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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2017 Klaus Spanderen
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license0/0 iee along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<https://www.quantlib.org/license.shtml>.
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 license for more details.
*/
/*! \file gmres.cpp
\brief generalized minimal residual method
*/
#include <ql/math/functional.hpp>
#include <ql/math/matrixutilities/gmres.hpp>
#include <ql/math/matrixutilities/qrdecomposition.hpp>
#include <numeric>
#include <utility>
namespace QuantLib {
GMRES::GMRES(GMRES::MatrixMult A, Size maxIter, Real relTol, GMRES::MatrixMult preConditioner)
: A_(std::move(A)), M_(std::move(preConditioner)), maxIter_(maxIter), relTol_(relTol) {
QL_REQUIRE(maxIter_ > 0, "maxIter must be greater than zero");
}
GMRESResult GMRES::solve(const Array& b, const Array& x0) const {
GMRESResult result = solveImpl(b, x0);
QL_REQUIRE(result.errors.back() < relTol_, "could not converge");
return result;
}
GMRESResult GMRES::solveWithRestart(
Size restart, const Array& b, const Array& x0) const {
GMRESResult result = solveImpl(b, x0);
std::list<Real> errors = result.errors;
for (Size i=0; i < restart-1 && result.errors.back() >= relTol_;++i) {
result = solveImpl(b, result.x);
errors.insert(
errors.end(), result.errors.begin(), result.errors.end());
}
QL_REQUIRE(errors.back() < relTol_, "could not converge");
result.errors = errors;
return result;
}
GMRESResult GMRES::solveImpl(const Array& b, const Array& x0) const {
const Real bn = Norm2(b);
if (bn == 0.0) {
GMRESResult result = { std::list<Real>(1, 0.0), b };
return result;
}
Array x = ((!x0.empty()) ? x0 : Array(b.size(), 0.0));
Array r = b - A_(x);
const Real g = Norm2(r);
if (g/bn < relTol_) {
GMRESResult result = { std::list<Real>(1, g/bn), x };
return result;
}
std::vector<Array> v(1, r/g);
std::vector<Array> h(1, Array(maxIter_, 0.0));
std::vector<Real> c(maxIter_+1), s(maxIter_+1), z(maxIter_+1);
z[0] = g;
std::list<Real> errors(1, g/bn);
for (Size j=0; j < maxIter_ && errors.back() >= relTol_; ++j) {
h.emplace_back(maxIter_, 0.0);
Array w = A_(!M_ ? v[j] : M_(v[j]));
for (Size i=0; i <= j; ++i) {
h[i][j] = DotProduct(w, v[i]);
w -= h[i][j] * v[i];
}
h[j+1][j] = Norm2(w);
if (h[j+1][j] < QL_EPSILON*QL_EPSILON)
break;
v.push_back(w / h[j+1][j]);
for (Size i=0; i < j; ++i) {
const Real h0 = c[i]*h[i][j] + s[i]*h[i+1][j];
const Real h1 =-s[i]*h[i][j] + c[i]*h[i+1][j];
h[i][j] = h0;
h[i+1][j] = h1;
}
const Real nu = std::sqrt(squared(h[j][j]) + squared(h[j+1][j]));
c[j] = h[j][j]/nu;
s[j] = h[j+1][j]/nu;
h[j][j] = nu;
h[j+1][j] = 0.0;
z[j+1] = -s[j]*z[j];
z[j] = c[j] * z[j];
errors.push_back(std::fabs(z[j+1]/bn));
}
const Size k = v.size()-1;
Array y(k, 0.0);
y[k-1]=z[k-1]/h[k-1][k-1];
for (Integer i=k-2; i >= 0; --i) {
y[i] = (z[i] - std::inner_product(
h[i].begin()+i+1, h[i].begin()+k, y.begin()+i+1, Real(0.0)))/h[i][i];
}
Array xm = std::inner_product(
v.begin(), v.begin()+k, y.begin(), Array(x.size(), Real(0.0)));
xm = x + (!M_ ? xm : M_(xm));
GMRESResult result = { errors, xm };
return result;
}
}
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