NAG Library Routine Document

f08chf (dgerqf)


    1  Purpose
    7  Accuracy


f08chf (dgerqf) computes an RQ factorization of a real m by n matrix A.


Fortran Interface
Subroutine f08chf ( m, n, a, lda, tau, work, lwork, 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))
C Header Interface
#include nagmk26.h
void  f08chf_ (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 its LAPACK name dgerqf.


f08chf (dgerqf) forms the RQ factorization of an arbitrary rectangular real m by n matrix. If mn, the factorization is given by
A = 0 R Q ,  
where R is an m by m lower triangular matrix and Q is an n by n orthogonal matrix. If m>n the factorization is given by
A =RQ ,  
where R is an m by n upper trapezoidal matrix and Q is again an n by n orthogonal matrix. In the case where m<n the factorization can be expressed as
A = 0 R Q1 Q2 =RQ2 ,  
where Q1 consists of the first n-m rows of Q and Q2 the remaining m rows.
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 9).


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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore


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:     alda* – Real (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 upper triangle of the subarray a1:mn-m+1:n contains the m by m upper triangular matrix R.
If mn, the elements on and above the m-nth subdiagonal contain the m by n upper trapezoidal matrix R; the remaining elements, with the array tau, represent the orthogonal matrix Q as a product of minm,n elementary reflectors (see Section 3.3.6 in the F08 Chapter Introduction).
4:     lda – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08chf (dgerqf) is called.
Constraint: ldamax1,m.
5:     tau* – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array tau must be at least max1,minm,n.
On exit: the scalar factors of the elementary reflectors.
6:     workmax1,lwork – Real (Kind=nag_wp) arrayWorkspace
On exit: if info=0, work1 contains the minimum value of lwork required for optimal performance.
7:     lwork – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08chf (dgerqf) 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
On exit: info=0 unless the routine detects an error (see Section 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.


The computed factorization is the exact factorization of a nearby matrix A+E, where
E2 = Oε A2  
and ε is the machine precision.

Parallelism and Performance

f08chf (dgerqf) 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.

Further Comments

The total number of floating-point operations is approximately 23m23n-m if mn, or 23n23m-n if m>n.
To form the orthogonal matrix Q f08chf (dgerqf) may be followed by a call to f08cjf (dorgrq):
Call dorgrq(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 f08chf (dgerqf). When mn, it is often only the first m rows of Q that are required and they may be formed by the call:
Call dorgrq(m,n,m,a,lda,tau,work,lwork,info)
To apply Q to an arbitrary real rectangular matrix C, f08chf (dgerqf) may be followed by a call to f08ckf (dormrq). For example:
Call dormrq('Left','Transpose',n,p,min(m,n),a,lda,tau,c,ldc, &
forms C=QTC, where C is n by p.
The complex analogue of this routine is f08cvf (zgerqf).


This example finds the minimum norm solution to the underdetermined equations
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 1.63 -3.52 0.45 .  
The solution is obtained by first obtaining an RQ factorization of the matrix A.
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.

Program Text

Program Text (f08chfe.f90)

Program Data

Program Data (f08chfe.d)

Program Results

Program Results (f08chfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017