nag_dtzrzf (f08bhc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dtzrzf (f08bhc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dtzrzf (f08bhc) reduces the m by n (mn) real upper trapezoidal matrix A to upper triangular form by means of orthogonal transformations.

2  Specification

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

3  Description

The m by n (mn) real 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 orthogonal 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 http://www.netlib.org/lapack/lug

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]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 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 orthogonal 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.
Constraints:
  • 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,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

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 floating point operations is approximately 4m2n-m.
The complex analogue of this function is nag_ztzrzf (f08bvc).

9  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 = -0.09 0.14 -0.46 0.68 1.29 -1.56 0.20 0.29 1.09 0.51 -1.48 -0.43 0.89 -0.71 -0.96 -1.09 0.84 0.77 2.11 -1.27 0.08 0.55 -1.13 0.14 1.74 -1.59 -0.72 1.06 1.24 0.34   and   B= 7.4 2.7 4.2 -3.0 -8.3 -9.6 1.8 1.1 8.6 4.0 2.1 -5.7 .
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.

9.1  Program Text

Program Text (f08bhce.c)

9.2  Program Data

Program Data (f08bhce.d)

9.3  Program Results

Program Results (f08bhce.r)


nag_dtzrzf (f08bhc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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