RWHessenbergDecomp< TypeT > Class Template Reference

Encapsulates a Hessenberg decomposition as well as an optional balance transformation. More...

#include <rw/lapack/hess.h>

Public Member Functions
	RWHessenbergDecomp ()
	RWHessenbergDecomp (const RWBalanceDecomp< TypeT > &)
	RWHessenbergDecomp (const RWGenMat< TypeT > &A, bool permute=true, bool scale=true)
RWGenMat< TypeT >	B () const
RWBalanceTransform< TypeT >	balanceTransform () const
RWGenMat< TypeT >	BInvTX (const RWGenMat< TypeT > &X) const
RWMathVec< TypeT >	BInvTx (const RWMathVec< TypeT > &x) const
RWGenMat< TypeT >	BX (const RWGenMat< TypeT > &X) const
RWMathVec< TypeT >	Bx (const RWMathVec< TypeT > &x) const
unsigned	cols () const
void	factor (const RWBalanceDecomp< TypeT > &)
void	factor (const RWGenMat< TypeT > &A, bool permute=true, bool scale=true)
RWGenMat< TypeT >	Q () const
RWGenMat< TypeT >	QTX (const RWGenMat< TypeT > &X) const
RWMathVec< TypeT >	QTx (const RWMathVec< TypeT > &) const
RWGenMat< TypeT >	QX (const RWGenMat< TypeT > &X) const
RWMathVec< TypeT >	Qx (const RWMathVec< TypeT > &x) const

Detailed Description

template<class TypeT>
class RWHessenbergDecomp< TypeT >

A Hessenberg decomposition uses orthogonal transformations to reduce a matrix A to a matrix H that is 0 below the first subdiagonal. Computation of the Hessenberg decomposition is the first step in the QR method for computing eigenvalues of a nonsymmetric matrix.

The class RWHessenbergDecomp encapsulates a Hessenberg decomposition as well as an optional balance transformation. The decomposition of a matrix A is:

\[ A = BQHQ'B^{-1} \]

where Q is orthogonal, H is 0 below the first subdiagonal, and B is a balance transformation. See the entry RWBalanceTransform.

Synopsis

#include <rw/lapack/hess.h>
 
RWHessenbergDecomp<double> hess(A);  // A is a RWGenMat<double>

Example

#include <rw/lapack/hess.h>
#include <iostream>
 
int main() {
  RWGenMat<double> A;
  std::cin >> A;
  RWHessenbergDecomp<double> hess(A);
  std::cout << "Input matrix: " << A << std::endl;
  std::cout << "B: " << hess.B() << std::endl;
  std::cout << "Q: " << hess.Q() << std::endl;
  std::cout << "H: " << hess.H() << std::endl;
  return 0;
}

Constructor & Destructor Documentation

RWHessenbergDecomp() [1/3]

template<class TypeT >

RWHessenbergDecomp< TypeT >::RWHessenbergDecomp

(

)

Default constructor. Builds a decomposition of size 0 x 0.

RWHessenbergDecomp() [2/3]

template<class TypeT >

RWHessenbergDecomp< TypeT >::RWHessenbergDecomp	(	const RWGenMat< TypeT > &	A,
		bool	permute = true,
		bool	scale = true )

Constructs the Hessenberg decomposition of the matrix A. The boolean parameters determine whether or not the permutation or scaling parts of the balance transformation are applied prior to forming the Hessenberg decomposition.

RWHessenbergDecomp() [3/3]

template<class TypeT >

RWHessenbergDecomp< TypeT >::RWHessenbergDecomp

(

const RWBalanceDecomp< TypeT > &

)

Builds a Hessenberg decomposition of the matrix represented by the balance decomposition.

Member Function Documentation

B()

template<class TypeT >

RWGenMat< TypeT > RWHessenbergDecomp< TypeT >::B

(

)

const

Computes an explicit representation of the balance transformation.

balanceTransform()

template<class TypeT >

RWBalanceTransform< TypeT > RWHessenbergDecomp< TypeT >::balanceTransform ( ) const

inline

Returns an object which represents the balance transformation.

BInvTX()

template<class TypeT >

RWGenMat< TypeT > RWHessenbergDecomp< TypeT >::BInvTX

(

const RWGenMat< TypeT > &

X

)

const

Computes the (conjugate) transpose inverse of the balance transformation B and the matrix X.

BInvTx()

template<class TypeT >

RWMathVec< TypeT > RWHessenbergDecomp< TypeT >::BInvTx

(

const RWMathVec< TypeT > &

x

)

const

Computes the (conjugate) transpose inverse of the balance transformation B and the vector x.

BX()

template<class TypeT >

RWGenMat< TypeT > RWHessenbergDecomp< TypeT >::BX

(

const RWGenMat< TypeT > &

X

)

const

Computes the inner product of the balance transformation B and the matrix X.

Bx()

template<class TypeT >

RWMathVec< TypeT > RWHessenbergDecomp< TypeT >::Bx

(

const RWMathVec< TypeT > &

x

)

const

Computes the inner product of the balance transformation B and the vector x.

cols()

template<class TypeT >

unsigned RWHessenbergDecomp< TypeT >::cols ( ) const

inline

Returns the number of columns in the matrix that the decomposition represents.

factor() [1/2]

template<class TypeT >

void RWHessenbergDecomp< TypeT >::factor

(

const RWBalanceDecomp< TypeT > &

)

Replaces the current decomposition with the Hessenberg decomposition of the matrix represented by the balance decomposition.

factor() [2/2]

template<class TypeT >

void RWHessenbergDecomp< TypeT >::factor	(	const RWGenMat< TypeT > &	A,
		bool	permute = true,
		bool	scale = true )

Replaces the current decomposition with the Hessenberg decomposition of the matrix A. The boolean parameters determine whether or not the permutation or scaling parts of the balance transformation are applied prior to forming the Hessenberg decomposition.

Q()

template<class TypeT >

RWGenMat< TypeT > RWHessenbergDecomp< TypeT >::Q

(

)

const

Computes an explicit representation of the orthogonal matrix Q.

QTX()

template<class TypeT >

RWGenMat< TypeT > RWHessenbergDecomp< TypeT >::QTX

(

const RWGenMat< TypeT > &

X

)

const

Computes the (conjugate) transpose of the orthogonal matrix Q and the matrix X.

QTx()

template<class TypeT >

RWMathVec< TypeT > RWHessenbergDecomp< TypeT >::QTx

(

const RWMathVec< TypeT > &

)

const

Computes the (conjugate) transpose of the orthogonal matrix Q and the vector x.

QX()

template<class TypeT >

RWGenMat< TypeT > RWHessenbergDecomp< TypeT >::QX

(

const RWGenMat< TypeT > &

X

)

const

Computes the inner product of the orthogonal matrix Q and the matrix X.

Qx()

template<class TypeT >

RWMathVec< TypeT > RWHessenbergDecomp< TypeT >::Qx

(

const RWMathVec< TypeT > &

x

)

const

Computes the inner product of the orthogonal matrix Q and the vector x.