NAG CL Interface
f08bqc (ztpmqrt)

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1 Purpose

f08bqc multiplies an arbitrary complex matrix C by the complex unitary matrix Q from a QR factorization computed by f08bpc.

2 Specification

#include <nag.h>
void  f08bqc (Nag_OrderType order, Nag_SideType side, Nag_TransType trans, Integer m, Integer n, Integer k, Integer l, Integer nb, const Complex v[], Integer pdv, const Complex t[], Integer pdt, Complex c1[], Integer pdc1, Complex c2[], Integer pdc2, NagError *fail)
The function may be called by the names: f08bqc, nag_lapackeig_ztpmqrt or nag_ztpmqrt.

3 Description

f08bqc is intended to be used after a call to f08bpc which performs a QR factorization of a triangular-pentagonal matrix containing an upper triangular matrix A over a pentagonal matrix B. The unitary matrix Q is represented as a product of elementary reflectors.
This function may be used to form the matrix products
QC , QHC , CQ ​ or ​ CQH ,  
where the complex rectangular mc×nc matrix C is split into component matrices C1 and C2.
If Q is being applied from the left (QC or QHC) then
C = ( C1 C2 )  
where C1 is k×nc, C2 is mv×nc, mc=k+mv is fixed and mv is the number of rows of the matrix V containing the elementary reflectors (i.e., m as passed to f08bpc); the number of columns of V is nv (i.e., n as passed to f08bpc).
If Q is being applied from the right (CQ or CQH) then
C = ( C1 C2 )  
where C1 is mc×k, and C2 is mc×mv and nc=k+mv is fixed.
The matrices C1 and C2 are overwriten by the result of the matrix product.
A common application of this routine is in updating the solution of a linear least squares problem as illustrated in Section 10 in f08bpc.

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: the number of rows of the matrix C2, that is,
if side=Nag_LeftSide
then mv, the number of rows of the matrix V;
if side=Nag_RightSide
then mc, the number of rows of the matrix C.
Constraint: m0.
5: n Integer Input
On entry: the number of columns of the matrix C2, that is,
if side=Nag_LeftSide
then nc, the number of columns of the matrix C;
if side=Nag_RightSide
then nv, the number of columns of the matrix V.
Constraint: n0.
6: k Integer Input
On entry: k, the number of elementary reflectors whose product defines the matrix Q.
Constraint: k0.
7: l Integer Input
On entry: l, the number of rows of the upper trapezoidal part of the pentagonal composite matrix V, passed (as b) in a previous call to f08bpc. This must be the same value used in the previous call to f08bpc (see l in f08bpc).
Constraint: 0lk.
8: nb Integer Input
On entry: nb, the blocking factor used in a previous call to f08bpc to compute the QR factorization of a triangular-pentagonal matrix containing composite matrices A and B.
Constraints:
  • nb1;
  • if k>0, nbk.
9: 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.
The (i,j)th element of the matrix V is stored in
  • v[(j-1)×pdv+i-1] when order=Nag_ColMajor;
  • v[(i-1)×pdv+j-1] when order=Nag_RowMajor.
On entry: the mv×nv matrix V; this should remain unchanged from the array b returned by a previous call to f08bpc.
10: 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).
11: 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: this must remain unchanged from a previous call to f08bpc (see t in f08bpc).
12: 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).
13: c1[dim] Complex Input/Output
Note: the dimension, dim, of the array c1 must be at least
  • max(1,pdc1×n) when side=Nag_LeftSide and order=Nag_ColMajor;
  • max(1,k×pdc1) when side=Nag_LeftSide and order=Nag_RowMajor;
  • max(1,pdc1×k) when side=Nag_RightSide and order=Nag_ColMajor;
  • max(1,m×pdc1) when side=Nag_RightSide and order=Nag_RowMajor.
On entry: C1, the first part of the composite matrix C:
if side=Nag_LeftSide
then c1 contains the first k rows of C;
if side=Nag_RightSide
then c1 contains the first k columns of C.
On exit: c1 is overwritten by the corresponding block of QC or QHC or CQ or CQH.
14: pdc1 Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array c1.
Constraints:
  • if order=Nag_ColMajor,
    • if side=Nag_LeftSide, pdc1 max(1,k) ;
    • if side=Nag_RightSide, pdc1 max(1,m) ;
  • if order=Nag_RowMajor,
    • if side=Nag_LeftSide, pdc1max(1,n);
    • if side=Nag_RightSide, pdc1max(1,k).
15: c2[dim] Complex Input/Output
Note: the dimension, dim, of the array c2 must be at least
  • max(1,pdc2×n) when order=Nag_ColMajor;
  • max(1,m×pdc2) when order=Nag_RowMajor.
On entry: C2, the second part of the composite matrix C.
if side=Nag_LeftSide
then c2 contains the remaining mv rows of C;
if side=Nag_RightSide
then c2 contains the remaining mv columns of C;
On exit: c2 is overwritten by the corresponding block of QC or QHC or CQ or CQH.
16: pdc2 Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array c2.
Constraints:
  • if order=Nag_ColMajor, pdc2 max(1,m) ;
  • if order=Nag_RowMajor, pdc2max(1,n).
17: 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, k=value, m=value and pdc1=value.
Constraint: if side=Nag_LeftSide, pdc1 max(1,k) ;
if side=Nag_RightSide, pdc1 max(1,m) .
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) .
On entry, side=value, pdc1=value, n=value and k=value.
Constraint: if side=Nag_LeftSide, pdc1max(1,n);
if side=Nag_RightSide, pdc1max(1,k).
NE_INT
On entry, k=value.
Constraint: k0.
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
NE_INT_2
On entry, l=value and k=value.
Constraint: 0lk.
On entry, m=value and pdc2=value.
Constraint: pdc2 max(1,m) .
On entry, nb=value and k=value.
Constraint: nb1 and
if k>0, nbk.
On entry, pdc2=value and n=value.
Constraint: pdc2max(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

f08bqc 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 2nk (2m-k) if side=Nag_LeftSide and 2mk (2n-k) if side=Nag_RightSide.
The real analogue of this function is f08bcc.

10 Example

See f08bpc.