NAG CL Interface
f08aqc (zgemqrt)

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1 Purpose

f08aqc multiplies an arbitrary complex matrix C by the complex unitary matrix Q from a QR factorization computed by f08apc.

2 Specification

#include <nag.h>
void  f08aqc (Nag_OrderType order, Nag_SideType side, Nag_TransType trans, Integer m, Integer n, Integer k, Integer nb, const Complex v[], Integer pdv, const Complex t[], Integer pdt, Complex c[], Integer pdc, NagError *fail)
The function may be called by the names: f08aqc, nag_lapackeig_zgemqrt or nag_zgemqrt.

3 Description

f08aqc is intended to be used after a call to f08apc, which performs a QR factorization of a complex matrix A. The unitary matrix Q is represented as a product of elementary reflectors.
This function may be used to form one of the matrix products
QC , QHC , CQ ​ or ​ CQH ,  
overwriting the result on C (which may be any complex rectangular matrix).
A common application of this function is in solving linear least squares problems, as described in the F08 Chapter Introduction and illustrated in Section 10 in f08apc.

4 References

Golub G H and Van Loan C F (2012) Matrix Computations (4th 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. Usually k=min(mA,nA) where mA, nA are the dimensions of the matrix A supplied in a previous call to f08apc.
Constraints:
  • if side=Nag_LeftSide, m k 0 ;
  • if side=Nag_RightSide, n k 0 .
7: nb Integer Input
On entry: the block size used in the QR factorization performed in a previous call to f08apc; this value must remain unchanged from that call.
Constraints:
  • nb1;
  • if k>0, nbk.
8: v[dim] const Complex Input
Note: the dimension, dim, of the array v must be at least
  • max(1,pdv×k) when order=Nag_ColMajor;
  • max(1,m×pdv) when order=Nag_RowMajor and side=Nag_LeftSide;
  • max(1,n×pdv) when order=Nag_RowMajor and side=Nag_RightSide.
On entry: details of the vectors which define the elementary reflectors, as returned by f08apc in the first k columns of its array argument a.
9: pdv Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array v.
Constraints:
  • if order=Nag_ColMajor,
    • if side=Nag_LeftSide, pdv max(1,m) ;
    • if side=Nag_RightSide, pdv max(1,n) ;
  • if order=Nag_RowMajor, pdvmax(1,k).
10: t[dim] const Complex Input
Note: the dimension, dim, of the array t must be at least
  • max(1,pdt×k) when order=Nag_ColMajor;
  • max(1,nb×pdt) when order=Nag_RowMajor.
The (i,j)th element of the matrix T is stored in
  • t[(j-1)×pdt+i-1] when order=Nag_ColMajor;
  • t[(i-1)×pdt+j-1] when order=Nag_RowMajor.
On entry: further details of the unitary matrix Q as returned by f08apc. The number of blocks is b=knb, where k=min(m,n) and each block is of order nb except for the last block, which is of order k-(b-1)×nb. For the b blocks the upper triangular block reflector factors T1,T2,,Tb are stored in the nb×n matrix T as T=[T1|T2||Tb].
11: pdt Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array t.
Constraints:
  • if order=Nag_ColMajor, pdtnb;
  • if order=Nag_RowMajor, pdtmax(1,k).
12: 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.
13: 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).
14: 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, m=value, n=value and pdv=value.
Constraint: if side=Nag_LeftSide, pdv max(1,m) ;
if side=Nag_RightSide, pdv max(1,n) .
NE_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
NE_INT_2
On entry, nb=value and k=value.
Constraint: nb1 and
if k>0, nbk.
On entry, pdc=value and m=value.
Constraint: pdcmax(1,m).
On entry, pdc=value and n=value.
Constraint: pdcmax(1,n).
On entry, pdt=value and k=value.
Constraint: pdtmax(1,k).
On entry, pdt=value and nb=value.
Constraint: pdtnb.
On entry, pdv=value and k=value.
Constraint: pdvmax(1,k).
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 = O(ε) C2 ,  
where ε is the machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08aqc 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 real 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 f08acc.

10 Example

See f08apc.