NAG C Library Function Document

nag_dormhr (f08ngc)

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:     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 3.3.1.3 in How to Use the NAG Library and its Documentation 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.
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:     trans Nag_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:     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.
5:     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.
6:     ilo IntegerInput
7:     ihi IntegerInput
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:     pda IntegerInput
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:   pdc IntegerInput
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:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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_dormhr (f08ngc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dormhr (f08ngc) 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 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).

10
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.

10.1
Program Text

Program Text (f08ngce.c)

10.2
Program Data

Program Data (f08ngce.d)

10.3
Program Results

Program Results (f08ngce.r)