nag_dgeqlf (f08cec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dgeqlf (f08cec)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgeqlf (f08cec) computes a QL factorization of a real m by n matrix A.

2  Specification

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

3  Description

nag_dgeqlf (f08cec) forms the QL factorization of an arbitrary rectangular real m by n matrix.
If mn, the factorization is given by:
A = Q 0 L ,
where L is an n by n lower triangular matrix and Q is an m by m orthogonal matrix. If m<n the factorization is given by
A = QL ,
where L is an m by n lower trapezoidal matrix and Q is again an m by m orthogonal matrix. In the case where m>n the factorization can be expressed as
A = Q1 Q2 0 L = Q2 L ,
where Q1 consists of the first m-n columns of Q, and Q2 the remaining n columns.
The matrix Q is not formed explicitly but is represented as a product of minm,n elementary reflectors (see Section 3.3.6 in 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 last k columns of the array a represents a QL factorization of the last k  columns of the original matrix A.

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  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.
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 lower triangle of the subarray Am-n+1:m,1:n contains the n by n lower triangular matrix L.
If mn, the elements on and below the n-mth superdiagonal contain the m by n lower trapezoidal matrix L. The remaining elements, with the array tau, represent the orthogonal matrix Q as a product of elementary reflectors (see Section 3.3.6 in the f08 Chapter Introduction).
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: the scalar factors of the elementary reflectors (see Section 8).
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_dgeqlf (f08cec) may be followed by a call to nag_dorgql (f08cfc):
nag_dorgql(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_dgeqlf (f08cec).
When mn, it is often only the first n columns of Q that are required, and they may be formed by the call:
nag_dorgql(order,m,n,n,&a,pda,tau,&fail)
To apply Q to an arbitrary real rectangular matrix C, nag_dgeqlf (f08cec) may be followed by a call to nag_dormql (f08cgc). For example,
nag_dormql(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.
The complex analogue of this function is nag_zgeqlf (f08csc).

9  Example

This example solves the linear least squares problems
minx bj - Axj 2 , ​ j=1,2
for x1 and x2, where bj is the jth column 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= -2.67 0.41 -0.55 -3.10 3.34 -4.01 -0.77 2.76 0.48 -6.17 4.10 0.21 .
The solution is obtained by first obtaining a QL factorization of the matrix A.

9.1  Program Text

Program Text (f08cece.c)

9.2  Program Data

Program Data (f08cece.d)

9.3  Program Results

Program Results (f08cece.r)


nag_dgeqlf (f08cec) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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