NAG CL Interface
f08cwc (zungrq)

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1 Purpose

f08cwc generates all or part of the complex n×n unitary matrix Q from an RQ factorization computed by f08cvc.

2 Specification

#include <nag.h>
void  f08cwc (Nag_OrderType order, Integer m, Integer n, Integer k, Complex a[], Integer pda, const Complex tau[], NagError *fail)
The function may be called by the names: f08cwc, nag_lapackeig_zungrq or nag_zungrq.

3 Description

f08cwc is intended to be used following a call to f08cvc, which performs an RQ factorization of a complex matrix A and represents the unitary matrix Q as a product of k elementary reflectors of order n.
This function may be used to generate Q explicitly as a square matrix, or to form only its trailing rows.
Usually Q is determined from the RQ factorization of a p×n matrix A with pn. The whole of Q may be computed by :
(note that the matrix A must have at least n rows) or its trailing p rows by :
The rows of Q returned by the last call form an orthonormal basis for the space spanned by the rows of A; thus f08cvc followed by f08cwc can be used to orthogonalize the rows of A.
The information returned by f08cvc also yields the RQ factorization of the trailing k rows of A, where k<p. The unitary matrix arising from this factorization can be computed by :
or its leading k columns by :

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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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 Q.
Constraint: m0.
3: n Integer Input
On entry: n, the number of columns of the matrix Q.
Constraint: nm.
4: k Integer Input
On entry: k, the number of elementary reflectors whose product defines the matrix Q.
Constraint: mk0.
5: a[dim] Complex 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.
On entry: details of the vectors which define the elementary reflectors, as returned by f08cvc.
On exit: the m×n matrix Q.
6: pda Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
  • if order=Nag_ColMajor, pdamax(1,m);
  • if order=Nag_RowMajor, pdamax(1,n).
7: tau[dim] const Complex Input
Note: the dimension, dim, of the array tau must be at least max(1,k).
On entry: tau[i-1] must contain the scalar factor of the elementary reflector Hi, as returned by f08cvc.
8: 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

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, m=value.
Constraint: m0.
On entry, pda=value.
Constraint: pda>0.
On entry, m=value and k=value.
Constraint: mk0.
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).
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.
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 matrix Q differs from an exactly unitary matrix by a matrix E such that
E2 = Oε  
and ε is the machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08cwc 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 16mnk-8(m+n)k2+163k3; when m=k this becomes 83m2(3n-m).
The real analogue of this function is f08cjc.

10 Example

This example generates the first four rows of the matrix Q of the RQ factorization of A as returned by f08cvc, where
A = ( 0.96-0.81i -0.98+1.98i 0.62-0.46i -0.37+0.38i 0.83+0.51i 1.08-0.28i -0.03+0.96i -1.20+0.19i 1.01+0.02i 0.19-0.54i 0.20+0.01i 0.20-0.12i -0.91+2.06i -0.66+0.42i 0.63-0.17i -0.98-0.36i -0.17-0.46i -0.07+1.23i -0.05+0.41i -0.81+0.56i -1.11+0.60i 0.22-0.20i 1.47+1.59i 0.26+0.26i ) .  

10.1 Program Text

Program Text (f08cwce.c)

10.2 Program Data

Program Data (f08cwce.d)

10.3 Program Results

Program Results (f08cwce.r)