NAG C Library Function Document

nag_zgemqrt (f08aqc)

1
Purpose

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

2
Specification

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

3
Description

nag_zgemqrt (f08aqc) is intended to be used after a call to nag_zgeqrt (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 nag_zgeqrt (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_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: m, the number of rows of the matrix C.
Constraint: m0.
5:     n IntegerInput
On entry: n, the number of columns of the matrix C.
Constraint: n0.
6:     k IntegerInput
On entry: k, the number of elementary reflectors whose product defines the matrix Q. Usually k=minmA,nA where mA, nA are the dimensions of the matrix A supplied in a previous call to nag_zgeqrt (f08apc).
Constraints:
  • if side=Nag_LeftSide, m k 0 ;
  • if side=Nag_RightSide, n k 0 .
7:     nb IntegerInput
On entry: the block size used in the QR factorization performed in a previous call to nag_zgeqrt (f08apc); this value must remain unchanged from that call.
Constraints:
  • nb1;
  • if k>0, nbk.
8:     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.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_zgeqrt (f08apc) in the first k columns of its array argument a.
9:     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.
10:   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: further details of the unitary matrix Q as returned by nag_zgeqrt (f08apc). The number of blocks is b=knb, where k=minm,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 by n matrix T as T=T1|T2||Tb.
11:   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.
12:   c[dim] ComplexInput/Output
Note: the dimension, dim, of the array c must be at least
  • max1,pdc×n when order=Nag_ColMajor;
  • max1,m×pdc when order=Nag_RowMajor.
The i,jth 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 by n matrix C.
On exit: c is overwritten by QC or QHC or CQ or CQH as specified by side and trans.
13:   pdc IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraints:
  • if order=Nag_ColMajor, pdcmax1,m;
  • if order=Nag_RowMajor, pdcmax1,n.
14:   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, 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 max1,m ;
if side=Nag_RightSide, pdv max1,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: pdcmax1,m.
On entry, pdc=value and n=value.
Constraint: pdcmax1,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_zgemqrt (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 nag_dgemqrt (f08acc).

10
Example

See Section 10 in nag_zgeqrt (f08apc).