F08CHF (DGERQF) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


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

F08CHF (DGERQF) computes an RQ factorization of a real m by n matrix A.

2  Specification

REAL (KIND=nag_wp)  A(LDA,*), TAU(*), WORK(max(1,LWORK))
The routine may be called by its LAPACK name dgerqf.

3  Description

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 8).

4  References

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 http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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,*) – 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:     WORK(max1,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).

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
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 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):
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:
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).

9  Example

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.

9.1  Program Text

Program Text (f08chfe.f90)

9.2  Program Data

Program Data (f08chfe.d)

9.3  Program Results

Program Results (f08chfe.r)

F08CHF (DGERQF) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012