NAG CL Interface
f08kgc (dormbr)

Settings help

CL Name Style:


1 Purpose

f08kgc multiplies an arbitrary real m×n matrix C by one of the real orthogonal matrices Q or P which were determined by f08kec when reducing a real matrix to bidiagonal form.

2 Specification

#include <nag.h>
void  f08kgc (Nag_OrderType order, Nag_VectType vect, Nag_SideType side, Nag_TransType trans, Integer m, Integer n, Integer k, const double a[], Integer pda, const double tau[], double c[], Integer pdc, NagError *fail)
The function may be called by the names: f08kgc, nag_lapackeig_dormbr or nag_dormbr.

3 Description

f08kgc is intended to be used after a call to f08kec, which reduces a real rectangular matrix A to bidiagonal form B by an orthogonal transformation: A=QBPT. f08kec represents the matrices Q and PT as products of elementary reflectors.
This function may be used to form one of the matrix products
QC , QTC , CQ , CQT , PC , PTC , CP ​ or ​ CPT ,  
overwriting the result on C (which may be any real rectangular matrix).

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

Note: in the descriptions below, r denotes the order of Q or PT: if side=Nag_LeftSide, r=m and if side=Nag_RightSide, r=n.
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: vect Nag_VectType Input
On entry: indicates whether Q or QT or P or PT is to be applied to C.
vect=Nag_ApplyQ
Q or QT is applied to C.
vect=Nag_ApplyP
P or PT is applied to C.
Constraint: vect=Nag_ApplyQ or Nag_ApplyP.
3: side Nag_SideType Input
On entry: indicates how Q or QT or P or PT is to be applied to C.
side=Nag_LeftSide
Q or QT or P or PT is applied to C from the left.
side=Nag_RightSide
Q or QT or P or PT is applied to C from the right.
Constraint: side=Nag_LeftSide or Nag_RightSide.
4: trans Nag_TransType Input
On entry: indicates whether Q or P or QT or PT is to be applied to C.
trans=Nag_NoTrans
Q or P is applied to C.
trans=Nag_Trans
QT or PT is applied to C.
Constraint: trans=Nag_NoTrans or Nag_Trans.
5: m Integer Input
On entry: m, the number of rows of the matrix C.
Constraint: m0.
6: n Integer Input
On entry: n, the number of columns of the matrix C.
Constraint: n0.
7: k Integer Input
On entry: if vect=Nag_ApplyQ, the number of columns in the original matrix A.
If vect=Nag_ApplyP, the number of rows in the original matrix A.
Constraint: k0.
8: a[dim] const double Input
Note: the dimension, dim, of the array a must be at least
  • max(1,pda× min(r,k) ) when vect=Nag_ApplyQ and order=Nag_ColMajor;
  • max(1,r×pda) when vect=Nag_ApplyQ and order=Nag_RowMajor;
  • max(1,pda×r) when vect=Nag_ApplyP and order=Nag_ColMajor;
  • max(1,min(r,k)×pda) when vect=Nag_ApplyP and order=Nag_RowMajor.
On entry: details of the vectors which define the elementary reflectors, as returned by f08kec.
9: 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,
    • if vect=Nag_ApplyQ, pda max(1,r) ;
    • if vect=Nag_ApplyP, pda max(1,min(r,k)) ;
  • if order=Nag_RowMajor,
    • if vect=Nag_ApplyQ, pda max(1,min(r,k)) ;
    • if vect=Nag_ApplyP, pdamax(1,r).
10: tau[dim] const double Input
Note: the dimension, dim, of the array tau must be at least max(1,min(r,k)).
On entry: further details of the elementary reflectors, as returned by f08kec in its argument tauq if vect=Nag_ApplyQ, or in its argument taup if vect=Nag_ApplyP.
11: c[dim] double 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 matrix C.
On exit: c is overwritten by QC or QTC or CQ or CTQ or PC or PTC or CP or CTP as specified by vect, side and trans.
12: 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).
13: 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_2
On entry, vect=value, pda=value and k=value.
Constraint: if vect=Nag_ApplyQ, pda max(1,min(r,k)) ;
if vect=Nag_ApplyP, pdamax(1,r).
On entry, vect=value, pda=value and k=value.
Constraint: if vect=Nag_ApplyQ, pda max(1,r) ;
if vect=Nag_ApplyP, pda max(1,min(r,k)) .
NE_INT
On entry, k=value.
Constraint: k0.
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, 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 = O(ε) C2 ,  
where ε is the machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08kgc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kgc 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 where k is the value of the argument k.
The complex analogue of this function is f08kuc.

10 Example

For this function two examples are presented. Both illustrate how the reduction to bidiagonal form of a matrix A may be preceded by a QR or LQ factorization of A.
In the first example, m>n, and
A = ( -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 ) .  
The function first performs a QR factorization of A as A=QaR and then reduces the factor R to bidiagonal form B: R=QbBPT. Finally it forms Qa and calls f08kgc to form Q=QaQb.
In the second example, m<n, and
A = ( -5.42 3.28 -3.68 0.27 2.06 0.46 -1.65 -3.40 -3.20 -1.03 -4.06 -0.01 -0.37 2.35 1.90 4.31 -1.76 1.13 -3.15 -0.11 1.99 -2.70 0.26 4.50 ) .  
The function first performs an LQ factorization of A as A=LPaT and then reduces the factor L to bidiagonal form B: L=QBPbT. Finally it forms PbT and calls f08kgc to form PT=PbTPaT.

10.1 Program Text

Program Text (f08kgce.c)

10.2 Program Data

Program Data (f08kgce.d)

10.3 Program Results

Program Results (f08kgce.r)