NAG C Library Function Document

nag_dgerqf (f08chc)


nag_dgerqf (f08chc) computes an RQ factorization of a real m by n matrix A.


#include <nag.h>
#include <nagf08.h>
void  nag_dgerqf (Nag_OrderType order, Integer m, Integer n, double a[], Integer pda, double tau[], NagError *fail)


nag_dgerqf (f08chc) 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). Functions 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:     order Nag_OrderTypeInput
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 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     m IntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
3:     n IntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: n0.
4:     a[dim] doubleInput/Output
Note: the dimension, dim, of the array a must be at least
  • max1,pda×n when order=Nag_ColMajor;
  • max1,m×pda when order=Nag_RowMajor.
Where Ai,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 by n matrix A.
On exit: if mn, the upper triangle of the subarray A1:m,n-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).
5:     pda IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
  • if order=Nag_ColMajor, pdamax1,m;
  • if order=Nag_RowMajor, pdamax1,n.
6:     tau[dim] doubleOutput
Note: the dimension, dim, of the array tau must be at least max1,minm,n.
On exit: the scalar factors of the elementary reflectors.
7:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation 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: pdamax1,m.
On entry, pda=value and n=value.
Constraint: pdamax1,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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.


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

nag_dgerqf (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.

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 nag_dgerqf (f08chc) may be followed by a call to nag_dorgrq (f08cjc):
nag_dorgrq(order, n, n, minmn, a, pda, tau, &fail)
where minmn=minm,n, but note that the first dimension of the array a must be at least n, which may be larger than was required by nag_dgerqf (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_dorgrq(order, m, n, m, a, pda, tau, c, pdc, &fail)
To apply Q to an arbitrary real rectangular matrix C, nag_dgerqf (f08chc) may be followed by a call to nag_dormrq (f08ckc). For example:
nag_dormrq(Nag_LeftSide, Nag_Trans, n, p, minmn, a, pda, tau, c, pdc, &fail)
forms C=QTC, where C is n by p.
The complex analogue of this function is nag_zgerqf (f08cvc).


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.

Program Text

Program Text (f08chce.c)

Program Data

Program Data (f08chce.d)

Program Results

Program Results (f08chce.r)