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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2007 Mark Joshi
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
<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.
*/
#include <ql/math/optimization/spherecylinder.hpp>
#include <ql/errors.hpp>
#include <algorithm>
namespace QuantLib {
namespace {
template<class F>
Real BrentMinimize(Real low,
Real mid,
Real high,
Real tolerance,
Size maxIt,
const F& objectiveFunction) {
Real W = 0.5*(3.0-std::sqrt(5.0));
Real x = W*low+(1-W)*high;
if (mid > low && mid < high)
x = mid;
Real midValue = objectiveFunction(x);
Size iterations = 0;
while (high-low > tolerance && iterations < maxIt) {
if (x - low > high -x) { // left interval is bigger
Real tentativeNewMid = W*low+(1-W)*x;
Real tentativeNewMidValue =
objectiveFunction(tentativeNewMid);
if (tentativeNewMidValue < midValue) { // go left
high =x;
x = tentativeNewMid;
midValue = tentativeNewMidValue;
} else { // go right
low = tentativeNewMid;
}
} else {
Real tentativeNewMid = W*x+(1-W)*high;
Real tentativeNewMidValue =
objectiveFunction(tentativeNewMid);
if (tentativeNewMidValue < midValue) { // go right
low =x;
x = tentativeNewMid;
midValue = tentativeNewMidValue;
} else { // go left
high = tentativeNewMid;
}
}
++iterations;
}
return x;
}
}
SphereCylinderOptimizer::SphereCylinderOptimizer(Real r,
Real s,
Real alpha,
Real z1,
Real z2,
Real z3,
Real zweight)
: r_(r), s_(s), alpha_(alpha), z1_(z1), z2_(z2), z3_(z3), zweight_(zweight)
{
QL_REQUIRE(r > 0, "sphere must have positive radius");
s = std::max(s, 0.0);
QL_REQUIRE(alpha > 0, "cylinder centre must have positive coordinate");
nonEmpty_ = std::fabs(alpha - s) <= r;
Real cylinderInside = r * r - (s + alpha) * (s + alpha);
if (cylinderInside > 0.0) {
topValue_ = alpha + s;
bottomValue_ = alpha - s;
} else {
bottomValue_ = alpha - s;
Real tmp = r * r - (s * s + alpha * alpha);
if (tmp <= 0) { // max to left of maximum
Real topValue2 = std::sqrt(s * s - tmp * tmp / (4 * alpha * alpha));
topValue_ = alpha - std::sqrt(s * s - topValue2 * topValue2);
} else {
topValue_ = alpha + tmp / (2.0 * alpha);
}
}
}
bool SphereCylinderOptimizer::isIntersectionNonEmpty() const {
return nonEmpty_;
}
void SphereCylinderOptimizer::findClosest(Size maxIterations,
Real tolerance,
Real& y1,
Real& y2,
Real& y3) const
{
Real x1,x2,x3;
findByProjection(x1,x2,x3);
y1 = BrentMinimize(
bottomValue_, x1, topValue_,tolerance, maxIterations,
[&](Real x){ return objectiveFunction(x); });
y2 =std::sqrt(s_*s_ - (y1-alpha_)*(y1-alpha_));
y3= std::sqrt(r_*r_ - y1*y1-y2*y2);
}
Real SphereCylinderOptimizer::objectiveFunction(Real x1) const
{
// Real x1 = alpha_ - std::sqrt(s_*s_-x2*x2);
Real x2sq = s_*s_ - (x1-alpha_)*(x1-alpha_);
// a negative number will be minuscule and a result of rounding error
Real x2 = x2sq >= 0.0 ? Real(std::sqrt(x2sq)) : 0.0;
Real x3= std::sqrt(r_*r_ - x1*x1-x2*x2);
Real err=0.0;
err+= (x1-z1_)*(x1-z1_);
err+= (x2-z2_)*(x2-z2_);
err+= (x3-z3_)*(x3-z3_)*zweight_;
return err;
}
bool SphereCylinderOptimizer::findByProjection(Real& y1,
Real& y2,
Real& y3) const {
Real z1moved = z1_-alpha_;
Real distance = std::sqrt( z1moved*z1moved + z2_*z2_);
Real scale = s_/distance;
Real y1moved = z1moved*scale;
y1 = alpha_+ y1moved;
y2 = scale*z2_;
Real residual = r_*r_ - y1*y1 -y2*y2;
if (residual >=0.0) {
y3 = std::sqrt(residual);
return true;
}
// we are outside the sphere
if (!isIntersectionNonEmpty()) {
y3=0.0;
return false;
}
// intersection is non-empty but projection point is outside sphere
// so take rightmost point
y3 = 0.0;
y1 = topValue_;
y2 = std::sqrt(r_*r_ -y1*y1);
return true;
}
std::vector<Real> sphereCylinderOptimizerClosest(Real r,
Real s,
Real alpha,
Real z1,
Real z2,
Real z3,
Natural maxIterations,
Real tolerance,
Real zweight)
{
SphereCylinderOptimizer optimizer(r, s, alpha, z1, z2, z3, zweight);
std::vector<Real> y(3);
QL_REQUIRE(optimizer.isIntersectionNonEmpty(),
"intersection empty so no solution");
if (maxIterations ==0)
optimizer.findByProjection(y[0], y[1], y[2]);
else
optimizer.findClosest(maxIterations, tolerance, y[0], y[1], y[2]);
return y;
}
}
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