nag_zgeqlf (f08csc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_zgeqlf (f08csc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zgeqlf (f08csc) computes a QL factorization of a complex m by n matrix A.

2  Specification

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

3  Description

nag_zgeqlf (f08csc) forms the QL factorization of an arbitrary rectangular complex 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 unitary 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 unitary 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 9).
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
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 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.
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 unitary 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.
  • 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: the scalar factors of the elementary reflectors (see Section 9).
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_zgeqlf (f08csc) is not threaded by NAG in any implementation.
nag_zgeqlf (f08csc) 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 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_zgeqlf (f08csc) may be followed by a call to nag_zungql (f08ctc):
but note that the second dimension of the array a must be at least m, which may be larger than was required by nag_zgeqlf (f08csc).
When mn, it is often only the first n columns of Q that are required, and they may be formed by the call:
To apply Q to an arbitrary complex rectangular matrix C, nag_zgeqlf (f08csc) may be followed by a call to nag_zunmql (f08cuc). For example,
forms C=QHC, where C is m by p.
The real analogue of this function is nag_dgeqlf (f08cec).

10  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 = 0.96-0.81i -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
B = -2.09+1.93i 3.26-2.70i 3.34-3.53i -6.22+1.16i -4.94-2.04i 7.94-3.13i 0.17+4.23i 1.04-4.26i -5.19+3.63i -2.31-2.12i 0.98+2.53i -1.39-4.05i .
The solution is obtained by first obtaining a QL factorization of the matrix A.

10.1  Program Text

Program Text (f08csce.c)

10.2  Program Data

Program Data (f08csce.d)

10.3  Program Results

Program Results (f08csce.r)

nag_zgeqlf (f08csc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

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