nag_dgelqf (f08ahc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_dgelqf (f08ahc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgelqf (f08ahc) computes the LQ factorization of a real m by n matrix.

2  Specification

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

3  Description

nag_dgelqf (f08ahc) forms the LQ factorization of an arbitrary rectangular real m by n matrix. No pivoting is performed.
If mn, the factorization is given by:
A = L 0 Q
where L is an m by m lower triangular matrix and Q is an n by n orthogonal matrix. It is sometimes more convenient to write the factorization as
A = L 0 Q1 Q2
which reduces to
A = LQ1 ,
where Q1 consists of the first m rows of Q, and Q2 the remaining n-m rows.
If m>n, L is trapezoidal, and the factorization can be written
A = L1 L2 Q
where L1 is lower triangular and L2 is rectangular.
The LQ factorization of A is essentially the same as the QR factorization of AT, since
A = L 0 QAT= QT LT 0 .
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).
Note also that for any k<m, the information returned in the first k rows of the array a represents an LQ factorization of the first k rows of the original matrix A.

4  References


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 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 above the diagonal are overwritten by details of the orthogonal matrix Q and the lower triangle is overwritten by the corresponding elements of the m by m lower triangular matrix L.
If m>n, the strictly upper 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 lower trapezoidal matrix L.
5:     pdaIntegerInput
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: 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

Dynamic memory allocation failed.
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.

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

nag_dgelqf (f08ahc) is not threaded by NAG in any implementation.
nag_dgelqf (f08ahc) 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 Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The total number of floating-point operations is approximately 23 m2 3n-m  if mn or 23 n2 3m-n  if m>n.
To form the orthogonal matrix Q nag_dgelqf (f08ahc) may be followed by a call to nag_dorglq (f08ajc):
but note that the first dimension of the array a, specified by the argument pda, must be at least n, which may be larger than was required by nag_dgelqf (f08ahc).
When mn, it is often only the first m rows of Q that are required, and they may be formed by the call:
To apply Q to an arbitrary real rectangular matrix C, nag_dgelqf (f08ahc) may be followed by a call to nag_dormlq (f08akc). For example,
forms the matrix product C=QTC, where C is m by p.
The complex analogue of this function is nag_zgelqf (f08avc).

10  Example

This example finds the minimum norm solutions of the under-determined systems of linear equations
Ax1= b1   and   Ax2= b2
where b1 and b2 are the columns of the matrix B,
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 -5.23 1.63 0.29 -3.52 4.76 0.45 -8.41 .

10.1  Program Text

Program Text (f08ahce.c)

10.2  Program Data

Program Data (f08ahce.d)

10.3  Program Results

Program Results (f08ahce.r)

nag_dgelqf (f08ahc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014