NAG CL Interface
f08cxc (zunmrq)

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1 Purpose

f08cxc multiplies a general complex m×n matrix C by the complex unitary matrix Q from an RQ factorization computed by f08cvc.

2 Specification

#include <nag.h>
void  f08cxc (Nag_OrderType order, Nag_SideType side, Nag_TransType trans, Integer m, Integer n, Integer k, Complex a[], Integer pda, const Complex tau[], Complex c[], Integer pdc, NagError *fail)
The function may be called by the names: f08cxc, nag_lapackeig_zunmrq or nag_zunmrq.

3 Description

f08cxc 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 elementary reflectors.
This function may be used to form one of the matrix products
QC ,   QHC ,   CQ ,   CQH ,  
overwriting the result on C, which may be any complex rectangular m×n matrix.
A common application of this function is in solving underdetermined linear least squares problems, as described in the F08 Chapter Introduction, and illustrated in Section 10 in f08cvc.

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
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: side Nag_SideType Input
On entry: indicates how Q or QH is to be applied to C.
side=Nag_LeftSide
Q or QH is applied to C from the left.
side=Nag_RightSide
Q or QH is applied to C from the right.
Constraint: side=Nag_LeftSide or Nag_RightSide.
3: trans Nag_TransType Input
On entry: indicates whether Q or QH is to be applied to C.
trans=Nag_NoTrans
Q is applied to C.
trans=Nag_ConjTrans
QH is applied to C.
Constraint: trans=Nag_NoTrans or Nag_ConjTrans.
4: m Integer Input
On entry: m, the number of rows of the matrix C.
Constraint: m0.
5: n Integer Input
On entry: n, the number of columns of the matrix C.
Constraint: n0.
6: k Integer Input
On entry: k, the number of elementary reflectors whose product defines the matrix Q.
Constraints:
  • if side=Nag_LeftSide, m k 0 ;
  • if side=Nag_RightSide, n k 0 .
7: a[dim] Complex Input/Output
Note: the dimension, dim, of the array a must be at least
  • max(1,pda×m) when side=Nag_LeftSide and order=Nag_ColMajor;
  • max(1,k×pda) when side=Nag_LeftSide and order=Nag_RowMajor;
  • max(1,pda×n) when side=Nag_RightSide and order=Nag_ColMajor;
  • max(1,k×pda) when side=Nag_RightSide and 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 ith row of a must contain the vector which defines the elementary reflector Hi, for i=1,2,,k, as returned by f08cvc.
On exit: is modified by f08cxc but restored on exit.
8: 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,k);
  • if order=Nag_RowMajor,
    • if side=Nag_LeftSide, pdamax(1,m);
    • if side=Nag_RightSide, pdamax(1,n).
9: 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.
10: c[dim] Complex Input/Output
Note: the dimension, dim, of the array c must be at least
  • max(1,pdc×n) when order=Nag_ColMajor;
  • max(1,m×pdc) when order=Nag_RowMajor.
The (i,j)th element of the matrix C is stored in
  • c[(j-1)×pdc+i-1] when order=Nag_ColMajor;
  • c[(i-1)×pdc+j-1] when order=Nag_RowMajor.
On entry: the m×n matrix C.
On exit: c is overwritten by QC or QHC or CQ or CQH as specified by side and trans.
11: pdc Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraints:
  • if order=Nag_ColMajor, pdcmax(1,m);
  • if order=Nag_RowMajor, pdcmax(1,n).
12: 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_ENUM_INT_3
On entry, side=value, m=value, n=value and k=value.
Constraint: if side=Nag_LeftSide, m k 0 ;
if side=Nag_RightSide, n k 0 .
On entry, side=value, pda=value, m=value and n=value.
Constraint: if side=Nag_LeftSide, pdamax(1,m);
if side=Nag_RightSide, pdamax(1,n).
NE_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdc=value.
Constraint: pdc>0.
NE_INT_2
On entry, pda=value and k=value.
Constraint: pdamax(1,k).
On entry, pdc=value and m=value.
Constraint: pdcmax(1,m).
On entry, pdc=value and n=value.
Constraint: pdcmax(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 result differs from the exact result by a matrix E such that
E2 = Oε C2  
where ε is the machine precision.

8 Parallelism and Performance

f08cxc 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 8nk(2m-k) if side=Nag_LeftSide and 8mk(2n-k) if side=Nag_RightSide.
The real analogue of this function is f08ckc.

10 Example

See f08cvc.