Sturm::Sequence< Real > Class Template Reference

Sturm sequence class. More...

#include <Sequence.hh>

Public Types
using	Vector
using	Interval

Public Member Functions
	Sequence ()
	Class constructor for the Sturm sequence.
	Sequence (const Poly< Real > &p)
	Class constructor for the Sturm sequence given a polynomial.
Real	a () const
	Get the lower bound of the interval containing the roots.
Real	b () const
	Get the upper bound of the interval containing the roots.
void	build (const Poly< Real > &p)
	Given the polynomial \( p(x) \) build its Sturm sequence.
Integer	length () const
	Get the length of the stored Sturm sequence.
const Poly< Real > &	get (Integer i) const
	Get the \( i \)-th polynomial of the stored Sturm sequence.
Integer	sign_variations (Real x, bool &on_root) const
Integer	separate_roots (Real a_in, Real b_in)
	Compute the subintervals containing a single root.
Integer	separate_roots ()
	Compute all the subintervals containing a single root in \( [a, b] \).
Integer	roots_number () const
	Get the number of roots found.
const Interval &	interval (Integer i) const
template<typename SolveFunction>
Vector	refine_roots (SolveFunction &&solve_function, bool verbose=false)
	Compute the roots in the intervals after the separation.

Private Attributes
std::vector< Poly< Real > >	m_sequence
std::vector< Interval >	m_intervals
Real	m_a {0.0}
Real	m_b {0.0}

Detailed Description

template<typename Real>
class Sturm::Sequence< Real >

This class implements the Sturm sequence for a given polynomial \( p(x) \). Such sequence is a sequence of polynomials \( p_0(x), p_1(x), \ldots, p_n(x) \) and allows to compute the number roots in a given interval \([a, b] \).

Template Parameters

Real	Scalar number type.

Member Typedef Documentation

Interval

template<typename Real>

using Sturm::Sequence< Real >::Interval

Initial value:

struct Interval {
     Real a;         
     Real b;         
     Integer va;     
     Integer vb;     
     bool a_on_root; 
     bool b_on_root; 
   }

Interval structure.

Vector

template<typename Real>

using Sturm::Sequence< Real >::Vector

Initial value:

 
Eigen::Vector<Real, Eigen::Dynamic>

Vector of real numbers.

Constructor & Destructor Documentation

Sequence() [1/2]

template<typename Real>

Sturm::Sequence< Real >::Sequence ( )

inline

Sequence() [2/2]

template<typename Real>

Sturm::Sequence< Real >::Sequence ( const Poly< Real > & p )

inline

Class constructor for the Sturm sequence given the polynomial \( p(x) \).

Parameters

[in]

p

Polynomial.

Member Function Documentation

a()

template<typename Real>

Real Sturm::Sequence< Real >::a ( ) const

inline

Returns

The lower bound of the interval containing the roots.

b()

template<typename Real>

Real Sturm::Sequence< Real >::b ( ) const

inline

Returns

The upper bound of the interval containing the roots.

build()

template<typename Real>

void Sturm::Sequence< Real >::build ( const Poly< Real > & p )

inline

get()

template<typename Real>

const Poly< Real > & Sturm::Sequence< Real >::get ( Integer i ) const

inline

Returns

The \( i \)-th polynomial of the stored Sturm sequence.

interval()

template<typename Real>

const Interval & Sturm::Sequence< Real >::interval ( Integer i ) const

inline

Get the \( i \)-th interval containing a single root.

Returns

The \( i \)-th interval containing a single root.

length()

template<typename Real>

Integer Sturm::Sequence< Real >::length ( ) const

inline

Returns

Length of the stored Sturm sequence.

refine_roots()

template<typename Real>

template<typename SolveFunction>

Vector Sturm::Sequence< Real >::refine_roots ( SolveFunction && solve_function, bool verbose = false )

inline

Parameters

[in]	solve_function	Lambda function for the root solver.
[in]	verbose	True if the function should print warnings.

Returns

A vector with the computed roots.

Template Parameters

Solver

Lambda function for the root solver with the signature bool(Real a, Real b, std::function<Real(Real)> f, Real & x).

roots_number()

template<typename Real>

Integer Sturm::Sequence< Real >::roots_number ( ) const

inline

Returns

The number of roots found.

separate_roots() [1/2]

template<typename Real>

Integer Sturm::Sequence< Real >::separate_roots ( )

inline

Compute an interval \( [a, b] \) that contains all the real roots and compute the subintervals containing a single root.

Returns

The numbers of intervals (roots) found.

separate_roots() [2/2]

template<typename Real>

Integer Sturm::Sequence< Real >::separate_roots ( Real a_in, Real b_in )

inline

Given an interval \( [a, b] \) compute the subintervals containing a single root.

Returns

The numbers of intervals (roots) found.

sign_variations()

template<typename Real>

Integer Sturm::Sequence< Real >::sign_variations ( Real x, bool & on_root ) const

inline

Compute the sign variations of the stored Sturm sequence at \( x \).

Returns

The number of sign variations.

Member Data Documentation

m_a

template<typename Real>

Real Sturm::Sequence< Real >::m_a {0.0}

private

Lower bound of the interval containing the roots.

m_b

template<typename Real>

Real Sturm::Sequence< Real >::m_b {0.0}

private

Upper bound of the interval containing the roots.

m_intervals

template<typename Real>

std::vector<Interval> Sturm::Sequence< Real >::m_intervals

private

Computed intervals.

m_sequence

template<typename Real>

std::vector<Poly<Real> > Sturm::Sequence< Real >::m_sequence

private

Sturm sequence.

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The documentation for this class was generated from the following file:

• core/include/Sturm/Sequence.hh