NAG Library Function Document

nag_ztpmqrt (f08bqc)

 Contents

    1  Purpose
    7  Accuracy
    10  Example

1
Purpose

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

2
Specification

#include <nag.h>
#include <nagf08.h>
void  nag_ztpmqrt (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)

3
Description

nag_ztpmqrt (f08bqc) is intended to be used after a call to nag_ztpqrt (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 by 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 by nc, C2 is mv by 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 nag_ztpqrt (f08bpc)); the number of columns of V is nv (i.e., n as passed to nag_ztpqrt (f08bpc)).
If Q is being applied from the right (CQ or CQH) then
C = C1 C2  
where C1 is mc by k, and C2 is mc by 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 nag_ztpqrt (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_OrderTypeInput
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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     side Nag_SideTypeInput
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_TransTypeInput
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 IntegerInput
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 IntegerInput
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 IntegerInput
On entry: k, the number of elementary reflectors whose product defines the matrix Q.
Constraint: k0.
7:     l IntegerInput
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 nag_ztpqrt (f08bpc). This must be the same value used in the previous call to nag_ztpqrt (f08bpc) (see l in nag_ztpqrt (f08bpc)).
Constraint: 0lk.
8:     nb IntegerInput
On entry: nb, the blocking factor used in a previous call to nag_ztpqrt (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 ComplexInput
Note: the dimension, dim, of the array v must be at least
  • max1,pdv×k when order=Nag_ColMajor;
  • max1,m×pdv when order=Nag_RowMajor and side=Nag_LeftSide;
  • max1,n×pdv when order=Nag_RowMajor and side=Nag_RightSide.
The i,jth 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 by nv matrix V; this should remain unchanged from the array b returned by a previous call to nag_ztpqrt (f08bpc).
10:   pdv IntegerInput
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 max1,m ;
    • if side=Nag_RightSide, pdv max1,n ;
  • if order=Nag_RowMajor, pdvmax1,k.
11:   t[dim] const ComplexInput
Note: the dimension, dim, of the array t must be at least
  • max1,pdt×k when order=Nag_ColMajor;
  • max1,nb×pdt when order=Nag_RowMajor.
The i,jth 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 nag_ztpqrt (f08bpc) (see t in nag_ztpqrt (f08bpc)).
12:   pdt IntegerInput
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, pdtmax1,k.
13:   c1[dim] ComplexInput/Output
Note: the dimension, dim, of the array c1 must be at least
  • max1,pdc1×n when side=Nag_LeftSide and order=Nag_ColMajor;
  • max1,k×pdc1 when side=Nag_LeftSide and order=Nag_RowMajor;
  • max1,pdc1×k when side=Nag_RightSide and order=Nag_ColMajor;
  • max1,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 IntegerInput
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 max1,k ;
    • if side=Nag_RightSide, pdc1 max1,m ;
  • if order=Nag_RowMajor,
    • if side=Nag_LeftSide, pdc1max1,n;
    • if side=Nag_RightSide, pdc1max1,k.
15:   c2[dim] ComplexInput/Output
Note: the dimension, dim, of the array c2 must be at least
  • max1,pdc2×n when order=Nag_ColMajor;
  • max1,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 IntegerInput
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 max1,m ;
  • if order=Nag_RowMajor, pdc2max1,n.
17:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 max1,k ;
if side=Nag_RightSide, pdc1 max1,m .
On entry, side=value, m=value, n=value and pdv=value.
Constraint: if side=Nag_LeftSide, pdv max1,m ;
if side=Nag_RightSide, pdv max1,n .
On entry, side=value, pdc1=value, n=value and k=value.
Constraint: if side=Nag_LeftSide, pdc1max1,n;
if side=Nag_RightSide, pdc1max1,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 max1,m .
On entry, nb=value and k=value.
Constraint: nb1 and
if k>0, nbk.
On entry, pdc2=value and n=value.
Constraint: pdc2max1,n.
On entry, pdt=value and k=value.
Constraint: pdtmax1,k.
On entry, pdt=value and nb=value.
Constraint: pdtnb.
On entry, pdv=value and k=value.
Constraint: pdvmax1,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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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

nag_ztpmqrt (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 nag_dtpmqrt (f08bcc).

10
Example

See Section 10 in nag_ztpqrt (f08bpc).
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017