NAG CL Interface
f08zfc (dggrqf)

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1 Purpose

f08zfc computes a generalized RQ factorization of a real matrix pair (A,B), where A is an m×n matrix and B is a p×n matrix.

2 Specification

#include <nag.h>
void  f08zfc (Nag_OrderType order, Integer m, Integer p, Integer n, double a[], Integer pda, double taua[], double b[], Integer pdb, double taub[], NagError *fail)
The function may be called by the names: f08zfc, nag_lapackeig_dggrqf or nag_dggrqf.

3 Description

f08zfc forms the generalized RQ factorization of an m×n matrix A and a p×n matrix B
A = RQ ,   B= ZTQ ,  
where Q is an n×n orthogonal matrix, Z is a p×p orthogonal matrix and R and T are of the form
R = { n-mmm0R12() ;   if ​ mn , nm-nR11nR21() ;   if ​ m>n ,  
with R12 or R21 upper triangular,
T = { nnT11p-n0() ;   if ​ pn , pn-ppT11T12() ;   if ​ p<n ,  
with T11 upper triangular.
In particular, if B is square and nonsingular, the generalized RQ factorization of A and B implicitly gives the RQ factorization of AB-1 as
AB-1= (RT-1) ZT .  

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
Anderson E, Bai Z and Dongarra J (1992) Generalized QR factorization and its applications Linear Algebra Appl. (Volume 162–164) 243–271
Hammarling S (1987) The numerical solution of the general Gauss-Markov linear model Mathematics in Signal Processing (eds T S Durrani, J B Abbiss, J E Hudson, R N Madan, J G McWhirter and T A Moore) 441–456 Oxford University Press
Paige C C (1990) Some aspects of generalized QR factorizations . In Reliable Numerical Computation (eds M G Cox and S Hammarling) 73–91 Oxford University Press

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: p Integer Input
On entry: p, the number of rows of the matrix B.
Constraint: p0.
4: n Integer Input
On entry: n, the number of columns of the matrices A and B.
Constraint: n0.
5: 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.
where A(i,j) appears in this document, it refers to the array element
  • a[(j-1)×pda+i-1] when order=Nag_ColMajor;
  • a[(i-1)×pda+j-1] when order=Nag_RowMajor.
On entry: the m×n matrix A.
On exit: if mn, the upper triangle of the subarray A(1:m,n-m+1:n) contains the m×m upper triangular matrix R12.
If mn, the elements on and above the (m-n)th subdiagonal contain the m×n upper trapezoidal matrix R; the remaining elements, with the array taua, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Section 3.4.6 in the F08 Chapter Introduction).
6: 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).
7: taua[min(m,n)] double Output
On exit: the scalar factors of the elementary reflectors which represent the orthogonal matrix Q.
8: b[dim] double Input/Output
Note: the dimension, dim, of the array b must be at least
  • max(1,pdb×n) when order=Nag_ColMajor;
  • max(1,p×pdb) when order=Nag_RowMajor.
The (i,j)th element of the matrix B is stored in
  • b[(j-1)×pdb+i-1] when order=Nag_ColMajor;
  • b[(i-1)×pdb+j-1] when order=Nag_RowMajor.
On entry: the p×n matrix B.
On exit: the elements on and above the diagonal of the array contain the min(p,n)×n upper trapezoidal matrix T (T is upper triangular if pn); the elements below the diagonal, with the array taub, represent the orthogonal matrix Z as a product of elementary reflectors (see Section 3.4.6 in the F08 Chapter Introduction).
9: pdb Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
  • if order=Nag_ColMajor, pdbmax(1,p);
  • if order=Nag_RowMajor, pdbmax(1,n).
10: taub[min(p,n)] double Output
On exit: the scalar factors of the elementary reflectors which represent the orthogonal matrix Z.
11: 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, n=value.
Constraint: n0.
On entry, p=value.
Constraint: p0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdb=value.
Constraint: pdb>0.
NE_INT_2
On entry, pda=value and m=value.
Constraint: pdamax(1,m).
On entry, pda=value and n=value.
Constraint: pdamax(1,n).
On entry, pdb=value and n=value.
Constraint: pdbmax(1,n).
On entry, pdb=value and p=value.
Constraint: pdbmax(1,p).
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 generalized RQ factorization is the exact factorization for nearby matrices (A+E) and (B+F), where
E2 = Oε A2   and   F2= Oε B2 ,  
and ε is the machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08zfc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08zfc 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 orthogonal matrices Q and Z may be formed explicitly by calls to f08cjc and f08afc respectively. f08ckc may be used to multiply Q by another matrix and f08agc may be used to multiply Z by another matrix.
The complex analogue of this function is f08ztc.

10 Example

This example solves the least squares problem
minimize x c-Ax2   subject to   Bx=d  
where
A = ( -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 ) ,   B= ( 1 0 −1 0 0 1 0 −1 ) ,  
c = ( -1.50 -2.14 1.23 -0.54 -1.68 0.82 )   and   d= ( 0 0 ) .  
The constraints Bx=d correspond to x1=x3 and x2=x4.
The solution is obtained by first computing a generalized RQ factorization of the matrix pair (B,A). The example illustrates the general solution process.

10.1 Program Text

Program Text (f08zfce.c)

10.2 Program Data

Program Data (f08zfce.d)

10.3 Program Results

Program Results (f08zfce.r)