nag_zgeqrf (f08asc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zgeqrf (f08asc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zgeqrf (f08asc) computes the QR factorization of a complex m by n matrix.

2  Specification

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

3  Description

nag_zgeqrf (f08asc) forms the QR factorization of an arbitrary rectangular complex 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 (with real diagonal elements) and Q is an m by m unitary 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]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.
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 unitary 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 unitary matrix Q and the remaining elements are overwritten by the corresponding elements of the m by n upper trapezoidal matrix R.
The diagonal elements of R are real.
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]ComplexOutput
Note: the dimension, dim, of the array tau must be at least max1,minm,n.
On exit: further details of the unitary 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 real floating point operations is approximately 83 n2 3m-n  if mn or 83 m2 3n-m  if m<n.
To form the unitary matrix Q nag_zgeqrf (f08asc) may be followed by a call to nag_zungqr (f08atc):
nag_zungqr(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_zgeqrf (f08asc).
When mn, it is often only the first n columns of Q that are required, and they may be formed by the call:
nag_zungqr(order,m,n,n,&a,pda,tau,&fail)
To apply Q to an arbitrary complex rectangular matrix C, nag_zgeqrf (f08asc) may be followed by a call to nag_zunmqr (f08auc). For example,
nag_zunmqr(order,Nag_LeftSide,Nag_ConjTrans,m,p,MIN(m,n),&a,pda,
  tau,&c,pdc,&fail)
forms C=QHC, where C is m by p.
To compute a QR factorization with column pivoting, use nag_zgeqpf (f08bsc).
The real analogue of this function is nag_dgeqrf (f08aec).

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 = 0.96-0.81i -0.03+0.96i -0.91+2.06i -0.05+0.41i -0.98+1.98i -1.20+0.19i -0.66+0.42i -0.81+0.56i 0.62-0.46i 1.01+0.02i 0.63-0.17i -1.11+0.60i -0.37+0.38i 0.19-0.54i -0.98-0.36i 0.22-0.20i 0.83+0.51i 0.20+0.01i -0.17-0.46i 1.47+1.59i 1.08-0.28i 0.20-0.12i -0.07+1.23i 0.26+0.26i
and
B = -1.54+0.76i 3.17-2.09i 0.12-1.92i -6.53+4.18i -9.08-4.31i 7.28+0.73i 7.49+3.65i 0.91-3.97i -5.63-2.12i -5.46-1.64i 2.37+8.03i -2.84-5.86i .

9.1  Program Text

Program Text (f08asce.c)

9.2  Program Data

Program Data (f08asce.d)

9.3  Program Results

Program Results (f08asce.r)


nag_zgeqrf (f08asc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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