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LA_library/gmres.h

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/*
LA: linear algebra C++ interface library
Copyright (C) 2008 Jiri Pittner <jiri.pittner@jh-inst.cas.cz> or <jiri@pittnerovi.com>
based on a routine originally written by Markus Warken <markus.warken@nsn.com>
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#ifndef _GMRES_H
#define _GMRES_H
#include "vec.h"
#include "smat.h"
#include "mat.h"
#include "sparsemat.h"
#include "nonclass.h"
#include <iomanip>
#include "auxstorage.h"
namespace LA {
//GMRES solution of a linear system
//matrix can be any class which has nrows(), ncols(), diagonalof() and gemv() available
//does not even have to be explicitly stored
/* GMRES-Algorithmus nach Schwarz, S. 552, original impl. M. Warken */
/* allows zeilen!= spalten*/
/* Matrix can be any class which provides nrows(), ncols(), gemv(), and diagonalof(), does not have to store elements explicitly */
template<class T>
void gmres_backsubstitute(const NRMat<T> &R, NRVec<T> &c, const NRVec<T> &d, const int k)
{
c.copyonwrite();
if(R(k,k)==0.) laerror("singular matrix in gmres triangular solution");
c[k] = d[k]/R(k,k);
for (int i=k-1;i>=0;i--) c[i] = (d[i]-xdot(k-i,&R(i,i+1),1,&c[i+1],1)) / R(i,i);
}
//x contains ev. initial guess and on return the solution
template<typename T, typename Matrix>
bool gmres(const Matrix &bigmat, const NRVec<T> &b, NRVec<T> &x, const bool doguess=1, const double eps=1e-7, const int MAXIT=50, const bool verbose=1, bool square=1,const bool precondition=1, int neustart=0, const int incore=1)
{
int zeilen=bigmat.nrows();
int spalten=bigmat.ncols();
if(spalten==1) laerror("gmres does not work for n==1, use conjgrad if you need this trivial case");
if(x.size()!=spalten || b.size() != zeilen) laerror("incompatible vectors and matrix sizes in GMRES");
if(zeilen!=spalten) square=0;
if(!neustart) neustart = zeilen/10;
if (neustart < 10) neustart = 10;
x.copyonwrite();
bool flag;
double beta,beta_0;
double d_alt=0;
AuxStorage<T> *st;
NRVec<T> *v;
NRVec<T> r_k(spalten),z(spalten);
NRVec<T> cci(MAXIT+1),ssi(MAXIT+1),c(MAXIT+1),d(MAXIT+1);
NRMat<T> H(MAXIT+1,MAXIT+1);
T ci,si;
v = new NRVec<T>[incore?MAXIT+1:1];
st = incore?NULL:new AuxStorage<T>;
if(doguess)
{
bigmat.gemv(0,x,'t',-1.,b); //x.gemv(0,bigmat,'t',-1.,b);
if(precondition) bigmat.diagonalof(x,true);
x.normalize();
}
neustart:
for (int l=0;l<neustart;l++) // main loop for restarts
{
if(square) // r_0 = b + A x_0
{
bigmat.gemv(0,r_k,'n',1,x); //r_k.gemv(0,bigmat,'n',1,x);
r_k -= b;
}
else //r_0 = A^t b + A^t A x_0
{
NRVec<T> dum(zeilen);
bigmat.gemv(0,dum,'n',1,x); //dum.gemv(0,bigmat,'n',1,x);
bigmat.gemv(0,r_k,'t',1,dum); //r_k.gemv(0,bigmat,'t',1,dum);
bigmat.gemv(0,z,'t',-1.,b); //z.gemv(0,bigmat,'t',-1.,b);
r_k += z;
}
if(precondition) bigmat.diagonalof(r_k,true);
beta = r_k.norm();
if(l==0) beta_0 = beta;
v[0] = r_k* (1./beta);
if(!incore) st->put(v[0],0);
// Iteration
for (int k=0;k<MAXIT;k++)
{
// *iter=l*MAXIT+k;
//if(dowarn) if (l>0) fprintf(stderr,"gmres: restart %d\n",l);
// Schritt 1
if(!incore) st->get(v[0],k);
if(square)
{
bigmat.gemv(0,z,'n',1,v[incore?k:0]); //z.gemv(0,bigmat,'n',1,v[incore?k:0]);
}
else
{
NRVec<T> dum(zeilen);
bigmat.gemv(0,dum,'n',1,v[incore?k:0]); //dum.gemv(0,bigmat,'n',1,v[incore?k:0]);
bigmat.gemv(0,z,'t',1,dum); //z.gemv(0,bigmat,'t',1,dum);
}
if(precondition) bigmat.diagonalof(z,true);
//Schritte 2 und 3
for (int i=0;i<=k;i++)
{
if(!incore) st->get(v[0],i);
H(i,k) = z*v[incore?i:0];
z.axpy(-H(i,k),v[incore?i:0]);
}
//Schritt 4
double tmp;
H(k+1,k) = tmp= z.norm();
if(tmp < 1.e-2*eps )
{
if(verbose) std::cerr <<("gmres restart performed\n");
// Abbruchbedingung, konstruiere x_k
for (int i=0;i<k;i++)
{
ci = cci[i];si = ssi[i];
for (int j=0;j<k;j++)
{
T a = H(i,j);
H(i,j) = ci*a+si*H(i+1,j);
H(i+1,j) = -si*a+ci*H(i+1,j);
}
}
// Loese R_k c = - d_k
d *= -1.;
gmres_backsubstitute(H,c,d,k-1);
for (int i=0;i<k-1;i++)
{
if(!incore) st->get(v[0],i);
x.axpy(c[i],v[incore?i:0]);
}
flag=0; goto neustart;
} // Ende Abbruch
v[incore?k+1:0] = z * (1./H(k+1,k));
if(!incore) st->put(v[0],k+1);
// Schritt 5 - berechne Phi_k
for (int j=0;j<k+2;j++) d[j] = H(j,k);
for (int i=0;i<k;i++)
{
ci = cci[i];
si = ssi[i];
T a = d[i];
d[i] = ci*a+si*d[i+1];
d[i+1] = -si*a+ci*d[i+1];
}
//phi[k]= atan(d[k+1]/d[k]);
ci=hypot(d[k],d[k+1]);
cci[k]=d[k]/ci;
ssi[k]=d[k+1]/ci;
//berechne neuen d-Vektor
d= 0.;
d[0]=beta;
for (int i=0;i<=k;i++)
{
ci = cci[i];si = ssi[i];
T a = d[i];
d[i] = ci*a+si*d[i+1];
d[i+1] = -si*a+ci*d[i+1];
}
//Schritt 6: Konvergenz?
if(verbose)
{
std::cout << "gmres iter "<<l<<" "<<k<<" resid "
<<std::setw(0)<<std::setiosflags(std::ios::scientific)<<std::setprecision(8)
<<std::abs(d[k+1])<< " thr "<<eps*beta_0<< " reduction "
<<std::setw(5)<<std::setprecision(2)<<std::resetiosflags(std::ios::scientific)
<<(d_alt - std::abs(d[k+1]))/d_alt*100<< "\n" <<std::setprecision(12);
std::cout.flush();
}
d_alt = abs(d[k+1]);
//*err= d_alt;
if (d_alt < eps*beta_0)
{
// konstruiere R_k
for (int i=0;i<k;i++)
{
ci = cci[i];
si = ssi[i];
for (int j=0;j<k;j++)
{
T a = H(i,j);
H(i,j) = ci*a+si*H(i+1,j);
H(i+1,j) = -si*a+ci*H(i+1,j);
}
}
// Loese R_k c = - d_k
d *= -1.;
gmres_backsubstitute(H,c,d,k-1);
for(int i=0;i<k;i++)
{
if(!incore) st->get(v[0],i);
x.axpy(c[i],v[incore?i:0]);
}
flag=0; goto myreturn;
}
} // k-Schleife
// zum Neustart: Konstruiere R_k
for (int i=0;i<MAXIT;i++)
{
ci = cci[i];si = ssi[i];
for (int j=0;j<MAXIT;j++)
{
T a = H(i,j);
H(i,j) = ci*a+si*H(i+1,j);
H(i+1,j) = -si*a+ci*H(i+1,j);
}
}
// Loese R_k c = - d_k
d *= -1.;
gmres_backsubstitute(H,c,d,MAXIT-1);
for(int i=0;i<MAXIT;i++)
{
if(!incore) st->get(v[0],i);
x.axpy(c[i],v[incore?i:0]);
}
} // l schleife
flag=1;
myreturn:
delete[] v;
if(!incore) delete st;
return !flag;
}
}//namespace
#endif
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