NAG CL Interface
f08ggc (dopmtr)

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1 Purpose

f08ggc multiplies an arbitrary real matrix C by the real orthogonal matrix Q which was determined by f08gec when reducing a real symmetric matrix to tridiagonal form.

2 Specification

#include <nag.h>
void  f08ggc (Nag_OrderType order, Nag_SideType side, Nag_UploType uplo, Nag_TransType trans, Integer m, Integer n, double ap[], const double tau[], double c[], Integer pdc, NagError *fail)
The function may be called by the names: f08ggc, nag_lapackeig_dopmtr or nag_dopmtr.

3 Description

f08ggc is intended to be used after a call to f08gec, which reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: A=QTQT. f08gec represents the orthogonal matrix Q as a product of elementary reflectors.
This function may be used to form one of the matrix products
QC , QTC , CQ ​ or ​ CQT ,  
overwriting the result on C (which may be any real rectangular matrix).
A common application of this function is to transform a matrix Z of eigenvectors of T to the matrix QZ of eigenvectors of A.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd 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 QT is to be applied to C.
Q or QT is applied to C from the left.
Q or QT is applied to C from the right.
Constraint: side=Nag_LeftSide or Nag_RightSide.
3: uplo Nag_UploType Input
On entry: this must be the same argument uplo as supplied to f08gec.
Constraint: uplo=Nag_Upper or Nag_Lower.
4: trans Nag_TransType Input
On entry: indicates whether Q or QT is to be applied to C.
Q is applied to C.
QT 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; m is also the order of Q if side=Nag_LeftSide.
Constraint: m0.
6: n Integer Input
On entry: n, the number of columns of the matrix C; n is also the order of Q if side=Nag_RightSide.
Constraint: n0.
7: ap[dim] double Input/Output
Note: the dimension, dim, of the array ap must be at least
  • max(1, m × (m+1) / 2 )  when side=Nag_LeftSide;
  • max(1, n × (n+1) / 2 )  when side=Nag_RightSide.
On entry: details of the vectors which define the elementary reflectors, as returned by f08gec.
On exit: is used as internal workspace prior to being restored and hence is unchanged.
8: tau[dim] const double Input
Note: the dimension, dim, of the array tau must be at least
  • max(1,m-1) when side=Nag_LeftSide;
  • max(1,n-1) when side=Nag_RightSide.
On entry: further details of the elementary reflectors, as returned by f08gec.
9: 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 m×n matrix C.
On exit: c is overwritten by QC or QTC or CQ or CQT as specified by side and trans.
10: pdc Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
  • if order=Nag_ColMajor, pdcmax(1,m);
  • if order=Nag_RowMajor, pdcmax(1,n).
11: 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

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, pdc=value.
Constraint: pdc>0.
On entry, pdc=value and m=value.
Constraint: pdcmax(1,m).
On entry, pdc=value and n=value.
Constraint: pdcmax(1,n).
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.
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.
f08ggc 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 2m2n if side=Nag_LeftSide and 2mn2 if side=Nag_RightSide.
The complex analogue of this function is f08guc.

10 Example

This example computes the two smallest eigenvalues, and the associated eigenvectors, of the matrix A, where
A = ( 2.07 3.87 4.20 -1.15 3.87 -0.21 1.87 0.63 4.20 1.87 1.15 2.06 -1.15 0.63 2.06 -1.81 ) ,  
using packed storage. Here A is symmetric and must first be reduced to tridiagonal form T by f08gec. The program then calls f08jjc to compute the requested eigenvalues and f08jkc to compute the associated eigenvectors of T. Finally f08ggc is called to transform the eigenvectors to those of A.

10.1 Program Text

Program Text (f08ggce.c)

10.2 Program Data

Program Data (f08ggce.d)

10.3 Program Results

Program Results (f08ggce.r)