nag_ztzrzf (f08bvc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_ztzrzf (f08bvc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_ztzrzf (f08bvc) reduces the m by n (mn) complex upper trapezoidal matrix A to upper triangular form by means of unitary transformations.

2  Specification

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

3  Description

The m by n (mn) complex upper trapezoidal matrix A given by
A = R1 R2 ,
where R1 is an m by m upper triangular matrix and R2 is an m by n-m matrix, is factorized as
A = R 0 Z ,
where R is also an m by m upper triangular matrix and Z is an n by n unitary matrix.

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

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.
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 leading m by n upper trapezoidal part of the array a must contain the matrix to be factorized.
On exit: the leading m by m upper triangular part of a contains the upper triangular matrix R, and elements m+1 to n of the first m rows of a, with the array tau, represent the unitary matrix Z as a product of m 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,m.
On exit: the scalar factors of the elementary reflectors.
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_ztzrzf (f08bvc) is not threaded by NAG in any implementation.
nag_ztzrzf (f08bvc) 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 16m2n-m.
The real analogue of this function is nag_dtzrzf (f08bhc).

10  Example

This example solves the linear least squares problems
minx bj - Axj 2 ,   j=1,2
for the minimum norm solutions x1 and x2, where bj is the jth column of the matrix B,
A = 0.47-0.34i -0.40+0.54i 0.60+0.01i 0.80-1.02i -0.32-0.23i -0.05+0.20i -0.26-0.44i -0.43+0.17i 0.35-0.60i -0.52-0.34i 0.87-0.11i -0.34-0.09i 0.89+0.71i -0.45-0.45i -0.02-0.57i 1.14-0.78i -0.19+0.06i 0.11-0.85i 1.44+0.80i 0.07+1.14i
B = -1.08-2.59i 2.22+2.35i -2.61-1.49i 1.62-1.48i 3.13-3.61i 1.65+3.43i 7.33-8.01i -0.98+3.08i 9.12+7.63i -2.84+2.78i .
The solution is obtained by first obtaining a QR factorization with column pivoting of the matrix A, and then the RZ factorization of the leading k by k part of R is computed, where k is the estimated rank of A. A tolerance of 0.01 is used to estimate the rank of A from the upper triangular factor, R.

10.1  Program Text

Program Text (f08bvce.c)

10.2  Program Data

Program Data (f08bvce.d)

10.3  Program Results

Program Results (f08bvce.r)

nag_ztzrzf (f08bvc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

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