NAG CL Interface
f08chc (dgerqf)

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1 Purpose

f08chc computes an RQ factorization of a real m×n matrix A.

2 Specification

#include <nag.h>
void  f08chc (Nag_OrderType order, Integer m, Integer n, double a[], Integer pda, double tau[], NagError *fail)
The function may be called by the names: f08chc, nag_lapackeig_dgerqf or nag_dgerqf.

3 Description

f08chc forms the RQ factorization of an arbitrary rectangular real m×n matrix. If mn, the factorization is given by
A = ( 0 R ) Q ,  
where R is an m×m lower triangular matrix and Q is an n×n orthogonal matrix. If m>n the factorization is given by
A =RQ ,  
where R is an m×n upper trapezoidal matrix and Q is again an n×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 min(m,n) elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with Q in this representation (see Section 9).

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

5 Arguments

1: order Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
3: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: n0.
4: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least
  • max(1,pda×n) when order=Nag_ColMajor;
  • max(1,m×pda) when order=Nag_RowMajor.
where A(i,j) appears in this document, it refers to the array element
  • a[(j-1)×pda+i-1] when order=Nag_ColMajor;
  • a[(i-1)×pda+j-1] when order=Nag_RowMajor.
On entry: the m×n matrix A.
On exit: if mn, the upper triangle of the subarray A(1:m,n-m+1:n) contains the m×m upper triangular matrix R.
If mn, the elements on and above the (m-n)th subdiagonal contain the m×n upper trapezoidal matrix R; the remaining elements, with the array tau, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Section 3.4.6 in the F08 Chapter Introduction).
5: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
  • if order=Nag_ColMajor, pdamax(1,m);
  • if order=Nag_RowMajor, pdamax(1,n).
6: tau[dim] double Output
Note: the dimension, dim, of the array tau must be at least max(1,min(m,n)).
On exit: the scalar factors of the elementary reflectors.
7: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pda=value and m=value.
Constraint: pdamax(1,m).
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

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 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08chc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of floating-point operations is approximately 23m2(3n-m) if mn, or 23n2(3m-n) if m>n.
To form the orthogonal matrix Q f08chc may be followed by a call to f08cjc :
nag_lapackeig_dorgrq(order, n, n, minmn, a, pda, tau, &fail)
where minmn=min(m,n), but note that the first dimension of the array a must be at least n, which may be larger than was required by f08chc. When mn, it is often only the first m rows of Q that are required and they may be formed by the call:
nag_lapackeig_dorgrq(order, m, n, m, a, pda, tau, c, pdc, &fail)
To apply Q to an arbitrary n×p real rectangular matrix C, f08chc may be followed by a call to f08ckc . For example:
nag_lapackeig_dormrq(Nag_LeftSide, Nag_Trans, n, p, minmn, a, pda, tau, c, pdc, &fail)
forms the matrix product C=QTC.
The complex analogue of this function is f08cvc.

10 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.

10.1 Program Text

Program Text (f08chce.c)

10.2 Program Data

Program Data (f08chce.d)

10.3 Program Results

Program Results (f08chce.r)