ax2= b2$
where ${b}_{1}$ and ${b}_{2}$ are the columns of the matrix $B$,
 $A = 0.28-0.36i 0.50-0.86i -0.77-0.48i 1.58+0.66i -0.50-1.10i -1.21+0.76i -0.32-0.24i -0.27-1.15i 0.36-0.51i -0.07+1.33i -0.75+0.47i -0.08+1.01i$
and
 $B = -1.35+0.19i 4.83-2.67i 9.41-3.56i -7.28+3.34i -7.57+6.93i 0.62+4.53i .$

### 10.1  Program Text

Program Text (f08avfe.f90)

### 10.2  Program Data

Program Data (f08avfe.d)

### 10.3  Program Results

Program Results (f08avfe.r)