F08AVF (ZGELQF) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08AVF (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.

+ Contents

    1  Purpose
    7  Accuracy

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 mn, 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 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 minm,n elementary reflectors (see the
F08 Chapter Introduction for details). Routines are provided to work with Q in this representation (see Section 8).
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

None.

5  Parameters

1:     M – INTEGERInput
On entry: m, the number of rows of the matrix A.
Constraint: M0.
2:     N – INTEGERInput
On entry: n, the number of columns of the matrix A.
Constraint: N0.
3:     A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,N.
On entry: the m by 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 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:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08AVF (ZGELQF) is called.
Constraint: LDAmax1,M.
5:     TAU(*) – COMPLEX (KIND=nag_wp) arrayOutput
Note: the dimension of the array TAU must be at least max1,minM,N.
On exit: further details of the unitary matrix Q.
6:     WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
7:     LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08AVF (ZGELQF) 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: LWORKmax1,M or LWORK=-1.
8:     INFO – INTEGEROutput

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
INFO<0
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 = Oε A2 ,
and ε is the machine precision.

8  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 (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 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 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=QHC, where C is m by p.
The real analogue of this routine is F08AHF (DGELQF).

9  Example

This example finds the minimum norm solutions of the under-determined 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
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 .

9.1  Program Text

Program Text (f08avfe.f90)

9.2  Program Data

Program Data (f08avfe.d)

9.3  Program Results

Program Results (f08avfe.r)


F08AVF (ZGELQF) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2011