NAG FL Interface
f08avf (zgelqf)

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1 Purpose

f08avf computes the LQ factorization of a complex m×n matrix.

2 Specification

Fortran Interface
Subroutine f08avf ( m, n, a, lda, tau, work, lwork, info)
Integer, Intent (In) :: m, n, lda, lwork
Integer, Intent (Out) :: info
Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), tau(*)
Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
C Header Interface
#include <nag.h>
void  f08avf_ (const Integer *m, const Integer *n, Complex a[], const Integer *lda, Complex tau[], Complex work[], const Integer *lwork, Integer *info)
The routine may be called by the names f08avf, nagf_lapackeig_zgelqf or its LAPACK name zgelqf.

3 Description

f08avf forms the LQ factorization of an arbitrary rectangular complex m×n matrix. No pivoting is performed.
If mn, the factorization is given by:
A = ( L 0 ) Q  
where L is an m×m lower triangular matrix (with real diagonal elements) and Q is an n×n unitary matrix. It is sometimes more convenient to write the factorization as
A = ( L 0 ) ( Q1 Q2 )  
which reduces to
A = LQ1 ,  
where Q1 consists of the first m rows of Q, and Q2 the remaining n-m rows.
If m>n, L is trapezoidal, and the factorization can be written
A = ( L1 L2 ) Q  
where L1 is lower triangular and L2 is rectangular.
The LQ factorization of A is essentially the same as the QR factorization of AH, since
A = ( L 0 ) QAH= QH ( LH 0 ) .  
The matrix Q is not formed explicitly but is represented as a product of min(m,n) 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<m, 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.

4 References


5 Arguments

1: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
2: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: n0.
3: a(lda,*) Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,n).
On entry: the m×n matrix A.
On exit: if mn, 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×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×n lower trapezoidal matrix L.
The diagonal elements of L are real.
4: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08avf is called.
Constraint: ldamax(1,m).
5: tau(*) Complex (Kind=nag_wp) array Output
Note: the dimension of the array tau must be at least max(1,min(m,n)).
On exit: further details of the unitary matrix Q.
6: work(max(1,lwork)) Complex (Kind=nag_wp) array Workspace
On exit: if info=0, the real part of work(1) contains the minimum value of lwork required for optimal performance.
7: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08avf is called.
If 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, lworkm×nb, where nb is the optimal block size.
Constraint: lworkmax(1,m) or lwork=−1.
8: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

If 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 ε is the machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08avf 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.

9 Further Comments

The total number of real floating-point operations is approximately 83 m2 (3n-m) if mn or 83 n2 (3m-n) if m>n.
To form the unitary matrix Q f08avf may be followed by a call to f08awf :
Call zunglq(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 f08avf.
When mn, 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 m×p complex rectangular matrix C, f08avf may be followed by a call to f08axf . For example,
Call zunmlq('Left','Conjugate Transpose',m,p,min(m,n),a,lda,tau, &
forms the matrix product C=QHC.
The real analogue of this routine is f08ahf.

10 Example

This example finds the minimum norm solutions of the underdetermined systems of linear equations
Ax1= b1   and   Ax2= b2  
where b1 and b2 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 )  
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)