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// $Id: eigen.h 31349 2013-10-20 19:07:06Z maier $
//
// Copyright (C) 2000 - 2013 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------
#ifndef __deal2__eigen_h
#define __deal2__eigen_h
#include <deal.II/base/config.h>
#include <deal.II/lac/shifted_matrix.h>
#include <deal.II/lac/solver.h>
#include <deal.II/lac/solver_control.h>
#include <deal.II/lac/solver_cg.h>
#include <deal.II/lac/solver_gmres.h>
#include <deal.II/lac/solver_minres.h>
#include <deal.II/lac/vector_memory.h>
#include <deal.II/lac/precondition.h>
#include <cmath>
DEAL_II_NAMESPACE_OPEN
/*!@addtogroup Solvers */
/*@{*/
/**
* Power method (von Mises) for eigenvalue computations.
*
* This method determines the largest eigenvalue of a matrix by
* applying increasing powers of this matrix to a vector. If there is
* an eigenvalue $l$ with dominant absolute value, the iteration vectors
* will become aligned to its eigenspace and $Ax = lx$.
*
* A shift parameter allows to shift the spectrum, so it is possible
* to compute the smallest eigenvalue, too.
*
* Convergence of this method is known to be slow.
*
* @author Guido Kanschat, 2000
*/
template <class VECTOR = Vector<double> >
class EigenPower : private Solver<VECTOR>
{
public:
/**
* Declare type of container size.
*/
typedef types::global_dof_index size_type;
/**
* Standardized data struct to
* pipe additional data to the
* solver.
*/
struct AdditionalData
{
/**
* Shift parameter. This
* parameter allows to shift
* the spectrum to compute a
* different eigenvalue.
*/
double shift;
/**
* Constructor. Set the shift parameter.
*/
AdditionalData (const double shift = 0.)
:
shift(shift)
{}
};
/**
* Constructor.
*/
EigenPower (SolverControl &cn,
VectorMemory<VECTOR> &mem,
const AdditionalData &data=AdditionalData());
/**
* Virtual destructor.
*/
virtual ~EigenPower ();
/**
* Power method. @p x is the
* (not necessarily normalized,
* but nonzero) start vector for
* the power method. After the
* iteration, @p value is the
* approximated eigenvalue and
* @p x is the corresponding
* eigenvector, normalized with
* respect to the l2-norm.
*/
template <class MATRIX>
void
solve (double &value,
const MATRIX &A,
VECTOR &x);
protected:
/**
* Shift parameter.
*/
AdditionalData additional_data;
};
/**
* Inverse iteration (Wieland) for eigenvalue computations.
*
* This class implements an adaptive version of the inverse iteration by Wieland.
*
* There are two choices for the stopping criterion: by default, the
* norm of the residual $A x - l x$ is computed. Since this might not
* converge to zero for non-symmetric matrices with non-trivial Jordan
* blocks, it can be replaced by checking the difference of successive
* eigenvalues. Use AdditionalData::use_residual for switching
* this option.
*
* Usually, the initial guess entering this method is updated after
* each step, replacing it with the new approximation of the
* eigenvalue. Using a parameter AdditionalData::relaxation
* between 0 and 1, this update can be damped. With relaxation
* parameter 0, no update is performed. This damping allows for slower
* adaption of the shift value to make sure that the method converges
* to the eigenvalue closest to the initial guess. This can be aided
* by the parameter AdditionalData::start_adaption, which
* indicates the first iteration step in which the shift value should
* be adapted.
*
* @author Guido Kanschat, 2000, 2003
*/
template <class VECTOR = Vector<double> >
class EigenInverse : private Solver<VECTOR>
{
public:
/**
* Declare type of container size.
*/
typedef types::global_dof_index size_type;
/**
* Standardized data struct to
* pipe additional data to the
* solver.
*/
struct AdditionalData
{
/**
* Damping of the updated shift value.
*/
double relaxation;
/**
* Start step of adaptive
* shift parameter.
*/
unsigned int start_adaption;
/**
* Flag for the stopping criterion.
*/
bool use_residual;
/**
* Constructor.
*/
AdditionalData (double relaxation = 1.,
unsigned int start_adaption = 6,
bool use_residual = true)
:
relaxation(relaxation),
start_adaption(start_adaption),
use_residual(use_residual)
{}
};
/**
* Constructor.
*/
EigenInverse (SolverControl &cn,
VectorMemory<VECTOR> &mem,
const AdditionalData &data=AdditionalData());
/**
* Virtual destructor.
*/
virtual ~EigenInverse ();
/**
* Inverse method. @p value is
* the start guess for the
* eigenvalue and @p x is the
* (not necessarily normalized,
* but nonzero) start vector for
* the power method. After the
* iteration, @p value is the
* approximated eigenvalue and
* @p x is the corresponding
* eigenvector, normalized with
* respect to the l2-norm.
*/
template <class MATRIX>
void
solve (double &value,
const MATRIX &A,
VECTOR &x);
protected:
/**
* Flags for execution.
*/
AdditionalData additional_data;
};
/*@}*/
//---------------------------------------------------------------------------
template <class VECTOR>
EigenPower<VECTOR>::EigenPower (SolverControl &cn,
VectorMemory<VECTOR> &mem,
const AdditionalData &data)
:
Solver<VECTOR>(cn, mem),
additional_data(data)
{}
template <class VECTOR>
EigenPower<VECTOR>::~EigenPower ()
{}
template <class VECTOR>
template <class MATRIX>
void
EigenPower<VECTOR>::solve (double &value,
const MATRIX &A,
VECTOR &x)
{
SolverControl::State conv=SolverControl::iterate;
deallog.push("Power method");
VECTOR *Vy = this->memory.alloc ();
VECTOR &y = *Vy;
y.reinit (x);
VECTOR *Vr = this->memory.alloc ();
VECTOR &r = *Vr;
r.reinit (x);
double length = x.l2_norm ();
double old_length = 0.;
x.scale(1./length);
A.vmult (y,x);
// Main loop
for (int iter=0; conv==SolverControl::iterate; iter++)
{
y.add(additional_data.shift, x);
// Compute absolute value of eigenvalue
old_length = length;
length = y.l2_norm ();
// do a little trick to compute the sign
// with not too much effect of round-off errors.
double entry = 0.;
size_type i = 0;
double thresh = length/x.size();
do
{
Assert (i<x.size(), ExcInternalError());
entry = y (i++);
}
while (std::fabs(entry) < thresh);
--i;
// Compute unshifted eigenvalue
value = (entry * x (i) < 0.) ? -length : length;
value -= additional_data.shift;
// Update normalized eigenvector
x.equ (1/length, y);
// Compute residual
A.vmult (y,x);
// Check the change of the eigenvalue
// Brrr, this is not really a good criterion
conv = this->control().check (iter, std::fabs(1./length-1./old_length));
}
this->memory.free(Vy);
this->memory.free(Vr);
deallog.pop();
// in case of failure: throw exception
if (this->control().last_check() != SolverControl::success)
AssertThrow(false, SolverControl::NoConvergence (this->control().last_step(),
this->control().last_value()));
// otherwise exit as normal
}
//---------------------------------------------------------------------------
template <class VECTOR>
EigenInverse<VECTOR>::EigenInverse (SolverControl &cn,
VectorMemory<VECTOR> &mem,
const AdditionalData &data)
:
Solver<VECTOR>(cn, mem),
additional_data(data)
{}
template <class VECTOR>
EigenInverse<VECTOR>::~EigenInverse ()
{}
template <class VECTOR>
template <class MATRIX>
void
EigenInverse<VECTOR>::solve (double &value,
const MATRIX &A,
VECTOR &x)
{
deallog.push("Wielandt");
SolverControl::State conv=SolverControl::iterate;
// Prepare matrix for solver
ShiftedMatrix <MATRIX> A_s(A, -value);
// Define solver
ReductionControl inner_control (5000, 1.e-16, 1.e-5, false, false);
PreconditionIdentity prec;
SolverGMRES<VECTOR>
solver(inner_control, this->memory);
// Next step for recomputing the shift
unsigned int goal = additional_data.start_adaption;
// Auxiliary vector
VECTOR *Vy = this->memory.alloc ();
VECTOR &y = *Vy;
y.reinit (x);
VECTOR *Vr = this->memory.alloc ();
VECTOR &r = *Vr;
r.reinit (x);
double length = x.l2_norm ();
double old_value = value;
x.scale(1./length);
// Main loop
for (size_type iter=0; conv==SolverControl::iterate; iter++)
{
solver.solve (A_s, y, x, prec);
// Compute absolute value of eigenvalue
length = y.l2_norm ();
// do a little trick to compute the sign
// with not too much effect of round-off errors.
double entry = 0.;
size_type i = 0;
double thresh = length/x.size();
do
{
Assert (i<x.size(), ExcInternalError());
entry = y (i++);
}
while (std::fabs(entry) < thresh);
--i;
// Compute unshifted eigenvalue
value = (entry * x (i) < 0.) ? -length : length;
value = 1./value;
value -= A_s.shift ();
if (iter==goal)
{
const double new_shift = - additional_data.relaxation * value
+ (1.-additional_data.relaxation) * A_s.shift();
A_s.shift(new_shift);
++goal;
}
// Update normalized eigenvector
x.equ (1./length, y);
// Compute residual
if (additional_data.use_residual)
{
y.equ (value, x);
A.vmult(r,x);
r.sadd(-1., value, x);
double res = r.l2_norm();
// Check the residual
conv = this->control().check (iter, res);
}
else
{
conv = this->control().check (iter, std::fabs(1./value-1./old_value));
}
old_value = value;
}
this->memory.free(Vy);
this->memory.free(Vr);
deallog.pop();
// in case of failure: throw
// exception
if (this->control().last_check() != SolverControl::success)
throw SolverControl::NoConvergence (this->control().last_step(),
this->control().last_value());
// otherwise exit as normal
}
DEAL_II_NAMESPACE_CLOSE
#endif
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