NAG FL Interface
f01qgf (real_​trapez_​rq)

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1 Purpose

f01qgf reduces the m×n (mn) real upper trapezoidal matrix A to upper triangular form by means of orthogonal transformations.

2 Specification

Fortran Interface
Subroutine f01qgf ( m, n, a, lda, zeta, ifail)
Integer, Intent (In) :: m, n, lda
Integer, Intent (Inout) :: ifail
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*)
Real (Kind=nag_wp), Intent (Out) :: zeta(m)
C Header Interface
#include <nag.h>
void  f01qgf_ (const Integer *m, const Integer *n, double a[], const Integer *lda, double zeta[], Integer *ifail)
The routine may be called by the names f01qgf or nagf_matop_real_trapez_rq.

3 Description

The m×n (mn) real upper trapezoidal matrix A given by
A= ( U X ) ,  
where U is an m×m upper triangular matrix, is factorized as
A=( R 0 ) PT,  
where P is an n×n orthogonal matrix and R is an m×m upper triangular matrix.
P is given as a sequence of Householder transformation matrices
the (m-k+1)th transformation matrix, Pk, being used to introduce zeros into the kth row of A. Pk has the form
Pk=( I 0 0 Tk ) ,  
Tk=I-ukukT, uk=( ζk 0 zk ),  
ζk is a scalar and zk is an (n-m) element vector. ζk and zk are chosen to annihilate the elements of the kth row of X.
The vector uk is returned in the kth element of the array zeta and in the kth row of a, such that ζk is in zeta(k) and the elements of zk are in a(k,m+1),,a(k,n). The elements of R are returned in the upper triangular part of a.
For further information on this factorization and its use see Section 6.5 of Golub and Van Loan (1996).

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford

5 Arguments

1: m Integer Input
On entry: m, the number of rows of the matrix A.
When m=0 then an immediate return is effected.
Constraint: m0.
2: n Integer Input
On entry: n, the number of columns of the matrix A.
Constraint: nm.
3: a(lda,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,n).
On entry: the leading m×n upper trapezoidal part of the array a must contain the matrix to be factorized.
On exit: the m×m upper triangular part of a will contain the upper triangular matrix R, and the m×(n-m) upper trapezoidal part of a will contain details of the factorization as described in Section 3.
4: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f01qgf is called.
Constraint: ldamax(1,m).
5: zeta(m) Real (Kind=nag_wp) array Output
On exit: zeta(k) contains the scalar ζk for the (m-k+1)th transformation. If Tk=I then zeta(k)=0.0, otherwise zeta(k) contains ζk as described in Section 3 and ζk is always in the range (1.0,2.0).
6: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 1 to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of 0 causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of −1 means that an error message is printed while a value of 1 means that it is not.
If halting is not appropriate, the value −1 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value 0 is recommended. When the value -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, lda=value and m=value.
Constraint: ldam.
On entry, m=value.
Constraint: m0.
On entry, n=value and m=value.
Constraint: nm.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The computed factors R and P satisfy the relation
Ecε A,  
ε is the machine precision (see x02ajf), c is a modest function of m and n and . denotes the spectral (two) norm.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f01qgf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The approximate number of floating-point operations is given by 2×m2(n-m).

10 Example

This example reduces the 3×5 matrix
A=( 2.4 0.8 -1.4 3.0 -0.8 0.0 1.6 0.8 0.4 -0.8 0.0 0.0 1.0 2.0 2.0 )  
to upper triangular form.

10.1 Program Text

Program Text (f01qgfe.f90)

10.2 Program Data

Program Data (f01qgfe.d)

10.3 Program Results

Program Results (f01qgfe.r)