| // This file is part of Eigen, a lightweight C++ template library |
| // for linear algebra. |
| // |
| // Copyright (C) 2010 Manuel Yguel <manuel.yguel@gmail.com> |
| // |
| // This Source Code Form is subject to the terms of the Mozilla |
| // Public License v. 2.0. If a copy of the MPL was not distributed |
| // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. |
| |
| #ifndef EIGEN_POLYNOMIAL_UTILS_H |
| #define EIGEN_POLYNOMIAL_UTILS_H |
| |
| // IWYU pragma: private |
| #include "./InternalHeaderCheck.h" |
| |
| namespace Eigen { |
| |
| /** \ingroup Polynomials_Module |
| * \returns the evaluation of the polynomial at x using Horner algorithm. |
| * |
| * \param[in] poly : the vector of coefficients of the polynomial ordered |
| * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial |
| * e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$. |
| * \param[in] x : the value to evaluate the polynomial at. |
| * |
| * \note for stability: |
| * \f$ |x| \le 1 \f$ |
| */ |
| template <typename Polynomials, typename T> |
| inline T poly_eval_horner(const Polynomials& poly, const T& x) { |
| T val = poly[poly.size() - 1]; |
| for (DenseIndex i = poly.size() - 2; i >= 0; --i) { |
| val = val * x + poly[i]; |
| } |
| return val; |
| } |
| |
| /** \ingroup Polynomials_Module |
| * \returns the evaluation of the polynomial at x using stabilized Horner algorithm. |
| * |
| * \param[in] poly : the vector of coefficients of the polynomial ordered |
| * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial |
| * e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$. |
| * \param[in] x : the value to evaluate the polynomial at. |
| */ |
| template <typename Polynomials, typename T> |
| inline T poly_eval(const Polynomials& poly, const T& x) { |
| typedef typename NumTraits<T>::Real Real; |
| |
| if (numext::abs2(x) <= Real(1)) { |
| return poly_eval_horner(poly, x); |
| } else { |
| T val = poly[0]; |
| T inv_x = T(1) / x; |
| for (DenseIndex i = 1; i < poly.size(); ++i) { |
| val = val * inv_x + poly[i]; |
| } |
| |
| return numext::pow(x, (T)(poly.size() - 1)) * val; |
| } |
| } |
| |
| /** \ingroup Polynomials_Module |
| * \returns a maximum bound for the absolute value of any root of the polynomial. |
| * |
| * \param[in] poly : the vector of coefficients of the polynomial ordered |
| * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial |
| * e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$. |
| * |
| * \pre |
| * the leading coefficient of the input polynomial poly must be non zero |
| */ |
| template <typename Polynomial> |
| inline typename NumTraits<typename Polynomial::Scalar>::Real cauchy_max_bound(const Polynomial& poly) { |
| using std::abs; |
| typedef typename Polynomial::Scalar Scalar; |
| typedef typename NumTraits<Scalar>::Real Real; |
| |
| eigen_assert(Scalar(0) != poly[poly.size() - 1]); |
| const Scalar inv_leading_coeff = Scalar(1) / poly[poly.size() - 1]; |
| Real cb(0); |
| |
| for (DenseIndex i = 0; i < poly.size() - 1; ++i) { |
| cb += abs(poly[i] * inv_leading_coeff); |
| } |
| return cb + Real(1); |
| } |
| |
| /** \ingroup Polynomials_Module |
| * \returns a minimum bound for the absolute value of any non zero root of the polynomial. |
| * \param[in] poly : the vector of coefficients of the polynomial ordered |
| * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial |
| * e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$. |
| */ |
| template <typename Polynomial> |
| inline typename NumTraits<typename Polynomial::Scalar>::Real cauchy_min_bound(const Polynomial& poly) { |
| using std::abs; |
| typedef typename Polynomial::Scalar Scalar; |
| typedef typename NumTraits<Scalar>::Real Real; |
| |
| DenseIndex i = 0; |
| while (i < poly.size() - 1 && Scalar(0) == poly(i)) { |
| ++i; |
| } |
| if (poly.size() - 1 == i) { |
| return Real(1); |
| } |
| |
| const Scalar inv_min_coeff = Scalar(1) / poly[i]; |
| Real cb(1); |
| for (DenseIndex j = i + 1; j < poly.size(); ++j) { |
| cb += abs(poly[j] * inv_min_coeff); |
| } |
| return Real(1) / cb; |
| } |
| |
| /** \ingroup Polynomials_Module |
| * Given the roots of a polynomial compute the coefficients in the |
| * monomial basis of the monic polynomial with same roots and minimal degree. |
| * If RootVector is a vector of complexes, Polynomial should also be a vector |
| * of complexes. |
| * \param[in] rv : a vector containing the roots of a polynomial. |
| * \param[out] poly : the vector of coefficients of the polynomial ordered |
| * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial |
| * e.g. \f$ 3 + x^2 \f$ is stored as a vector \f$ [ 3, 0, 1 ] \f$. |
| */ |
| template <typename RootVector, typename Polynomial> |
| void roots_to_monicPolynomial(const RootVector& rv, Polynomial& poly) { |
| typedef typename Polynomial::Scalar Scalar; |
| |
| poly.setZero(rv.size() + 1); |
| poly[0] = -rv[0]; |
| poly[1] = Scalar(1); |
| for (DenseIndex i = 1; i < rv.size(); ++i) { |
| for (DenseIndex j = i + 1; j > 0; --j) { |
| poly[j] = poly[j - 1] - rv[i] * poly[j]; |
| } |
| poly[0] = -rv[i] * poly[0]; |
| } |
| } |
| |
| } // end namespace Eigen |
| |
| #endif // EIGEN_POLYNOMIAL_UTILS_H |