nag_dormtr (f08fgc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dormtr (f08fgc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dormtr (f08fgc) multiplies an arbitrary real matrix C by the real orthogonal matrix Q which was determined by nag_dsytrd (f08fec) when reducing a real symmetric matrix to tridiagonal form.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dormtr (Nag_OrderType order, Nag_SideType side, Nag_UploType uplo, Nag_TransType trans, Integer m, Integer n, const double a[], Integer pda, const double tau[], double c[], Integer pdc, NagError *fail)

3  Description

nag_dormtr (f08fgc) is intended to be used after a call to nag_dsytrd (f08fec), which reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation: A=QTQT. nag_dsytrd (f08fec) 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:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     sideNag_SideTypeInput
On entry: indicates how Q or QT is to be applied to C.
side=Nag_LeftSide
Q or QT is applied to C from the left.
side=Nag_RightSide
Q or QT is applied to C from the right.
Constraint: side=Nag_LeftSide or Nag_RightSide.
3:     uploNag_UploTypeInput
On entry: this must be the same argument uplo as supplied to nag_dsytrd (f08fec).
Constraint: uplo=Nag_Upper or Nag_Lower.
4:     transNag_TransTypeInput
On entry: indicates whether Q or QT is to be applied to C.
trans=Nag_NoTrans
Q is applied to C.
trans=Nag_Trans
QT is applied to C.
Constraint: trans=Nag_NoTrans or Nag_Trans.
5:     mIntegerInput
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:     nIntegerInput
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:     a[dim]const doubleInput
Note: the dimension, dim, of the array a must be at least
  • max1,pda×m when side=Nag_LeftSide;
  • max1,pda×n when side=Nag_RightSide.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_dsytrd (f08fec).
8:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array a.
Constraints:
  • if side=Nag_LeftSide, pda max1,m ;
  • if side=Nag_RightSide, pda max1,n .
9:     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_dsytrd (f08fec).
10:   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.
11:   pdcIntegerInput
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.
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_ENUM_INT_3
On entry, side=value, m=value, n=value and pda=value.
Constraint: if side=Nag_LeftSide, pda max1,m ;
if side=Nag_RightSide, pda max1,n .
NE_INT
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: pdcmax1,m.
On entry, pdc=value and n=value.
Constraint: pdcmax1,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.

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  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_zunmtr (f08fuc).

9  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 .
Here A is symmetric and must first be reduced to tridiagonal form T by nag_dsytrd (f08fec). 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_dormtr (f08fgc) is called to transform the eigenvectors to those of A.

9.1  Program Text

Program Text (f08fgce.c)

9.2  Program Data

Program Data (f08fgce.d)

9.3  Program Results

Program Results (f08fgce.r)


nag_dormtr (f08fgc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012