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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2005 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 license 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 tqreigendecomposition.hpp
\brief tridiag. QR eigen decompositions with implicit shift
*/
#include <ql/math/matrixutilities/tqreigendecomposition.hpp>
#include <vector>
namespace QuantLib {
TqrEigenDecomposition::TqrEigenDecomposition(const Array& diag,
const Array& sub,
EigenVectorCalculation calc,
ShiftStrategy strategy)
: iter_(0), d_(diag),
ev_((calc == WithEigenVector)? d_.size() :
(calc == WithoutEigenVector)? 0 : 1, d_.size(), 0)
{
Size n = diag.size();
QL_REQUIRE(n == sub.size()+1, "Wrong dimensions");
Array e(n, 0.0);
std::copy(sub.begin(),sub.end(),e.begin()+1);
Size i;
for (i=0; i < ev_.rows(); ++i) {
ev_[i][i] = 1.0;
}
for (Size k=n-1; k >=1; --k) {
while (!offDiagIsZero(k, e)) {
Size l = k;
while (--l > 0 && !offDiagIsZero(l,e));
iter_++;
Real q = d_[l];
if (strategy != NoShift) {
// calculated eigenvalue of 2x2 sub matrix of
// [ d_[k-1] e_[k] ]
// [ e_[k] d_[k] ]
// which is closer to d_[k+1].
// FLOATING_POINT_EXCEPTION
const Real t1 = std::sqrt(
0.25*(d_[k]*d_[k] + d_[k-1]*d_[k-1])
- 0.5*d_[k-1]*d_[k] + e[k]*e[k]);
const Real t2 = 0.5*(d_[k]+d_[k-1]);
const Real lambda =
(std::fabs(t2+t1 - d_[k]) < std::fabs(t2-t1 - d_[k]))?
t2+t1 : t2-t1;
if (strategy == CloseEigenValue) {
q-=lambda;
} else {
q-=((k==n-1)? 1.25 : 1.0)*lambda;
}
}
// the QR transformation
Real sine = 1.0;
Real cosine = 1.0;
Real u = 0.0;
bool recoverUnderflow = false;
for (Size i=l+1; i <= k && !recoverUnderflow; ++i) {
const Real h = cosine*e[i];
const Real p = sine*e[i];
e[i-1] = std::sqrt(p*p+q*q);
if (e[i-1] != 0.0) {
sine = p/e[i-1];
cosine = q/e[i-1];
const Real g = d_[i-1]-u;
const Real t = (d_[i]-g)*sine+2*cosine*h;
u = sine*t;
d_[i-1] = g + u;
q = cosine*t - h;
for (Size j=0; j < ev_.rows(); ++j) {
const Real tmp = ev_[j][i-1];
ev_[j][i-1] = sine*ev_[j][i] + cosine*tmp;
ev_[j][i] = cosine*ev_[j][i] - sine*tmp;
}
} else {
// recover from underflow
d_[i-1] -= u;
e[l] = 0.0;
recoverUnderflow = true;
}
}
if (!recoverUnderflow) {
d_[k] -= u;
e[k] = q;
e[l] = 0.0;
}
}
}
// sort (eigenvalues, eigenvectors),
// code taken from symmetricSchureDecomposition.cpp
std::vector<std::pair<Real, std::vector<Real> > > temp(n);
std::vector<Real> eigenVector(ev_.rows());
for (i=0; i<n; i++) {
if (ev_.rows() > 0)
std::copy(ev_.column_begin(i),
ev_.column_end(i), eigenVector.begin());
temp[i] = std::make_pair(d_[i], eigenVector);
}
std::sort(temp.begin(), temp.end(),
std::greater<std::pair<Real, std::vector<Real> > >());
// first element is positive
for (i=0; i<n; i++) {
d_[i] = temp[i].first;
Real sign = 1.0;
if (ev_.rows() > 0 && temp[i].second[0]<0.0)
sign = -1.0;
for (Size j=0; j<ev_.rows(); ++j) {
ev_[j][i] = sign * temp[i].second[j];
}
}
}
// see NR for abort assumption as it is
// not part of the original Wilkinson algorithm
bool TqrEigenDecomposition::offDiagIsZero(Size k, Array& e) {
return std::fabs(d_[k-1])+std::fabs(d_[k])
== std::fabs(d_[k-1])+std::fabs(d_[k])+std::fabs(e[k]);
}
}
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