Polynomials module

This module provides a QR based polynomial solver. More...

Classes
class	Eigen::PolynomialSolver< Scalar_, Deg_ >
	A polynomial solver. More...
class	Eigen::PolynomialSolverBase< Scalar_, Deg_ >
	Defined to be inherited by polynomial solvers: it provides convenient methods such as. More...

Functions
template<typename Polynomial >
NumTraits< typename Polynomial::Scalar >::Real	Eigen::cauchy_max_bound (const Polynomial &poly)
template<typename Polynomial >
NumTraits< typename Polynomial::Scalar >::Real	Eigen::cauchy_min_bound (const Polynomial &poly)
template<typename Polynomials , typename T >
T	Eigen::poly_eval (const Polynomials &poly, const T &x)
template<typename Polynomials , typename T >
T	Eigen::poly_eval_horner (const Polynomials &poly, const T &x)
template<typename RootVector , typename Polynomial >
void	Eigen::roots_to_monicPolynomial (const RootVector &rv, Polynomial &poly)

Detailed Description

This module provides a QR based polynomial solver.

To use this module, add

#include <unsupported/Eigen/Polynomials>

at the start of your source file.

Function Documentation

cauchy_max_bound()

template<typename Polynomial >

NumTraits<typename Polynomial::Scalar>::Real Eigen::cauchy_max_bound ( const Polynomial &  poly )

inline

Returns

a maximum bound for the absolute value of any root of the polynomial.

Parameters

[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. \( 1 + 3x^2 \) is stored as a vector \( [ 1, 0, 3 ] \).

Precondition

the leading coefficient of the input polynomial poly must be non zero

Definition at line 77 of file PolynomialUtils.h.

{
  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);
}

cauchy_min_bound()

template<typename Polynomial >

NumTraits<typename Polynomial::Scalar>::Real Eigen::cauchy_min_bound ( const Polynomial &  poly )

inline

Returns

a minimum bound for the absolute value of any non zero root of the polynomial.

Parameters

[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. \( 1 + 3x^2 \) is stored as a vector \( [ 1, 0, 3 ] \).

Definition at line 100 of file PolynomialUtils.h.

{
  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;
}

poly_eval()

template<typename Polynomials , typename T >

T Eigen::poly_eval ( const Polynomials &  poly, const T &  x  )

inline

Returns

the evaluation of the polynomial at x using stabilized Horner algorithm.

Parameters

[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. \( 1 + 3x^2 \) is stored as a vector \( [ 1, 0, 3 ] \).
[in]	x	: the value to evaluate the polynomial at.

Definition at line 48 of file PolynomialUtils.h.

{
  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;
  }
}

poly_eval_horner()

template<typename Polynomials , typename T >

T Eigen::poly_eval_horner ( const Polynomials &  poly, const T &  x  )

inline

Returns

the evaluation of the polynomial at x using Horner algorithm.

Parameters

[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. \( 1 + 3x^2 \) is stored as a vector \( [ 1, 0, 3 ] \).
[in]	x	: the value to evaluate the polynomial at.

Note

for stability: \( |x| \le 1 \)

Definition at line 30 of file PolynomialUtils.h.

{
  T val=poly[poly.size()-1];
  for(DenseIndex i=poly.size()-2; i>=0; --i ){
    val = val*x + poly[i]; }
  return val;
}

roots_to_monicPolynomial()

template<typename RootVector , typename Polynomial >

void Eigen::roots_to_monicPolynomial	(	const RootVector &	rv,
		Polynomial &	poly
	)

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.

Parameters

[in]	rv	: a vector containing the roots of a polynomial.
[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. \( 3 + x^2 \) is stored as a vector \( [ 3, 0, 1 ] \).

Definition at line 129 of file PolynomialUtils.h.

{
 
  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];
  }
}