nag_dgeqpf (f08bec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dgeqpf (f08bec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgeqpf (f08bec) computes the QR factorization, with column pivoting, of a real m by n matrix.

2  Specification

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

3  Description

nag_dgeqpf (f08bec) forms the QR factorization, with column pivoting, of an arbitrary rectangular real m by n matrix.
If mn, the factorization is given by:
AP= Q R 0 ,
where R is an n by n upper triangular matrix, Q is an m by m orthogonal matrix and P is an n by n permutation matrix. It is sometimes more convenient to write the factorization as
AP= Q1 Q2 R 0 ,
which reduces to
AP= Q1 R ,
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
AP= 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 permuted matrix AP.
The function allows specified columns of A to be moved to the leading columns of AP at the start of the factorization and fixed there. The remaining columns are free to be interchanged so that at the ith stage the pivot column is chosen to be the column which maximizes the 2-norm of elements i to m over columns i to n.

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:     jpvt[dim]IntegerInput/Output
Note: the dimension, dim, of the array jpvt must be at least max1,n.
On entry: if jpvt[i-1]0, then the i th column of A is moved to the beginning of AP before the decomposition is computed and is fixed in place during the computation. Otherwise, the i th column of A is a free column (i.e., one which may be interchanged during the computation with any other free column).
On exit: details of the permutation matrix P. More precisely, if jpvt[i-1]=k, then the kth column of A is moved to become the i th column of AP; in other words, the columns of AP are the columns of A in the order jpvt[0],jpvt[1],,jpvt[n-1].
7:     tau[minm,n]doubleOutput
On exit: further details of the orthogonal matrix Q.
8:     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_dgeqpf (f08bec) 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_dgeqpf (f08bec).
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_dgeqpf (f08bec) 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 without column pivoting, use nag_dgeqrf (f08aec).
The complex analogue of this function is nag_zgeqpf (f08bsc).

9  Example

This example finds the basic solutions for the linear least squares problems
minimize Axi - bi 2 ,   i=1,2
where b1 and b2 are the columns of the matrix B,
A = -0.09 0.14 -0.46 0.68 1.29 -1.56 0.20 0.29 1.09 0.51 -1.48 -0.43 0.89 -0.71 -0.96 -1.09 0.84 0.77 2.11 -1.27 0.08 0.55 -1.13 0.14 1.74 -1.59 -0.72 1.06 1.24 0.34   and  B= -0.01 -0.04 0.04 -0.03 0.05 0.01 -0.03 -0.02 0.02 0.05 -0.06 0.07 .
Here A is approximately rank-deficient, and hence it is preferable to use nag_dgeqpf (f08bec) rather than nag_dgeqrf (f08aec).

9.1  Program Text

Program Text (f08bece.c)

9.2  Program Data

Program Data (f08bece.d)

9.3  Program Results

Program Results (f08bece.r)


nag_dgeqpf (f08bec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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