nag_zunglq (f08awc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zunglq (f08awc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zunglq (f08awc) generates all or part of the complex unitary matrix Q from an LQ factorization computed by nag_zgelqf (f08avc).

2  Specification

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

3  Description

nag_zunglq (f08awc) is intended to be used after a call to nag_zgelqf (f08avc), which performs an LQ factorization of a complex matrix A. The unitary matrix Q is represented as a product of elementary reflectors.
This function may be used to generate Q explicitly as a square matrix, or to form only its leading rows.
Usually Q is determined from the LQ factorization of a p by n matrix A with pn. The whole of Q may be computed by:
nag_zunglq(order,n,n,p,&a,pda,tau,&fail)
(note that the array a must have at least n rows) or its leading p rows by:
nag_zunglq(order,p,n,p,&a,pda,tau,&fail)
The rows of Q returned by the last call form an orthonormal basis for the space spanned by the rows of A; thus nag_zgelqf (f08avc) followed by nag_zunglq (f08awc) can be used to orthogonalise the rows of A.
The information returned by the LQ factorization functions also yields the LQ factorization of the leading k rows of A, where k<p. The unitary matrix arising from this factorization can be computed by:
nag_zunglq(order,n,n,k,&a,pda,tau,&fail)
or its leading k rows by:
nag_zunglq(order,k,n,k,&a,pda,tau,&fail)

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 Q.
Constraint: m0.
3:     nIntegerInput
On entry: n, the number of columns of the matrix Q.
Constraint: nm.
4:     kIntegerInput
On entry: k, the number of elementary reflectors whose product defines the matrix Q.
Constraint: mk0.
5:     a[dim]ComplexInput/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.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_zgelqf (f08avc).
On exit: the m by n matrix Q.
If order=Nag_ColMajor, the i,jth element of the matrix is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, the i,jth element of the matrix is stored in a[i-1×pda+j-1].
6:     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.
7:     tau[dim]const ComplexInput
Note: the dimension, dim, of the array tau must be at least max1,k.
On entry: further details of the elementary reflectors as returned by nag_zgelqf (f08avc).
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, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, m=value and k=value.
Constraint: mk0.
On entry, n=value and m=value.
Constraint: nm.
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 matrix Q differs from an exactly unitary matrix by a matrix E such that
E2 = Oε ,
where ε is the machine precision.

8  Further Comments

The total number of real floating point operations is approximately 16mnk-8 m+n k2 + 163 k3 ; when m=k, the number is approximately 83 m2 3n-m .
The real analogue of this function is nag_dorglq (f08ajc).

9  Example

This example forms the leading 4 rows of the unitary matrix Q from the LQ factorization of the matrix A, where
A = 0.28-0.36i 0.50-0.86i -0.77-0.48i 1.58+0.66i -0.50-1.10i -1.21+0.76i -0.32-0.24i -0.27-1.15i 0.36-0.51i -0.07+1.33i -0.75+0.47i -0.08+1.01i .
The rows of Q form an orthonormal basis for the space spanned by the rows of A.

9.1  Program Text

Program Text (f08awce.c)

9.2  Program Data

Program Data (f08awce.d)

9.3  Program Results

Program Results (f08awce.r)


nag_zunglq (f08awc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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