414 lines
14 KiB
C++
414 lines
14 KiB
C++
// -*- mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
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// vi: set et ts=4 sw=4 sts=4:
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/*
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This file is part of the Open Porous Media project (OPM).
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OPM is free software: you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation, either version 2 of the License, or
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(at your option) any later version.
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OPM is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with OPM. If not, see <http://www.gnu.org/licenses/>.
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Consult the COPYING file in the top-level source directory of this
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module for the precise wording of the license and the list of
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copyright holders.
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*/
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/*!
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* \file
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* \brief Provides free functions to invert polynomials of degree 1, 2 and 3.
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*/
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#ifndef OPM_POLYNOMIAL_UTILS_HH
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#define OPM_POLYNOMIAL_UTILS_HH
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#include <cmath>
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#include <algorithm>
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#include <opm/material/common/MathToolbox.hpp>
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namespace Opm {
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/*!
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* \ingroup Math
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* \brief Invert a linear polynomial analytically
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*
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* The polynomial is defined as
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* \f[ p(x) = a\; x + b \f]
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*
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* This method Returns the number of solutions which are in the real
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* numbers, i.e. 1 except if the slope of the line is 0.
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*
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* \param sol Container into which the solutions are written
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* \param a The coefficient for the linear term
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* \param b The coefficient for the constant term
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*/
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template <class Scalar, class SolContainer>
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unsigned invertLinearPolynomial(SolContainer& sol,
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Scalar a,
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Scalar b)
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{
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if (std::abs(scalarValue(a)) < 1e-30)
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return 0;
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sol[0] = -b/a;
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return 1;
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}
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/*!
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* \ingroup Math
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* \brief Invert a quadratic polynomial analytically
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*
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* The polynomial is defined as
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* \f[ p(x) = a\; x^2 + + b\;x + c \f]
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*
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* This method teturns the number of solutions which are in the real
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* numbers. The "sol" argument contains the real roots of the parabola
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* in order with the smallest root first.
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*
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* \param sol Container into which the solutions are written
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* \param a The coefficient for the quadratic term
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* \param b The coefficient for the linear term
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* \param c The coefficient for the constant term
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*/
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template <class Scalar, class SolContainer>
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unsigned invertQuadraticPolynomial(SolContainer& sol,
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Scalar a,
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Scalar b,
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Scalar c)
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{
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// check for a line
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if (std::abs(scalarValue(a)) < 1e-30)
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return invertLinearPolynomial(sol, b, c);
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// discriminant
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Scalar Delta = b*b - 4*a*c;
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if (Delta < 0)
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return 0; // no real roots
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Delta = sqrt(Delta);
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sol[0] = (- b + Delta)/(2*a);
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sol[1] = (- b - Delta)/(2*a);
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// sort the result
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if (sol[0] > sol[1])
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std::swap(sol[0], sol[1]);
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return 2; // two real roots
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}
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//! \cond SKIP_THIS
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template <class Scalar, class SolContainer>
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void invertCubicPolynomialPostProcess_(SolContainer& sol,
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int numSol,
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Scalar a,
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Scalar b,
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Scalar c,
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Scalar d)
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{
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// do one Newton iteration on the analytic solution if the
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// precision is increased
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for (int i = 0; i < numSol; ++i) {
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Scalar x = sol[i];
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Scalar fOld = d + x*(c + x*(b + x*a));
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Scalar fPrime = c + x*(2*b + x*3*a);
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if (std::abs(scalarValue(fPrime)) < 1e-30)
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continue;
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x -= fOld/fPrime;
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Scalar fNew = d + x*(c + x*(b + x*a));
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if (std::abs(scalarValue(fNew)) < std::abs(scalarValue(fOld)))
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sol[i] = x;
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}
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}
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//! \endcond
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/*!
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* \ingroup Math
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* \brief Invert a cubic polynomial analytically
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*
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* The polynomial is defined as
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* \f[ p(x) = a\; x^3 + + b\;x^3 + c\;x + d \f]
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*
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* This method teturns the number of solutions which are in the real
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* numbers. The "sol" argument contains the real roots of the cubic
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* polynomial in order with the smallest root first.
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*
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* \param sol Container into which the solutions are written
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* \param a The coefficient for the cubic term
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* \param b The coefficient for the quadratic term
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* \param c The coefficient for the linear term
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* \param d The coefficient for the constant term
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*/
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template <class Scalar, class SolContainer>
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unsigned invertCubicPolynomial(SolContainer* sol,
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Scalar a,
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Scalar b,
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Scalar c,
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Scalar d)
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{
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// reduces to a quadratic polynomial
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if (std::abs(scalarValue(a)) < 1e-30)
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return invertQuadraticPolynomial(sol, b, c, d);
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// normalize the polynomial
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b /= a;
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c /= a;
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d /= a;
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a = 1;
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// get rid of the quadratic term by subsituting x = t - b/3
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Scalar p = c - b*b/3;
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Scalar q = d + (2*b*b*b - 9*b*c)/27;
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if (std::abs(scalarValue(p)) > 1e-30 && std::abs(scalarValue(q)) > 1e-30) {
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// At this point
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//
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// t^3 + p*t + q = 0
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//
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// with p != 0 and q != 0 holds. Introducing the variables u and v
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// with the properties
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//
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// u + v = t and 3*u*v + p = 0
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//
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// leads to
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//
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// u^3 + v^3 + q = 0 .
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//
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// multiplying both sides with u^3 and taking advantage of the
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// fact that u*v = -p/3 leads to
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//
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// u^6 + q*u^3 - p^3/27 = 0
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//
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// Now, substituting u^3 = w yields
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//
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// w^2 + q*w - p^3/27 = 0
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//
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// This is a quadratic equation with the solutions
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//
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// w = -q/2 + sqrt(q^2/4 + p^3/27) and
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// w = -q/2 - sqrt(q^2/4 + p^3/27)
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//
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// Since w is equivalent to u^3 it is sufficient to only look at
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// one of the two cases. Then, there are still 2 cases: positive
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// and negative discriminant.
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Scalar wDisc = q*q/4 + p*p*p/27;
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if (wDisc >= 0) { // the positive discriminant case:
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// calculate the cube root of - q/2 + sqrt(q^2/4 + p^3/27)
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Scalar u = - q/2 + sqrt(wDisc);
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if (u < 0) u = - pow(-u, 1.0/3);
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else u = pow(u, 1.0/3);
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// at this point, u != 0 since p^3 = 0 is necessary in order
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// for u = 0 to hold, so
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sol[0] = u - p/(3*u) - b/3;
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// does not produce a division by zero. the remaining two
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// roots of u are rotated by +- 2/3*pi in the complex plane
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// and thus not considered here
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invertCubicPolynomialPostProcess_(sol, 1, a, b, c, d);
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return 1;
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}
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else { // the negative discriminant case:
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Scalar uCubedRe = - q/2;
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Scalar uCubedIm = sqrt(-wDisc);
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// calculate the cube root of - q/2 + sqrt(q^2/4 + p^3/27)
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Scalar uAbs = pow(sqrt(uCubedRe*uCubedRe + uCubedIm*uCubedIm), 1.0/3);
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Scalar phi = atan2(uCubedIm, uCubedRe)/3;
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// calculate the length and the angle of the primitive root
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// with the definitions from above it follows that
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//
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// x = u - p/(3*u) - b/3
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//
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// where x and u are complex numbers. Rewritten in polar form
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// this is equivalent to
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//
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// x = |u|*e^(i*phi) - p*e^(-i*phi)/(3*|u|) - b/3 .
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//
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// Factoring out the e^ terms and subtracting the additional
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// terms, yields
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//
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// x = (e^(i*phi) + e^(-i*phi))*(|u| - p/(3*|u|)) - y - b/3
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//
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// with
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//
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// y = - |u|*e^(-i*phi) + p*e^(i*phi)/(3*|u|) .
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//
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// The crucial observation is the fact that y is the conjugate
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// of - x + b/3. This means that after taking advantage of the
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// relation
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//
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// e^(i*phi) + e^(-i*phi) = 2*cos(phi)
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//
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// the equation
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//
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// x = 2*cos(phi)*(|u| - p / (3*|u|)) - conj(x) - 2*b/3
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//
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// holds. Since |u|, p, b and cos(phi) are real numbers, it
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// follows that Im(x) = - Im(x) and thus Im(x) = 0. This
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// implies
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//
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// Re(x) = x = cos(phi)*(|u| - p / (3*|u|)) - b/3 .
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//
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// Considering the fact that u is a cubic root, we have three
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// values for phi which differ by 2/3*pi. This allows to
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// calculate the three real roots of the polynomial:
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for (int i = 0; i < 3; ++i) {
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sol[i] = cos(phi)*(uAbs - p/(3*uAbs)) - b/3;
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phi += 2*M_PI/3;
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}
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// post process the obtained solution to increase numerical
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// precision
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invertCubicPolynomialPostProcess_(sol, 3, a, b, c, d);
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// sort the result
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std::sort(sol, sol + 3);
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return 3;
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}
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}
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// Handle some (pretty unlikely) special cases required to avoid
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// divisions by zero in the code above...
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else if (std::abs(scalarValue(p)) < 1e-30 && std::abs(scalarValue(q)) < 1e-30) {
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// t^3 = 0, i.e. triple root at t = 0
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sol[0] = sol[1] = sol[2] = 0.0 - b/3;
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return 3;
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}
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else if (std::abs(scalarValue(p)) < 1e-30 && std::abs(scalarValue(q)) > 1e-30) {
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// t^3 + q = 0,
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//
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// i. e. single real root at t=curt(q)
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Scalar t;
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if (-q > 0) t = pow(-q, 1./3);
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else t = - pow(q, 1./3);
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sol[0] = t - b/3;
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return 1;
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}
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assert(std::abs(scalarValue(p)) > 1e-30 && std::abs(scalarValue(q)) > 1e-30);
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// t^3 + p*t = 0 = t*(t^2 + p),
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//
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// i. e. roots at t = 0, t^2 + p = 0
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if (p > 0) {
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sol[0] = 0.0 - b/3;
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return 1; // only a single real root at t=0
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}
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// two additional real roots at t = sqrt(-p) and t = -sqrt(-p)
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sol[0] = -sqrt(-p) - b/3;;
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sol[1] = 0.0 - b/3;
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sol[2] = sqrt(-p) - b/3;
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return 3;
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}
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/*!
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* \ingroup Math
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* \brief Invert a cubic polynomial analytically
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*
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* The polynomial is defined as
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* \f[ p(x) = a\; x^3 + + b\;x^3 + c\;x + d \f]
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*
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* This method returns the number of solutions which are in the real
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* numbers. The "sol" argument contains the real roots of the cubic
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* polynomial in order with the smallest root first.
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*
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* \param sol Container into which the solutions are written
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* \param a The coefficient for the cubic term
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* \param b The coefficient for the quadratic term
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* \param c The coefficient for the linear term
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* \param d The coefficient for the constant term
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*/
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template <class Scalar, class SolContainer>
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unsigned cubicRoots(SolContainer* sol,
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Scalar a,
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Scalar b,
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Scalar c,
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Scalar d)
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{
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// reduces to a quadratic polynomial
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if (std::abs(scalarValue(a)) < 1e-30)
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return invertQuadraticPolynomial(sol, b, c, d);
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// We need to reduce the cubic equation to its "depressed cubic" form (however strange that sounds)
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// Depressed cubic form: t^3 + p*t + q, where x = t - b/3*a is the transform we use when we have
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// roots for t. p and q are defined below.
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// Formula for p and q:
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Scalar p = (3.0 * a * c - b * b) / (3.0 * a * a);
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Scalar q = (2.0 * b * b * b - 9.0 * a * b * c + 27.0 * d * a * a) / (27.0 * a * a * a);
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// Check if we have three or one real root by looking at the discriminant, and solve accordingly with
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// correct formula
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Scalar discr = 4.0 * p * p * p + 27.0 * q * q;
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if (discr < 0.0) {
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// Find three real roots of a depressed cubic, using the trigonometric method
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// Help calculation
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Scalar theta = (1.0 / 3.0) * acos( ((3.0 * q) / (2.0 * p)) * sqrt(-3.0 / p) );
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// Calculate the three roots
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sol[0] = 2.0 * sqrt(-p / 3.0) * cos( theta ) - b / (3.0 * a);
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sol[1] = 2.0 * sqrt(-p / 3.0) * cos( theta - ((2.0 * M_PI) / 3.0) ) - b / (3.0 * a);
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sol[2] = 2.0 * sqrt(-p / 3.0) * cos( theta - ((4.0 * M_PI) / 3.0) ) - b / (3.0 * a);
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// Sort in ascending order
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std::sort(sol, sol + 3);
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// Return confirmation of three roots
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return 3;
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}
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else if (discr > 0.0) {
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// Find one real root of a depressed cubic using hyperbolic method. Different solutions depending on
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// sign of p
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Scalar t = 0;
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if (p < 0) {
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// Help calculation
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Scalar theta = (1.0 / 3.0) * acosh( ((-3.0 * abs(q)) / (2.0 * p)) * sqrt(-3.0 / p) );
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// Root
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t = ( (-2.0 * abs(q)) / q ) * sqrt(-p / 3.0) * cosh(theta);
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}
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else if (p > 0) {
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// Help calculation
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Scalar theta = (1.0 / 3.0) * asinh( ((3.0 * q) / (2.0 * p)) * sqrt(3.0 / p) );
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// Root
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t = -2.0 * sqrt(p / 3.0) * sinh(theta);
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}
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else {
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std::runtime_error(" p = 0 in cubic root solver!");
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}
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// Transform t to output solution
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sol[0] = t - b / (3.0 * a);
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return 1;
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}
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else {
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// The discriminant, 4*p^3 + 27*q^2 = 0, thus we have simple (real) roots
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// If p = 0 then also q = 0, and t = 0 is a triple root
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if (p == 0) {
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sol[0] = sol[1] = sol[2] = 0.0 - b / (3.0 * a);
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}
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// If p != 0, the we have a simple root and a double root
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else {
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sol[0] = (3.0 * q / p) - b / (3.0 * a);
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sol[1] = sol[2] = (-3.0 * q) / (2.0 * p) - b / (3.0 * a);
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std::sort(sol, sol + 3);
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}
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return 3;
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}
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}
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} // end Opm
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#endif
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