NAG CL Interface
f08bhc (dtzrzf)

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1 Purpose

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

2 Specification

#include <nag.h>
void  f08bhc (Nag_OrderType order, Integer m, Integer n, double a[], Integer pda, double tau[], NagError *fail)
The function may be called by the names: f08bhc, nag_lapackeig_dtzrzf or nag_dtzrzf.

3 Description

The m×n (mn) real upper trapezoidal matrix A given by
A = ( R1 R2 ) ,  
where R1 is an m×m upper triangular matrix and R2 is an m×(n-m) matrix, is factorized as
A = ( R 0 ) Z ,  
where R is also an m×m upper triangular matrix and Z is an n×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 https://www.netlib.org/lapack/lug

5 Arguments

1: order Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: m Integer Input
On entry: m, the number of rows of the matrix A.
Constraint: m0.
3: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: nm.
4: a[dim] double Input/Output
Note: the dimension, dim, of the array a must be at least
  • max(1,pda×n) when order=Nag_ColMajor;
  • max(1,m×pda) when order=Nag_RowMajor.
The (i,j)th 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×n upper trapezoidal part of the array a must contain the matrix to be factorized.
On exit: the leading m×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 3.4.6 in the F08 Chapter Introduction).
5: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
  • if order=Nag_ColMajor, pdamax(1,m);
  • if order=Nag_RowMajor, pdamax(1,n).
6: tau[dim] double Output
Note: the dimension, dim, of the array tau must be at least max(1,m).
On exit: the scalar factors of the elementary reflectors.
7: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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, n=value and m=value.
Constraint: nm.
On entry, pda=value and m=value.
Constraint: pdamax(1,m).
On entry, pda=value and n=value.
Constraint: pdamax(1,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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

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

f08bhc 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 X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also 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 4m2(n-m).
The complex analogue of this function is f08bvc.

10 Example

This example solves the linear least squares problems
minx bj-Axj2 ,   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×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 (f08bhce.c)

10.2 Program Data

Program Data (f08bhce.d)

10.3 Program Results

Program Results (f08bhce.r)