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Diffstat (limited to 'unsupported/Eigen/src/Polynomials/PolynomialUtils.h')
-rw-r--r-- | unsupported/Eigen/src/Polynomials/PolynomialUtils.h | 141 |
1 files changed, 141 insertions, 0 deletions
diff --git a/unsupported/Eigen/src/Polynomials/PolynomialUtils.h b/unsupported/Eigen/src/Polynomials/PolynomialUtils.h new file mode 100644 index 000000000..c23204c65 --- /dev/null +++ b/unsupported/Eigen/src/Polynomials/PolynomialUtils.h @@ -0,0 +1,141 @@ +// 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 + +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$. + * \param[in] x : the value to evaluate the polynomial at. + * + * <i><b>Note for stability:</b></i> + * <dd> \f$ |x| \le 1 \f$ </dd> + */ +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$. + * \param[in] x : the value to evaluate the polynomial at. + */ +template <typename Polynomials, typename T> +inline +T poly_eval( const Polynomials& poly, const T& x ) +{ + typedef typename NumTraits<T>::Real Real; + + if( internal::abs2( x ) <= Real(1) ){ + return poly_eval_horner( poly, x ); } + else + { + T val=poly[0]; + T inv_x = T(1)/x; + for( DenseIndex i=1; i<poly.size(); ++i ){ + val = val*inv_x + poly[i]; } + + return std::pow(x,(T)(poly.size()-1)) * val; + } +} + +/** \ingroup Polynomials_Module + * \returns a maximum bound for the absolute value of any root of the polynomial. + * + * \param[in] poly : the vector of coefficients of the polynomial ordered + * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial + * e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$. + * + * <i><b>Precondition:</b></i> + * <dd> the leading coefficient of the input polynomial poly must be non zero </dd> + */ +template <typename Polynomial> +inline +typename NumTraits<typename Polynomial::Scalar>::Real cauchy_max_bound( const Polynomial& poly ) +{ + typedef typename Polynomial::Scalar Scalar; + typedef typename NumTraits<Scalar>::Real Real; + + assert( Scalar(0) != poly[poly.size()-1] ); + const Scalar inv_leading_coeff = Scalar(1)/poly[poly.size()-1]; + Real cb(0); + + for( DenseIndex i=0; i<poly.size()-1; ++i ){ + cb += internal::abs(poly[i]*inv_leading_coeff); } + return cb + Real(1); +} + +/** \ingroup Polynomials_Module + * \returns a minimum bound for the absolute value of any non zero root of the polynomial. + * \param[in] poly : the vector of coefficients of the polynomial ordered + * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial + * e.g. \f$ 1 + 3x^2 \f$ is stored as a vector \f$ [ 1, 0, 3 ] \f$. + */ +template <typename Polynomial> +inline +typename NumTraits<typename Polynomial::Scalar>::Real cauchy_min_bound( const Polynomial& poly ) +{ + typedef typename Polynomial::Scalar Scalar; + typedef typename NumTraits<Scalar>::Real Real; + + DenseIndex i=0; + while( i<poly.size()-1 && Scalar(0) == poly(i) ){ ++i; } + if( poly.size()-1 == i ){ + return Real(1); } + + const Scalar inv_min_coeff = Scalar(1)/poly[i]; + Real cb(1); + for( DenseIndex j=i+1; j<poly.size(); ++j ){ + cb += internal::abs(poly[j]*inv_min_coeff); } + return Real(1)/cb; +} + +/** \ingroup Polynomials_Module + * Given the roots of a polynomial compute the coefficients in the + * monomial basis of the monic polynomial with same roots and minimal degree. + * If RootVector is a vector of complexes, Polynomial should also be a vector + * of complexes. + * \param[in] rv : a vector containing the roots of a polynomial. + * \param[out] poly : the vector of coefficients of the polynomial ordered + * by degrees i.e. poly[i] is the coefficient of degree i of the polynomial + * e.g. \f$ 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 |