F08BHF (DTZRZF) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F08BHF (DTZRZF) reduces the m by n (mn) real upper trapezoidal matrix A to upper triangular form by means of orthogonal transformations.

2  Specification

REAL (KIND=nag_wp)  A(LDA,*), TAU(*), WORK(max(1,LWORK))
The routine may be called by its LAPACK name dtzrzf.

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  Parameters

1:     M – INTEGERInput
On entry: m, the number of rows of the matrix A.
Constraint: M0.
2:     N – INTEGERInput
On entry: n, the number of columns of the matrix A.
Constraint: N0.
3:     A(LDA,*) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array A must be at least max1,N.
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).
4:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F08BHF (DTZRZF) is called.
Constraint: LDAmax1,M.
5:     TAU(*) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the array TAU must be at least max1,M.
On exit: the scalar factors of the elementary reflectors.
6:     WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if INFO=0, WORK1 contains the minimum value of LWORK required for optimal performance.
7:     LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08BHF (DTZRZF) is called.
If LWORK=-1, a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued.
Suggested value: for optimal performance, LWORKM×nb, where nb is the optimal block size.
Constraint: LWORKmax1,M or LWORK=-1.
8:     INFO – INTEGEROutput
On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

Errors or warnings detected by the routine:
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

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 routine is F08BVF (ZTZRZF).

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.
Note that the block size (NB) of 64 assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

9.1  Program Text

Program Text (f08bhfe.f90)

9.2  Program Data

Program Data (f08bhfe.d)

9.3  Program Results

Program Results (f08bhfe.r)

F08BHF (DTZRZF) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

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