nag_dopmtr (f08ggc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_dopmtr (f08ggc)


    1  Purpose
    7  Accuracy

1  Purpose

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

2  Specification

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

3  Description

nag_dopmtr (f08ggc) is intended to be used after a call to nag_dsptrd (f08gec), which reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: A=QTQT. nag_dsptrd (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_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 in the Essential Introduction 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 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_UploTypeInput
On entry: this must be the same argument uplo as supplied to nag_dsptrd (f08gec).
Constraint: uplo=Nag_Upper or Nag_Lower.
4:     trans Nag_TransTypeInput
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 IntegerInput
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 IntegerInput
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] doubleInput/Output
Note: the dimension, dim, of the array ap must be at least
  • max1, m × m+1 / 2  when side=Nag_LeftSide;
  • max1, n × n+1 / 2  when side=Nag_RightSide.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_dsptrd (f08gec).
On exit: is used as internal workspace prior to being restored and hence is unchanged.
8:     tau[dim] const doubleInput
Note: the dimension, dim, of the array tau must be at least
  • max1,m-1 when side=Nag_LeftSide;
  • max1,n-1 when side=Nag_RightSide.
On entry: further details of the elementary reflectors, as returned by nag_dsptrd (f08gec).
9:     c[dim] doubleInput/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 QTC or CQ or CQT as specified by side and trans.
10:   pdc IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
  • if order=Nag_ColMajor, pdcmax1,m;
  • if order=Nag_RowMajor, pdcmax1,n.
11:   fail NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in the Essential Introduction 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: pdcmax1,m.
On entry, pdc=value and n=value.
Constraint: pdcmax1,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.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction 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_dopmtr (f08ggc) is not threaded by NAG in any implementation.
nag_dopmtr (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 nag_zupmtr (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 nag_dsptrd (f08gec). The program then calls nag_dstebz (f08jjc) to compute the requested eigenvalues and nag_dstein (f08jkc) to compute the associated eigenvectors of T. Finally nag_dopmtr (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)

nag_dopmtr (f08ggc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015