nag_dormhr (f08ngc) (PDF version)
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f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dormhr (f08ngc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dormhr (f08ngc) multiplies an arbitrary real matrix C by the real orthogonal matrix Q which was determined by nag_dgehrd (f08nec) when reducing a real general matrix to Hessenberg form.

2  Specification

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

3  Description

nag_dormhr (f08ngc) is intended to be used following a call to nag_dgehrd (f08nec), which reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: A=QHQT. nag_dgehrd (f08nec) represents the matrix Q as a product of ihi-ilo elementary reflectors. Here ilo and ihi are values determined by nag_dgebal (f08nhc) when balancing the matrix; if the matrix has not been balanced, ilo=1 and ihi=n.
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 V of eigenvectors of H to the matrix QV 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:     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.
4:     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.
5:     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.
6:     iloIntegerInput
7:     ihiIntegerInput
On entry: these must be the same arguments ilo and ihi, respectively, as supplied to nag_dgehrd (f08nec).
Constraints:
  • if side=Nag_LeftSide and m>0, 1 ilo ihi m ;
  • if side=Nag_LeftSide and m=0, ilo=1 and ihi=0;
  • if side=Nag_RightSide and n>0, 1 ilo ihi n ;
  • if side=Nag_RightSide and n=0, ilo=1 and ihi=0.
8:     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_dgehrd (f08nec).
9:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
  • if side=Nag_LeftSide, pda max1,m ;
  • if side=Nag_RightSide, pda max1,n .
10:   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_dgehrd (f08nec).
11:   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.
12:   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.
13:   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 .
On entry, side=value, pda=value, m=value and n=value.
Constraint: if side=Nag_LeftSide, pdamax1,m;
if side=Nag_RightSide, pdamax1,n.
NE_ENUM_INT_4
On entry, side=value, m=value, n=value, ilo=value and ihi=value.
Constraint: if side=Nag_LeftSide and m>0, 1 ilo ihi m ;
if side=Nag_LeftSide and m=0, ilo=1 and ihi=0;
if side=Nag_RightSide and n>0, 1 ilo ihi n ;
if side=Nag_RightSide and n=0, ilo=1 and ihi=0.
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 2nq2 if side=Nag_LeftSide and 2mq2 if side=Nag_RightSide, where q=ihi-ilo.
The complex analogue of this function is nag_zunmhr (f08nuc).

9  Example

This example computes all the eigenvalues of the matrix A, where
A = 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 ,
and those eigenvectors which correspond to eigenvalues λ such that Reλ<0. Here A is general and must first be reduced to upper Hessenberg form H by nag_dgehrd (f08nec). The program then calls nag_dhseqr (f08pec) to compute the eigenvalues, and nag_dhsein (f08pkc) to compute the required eigenvectors of H by inverse iteration. Finally nag_dormhr (f08ngc) is called to transform the eigenvectors of H back to eigenvectors of the original matrix A.

9.1  Program Text

Program Text (f08ngce.c)

9.2  Program Data

Program Data (f08ngce.d)

9.3  Program Results

Program Results (f08ngce.r)


nag_dormhr (f08ngc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

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