gmres.cpp

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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
 <http://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;
    }
 
}