nag_dgeqrf (f08aec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dgeqrf (f08aec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgeqrf (f08aec) computes the QR factorization of a real m by n matrix.

2  Specification

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

3  Description

nag_dgeqrf (f08aec) forms the QR factorization of an arbitrary rectangular real m by n matrix. No pivoting is performed.
If mn, the factorization is given by:
A = Q R 0 ,
where R is an n by n upper triangular matrix and Q is an m by m orthogonal matrix. It is sometimes more convenient to write the factorization as
A = Q1 Q2 R 0 ,
which reduces to
A = Q1R ,
where Q1 consists of the first n columns of Q, and Q2 the remaining m-n columns.
If m<n, R is trapezoidal, and the factorization can be written
A = Q R1 R2 ,
where R1 is upper triangular and R2 is rectangular.
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 8).
Note also that for any k<n, the information returned in the first k columns of the array a represents a QR factorization of the first k columns of the original matrix A.

4  References

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

5  Arguments

1:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     mIntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
3:     nIntegerInput
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.
The i,jth element of the matrix A is stored in
  • 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 elements below the diagonal are overwritten by details of the orthogonal matrix Q and the upper triangle is overwritten by the corresponding elements of the n by n upper triangular matrix R.
If m<n, the strictly lower triangular part is overwritten by details of the orthogonal matrix Q and the remaining elements are overwritten by the corresponding elements of the m by n upper trapezoidal matrix R.
5:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
  • 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: further details of the orthogonal matrix Q.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, pda=value and m=value.
Constraint: pdamax1,m.
On entry, pda=value and n=value.
Constraint: pdamax1,n.
NE_INTERNAL_ERROR
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.

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 23 n2 3m-n  if mn or 23 m2 3n-m  if m<n.
To form the orthogonal matrix Q nag_dgeqrf (f08aec) may be followed by a call to nag_dorgqr (f08afc):
nag_dorgqr(order,m,m,MIN(m,n),&a,pda,tau,&fail)
but note that the second dimension of the array a must be at least m, which may be larger than was required by nag_dgeqrf (f08aec).
When mn, it is often only the first n columns of Q that are required, and they may be formed by the call:
nag_dorgqr(order,m,n,n,&a,pda,tau,&fail)
To apply Q to an arbitrary real rectangular matrix C, nag_dgeqrf (f08aec) may be followed by a call to nag_dormqr (f08agc). For example,
nag_dormqr(order,Nag_LeftSide,Nag_Trans,m,p,MIN(m,n),&a,pda,tau,
+ &c,pdc,&fail)
forms C=QTC, where C is m by p.
To compute a QR factorization with column pivoting, use nag_dgeqpf (f08bec).
The complex analogue of this function is nag_zgeqrf (f08asc).

9  Example

This example solves the linear least squares problems
minimize Axi - bi 2 ,   i=1,2
where b1 and b2 are the columns of the matrix B,
A = -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50   and   B= -3.15 2.19 -0.11 -3.64 1.99 0.57 -2.70 8.23 0.26 -6.35 4.50 -1.48 .

9.1  Program Text

Program Text (f08aece.c)

9.2  Program Data

Program Data (f08aece.d)

9.3  Program Results

Program Results (f08aece.r)


nag_dgeqrf (f08aec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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