DGtal::LagrangeInterpolation< TEuclideanRing > Class Template Reference

Aim: This class implements Lagrange basis functions and Lagrange interpolation. More...

#include <DGtal/math/LagrangeInterpolation.h>

Inheritance diagram for DGtal::LagrangeInterpolation< TEuclideanRing >:

Public Types
typedef LagrangeInterpolation< TEuclideanRing >	Self
typedef TEuclideanRing	Ring
typedef DGtal::MPolynomial< 1, Ring >	Polynomial
	The monovariate polynomial type.
typedef std::size_t	Size

Public Member Functions
	BOOST_CONCEPT_ASSERT ((concepts::CEuclideanRing< Ring >))
	~LagrangeInterpolation ()=default
	LagrangeInterpolation ()
	LagrangeInterpolation (const std::vector< Ring > &xvalues)
	LagrangeInterpolation (const LagrangeInterpolation &other)=default
	LagrangeInterpolation (LagrangeInterpolation &&other)=default
LagrangeInterpolation &	operator= (const LagrangeInterpolation &other)=default
LagrangeInterpolation &	operator= (LagrangeInterpolation &&other)=default
Size	size () const
Size	degree () const
Ring	denominator () const
void	init (const std::vector< Ring > &xvalues)
Polynomial	polynomial (const std::vector< Ring > &yvalues)
Polynomial	basis (Size i) const
void	selfDisplay (std::ostream &that_stream) const
bool	OK () const

Protected Attributes
std::vector< Ring >	myX
	The vector of X-values (abscissa)
Ring	myDeterminant
	The determinant of the Vandermonde matrix corresponding to X-values.
std::vector< Polynomial >	myLagrangeBasis
	The Lagrange polynomial basis corresponding to X-values.

Detailed Description

template<typename TEuclideanRing>
class DGtal::LagrangeInterpolation< TEuclideanRing >

Aim: This class implements Lagrange basis functions and Lagrange interpolation.

Description of class 'LagrangeInterpolation'

See also

testLagrangeInterpolation.cpp

Template Parameters

TEuclideanRing

any model of CEuclideanRing like int, double, etc.

Definition at line 62 of file LagrangeInterpolation.h.

Member Typedef Documentation

Polynomial

template<typename TEuclideanRing>

typedef DGtal::MPolynomial< 1, Ring > DGtal::LagrangeInterpolation< TEuclideanRing >::Polynomial

The monovariate polynomial type.

Definition at line 72 of file LagrangeInterpolation.h.

Ring

template<typename TEuclideanRing>

typedef TEuclideanRing DGtal::LagrangeInterpolation< TEuclideanRing >::Ring

Definition at line 68 of file LagrangeInterpolation.h.

Self

template<typename TEuclideanRing>

typedef LagrangeInterpolation< TEuclideanRing > DGtal::LagrangeInterpolation< TEuclideanRing >::Self

Definition at line 67 of file LagrangeInterpolation.h.

Size

template<typename TEuclideanRing>

typedef std::size_t DGtal::LagrangeInterpolation< TEuclideanRing >::Size

Definition at line 73 of file LagrangeInterpolation.h.

Constructor & Destructor Documentation

~LagrangeInterpolation()

template<typename TEuclideanRing>

DGtal::LagrangeInterpolation< TEuclideanRing >::~LagrangeInterpolation ( )

default

Destructor.

LagrangeInterpolation() [1/4]

template<typename TEuclideanRing>

DGtal::LagrangeInterpolation< TEuclideanRing >::LagrangeInterpolation ( )

inline

Constructor. The object is invalid.

Definition at line 86 of file LagrangeInterpolation.h.

      : myDeterminant( NumberTraits< Ring >::ZERO )
    {}

LagrangeInterpolation() [2/4]

template<typename TEuclideanRing>

DGtal::LagrangeInterpolation< TEuclideanRing >::LagrangeInterpolation ( const std::vector< Ring > & xvalues )

inline

Constructs the Lagrange interpolation object with its x-values

Parameters

xvalues

the vector of X-values, corresponding to the abscissa where the interpolation will take place.

Definition at line 94 of file LagrangeInterpolation.h.

    {
      init( xvalues );
    }

LagrangeInterpolation() [3/4]

template<typename TEuclideanRing>

DGtal::LagrangeInterpolation< TEuclideanRing >::LagrangeInterpolation ( const LagrangeInterpolation< TEuclideanRing > & other )

default

Copy constructor.

Parameters

other

the object to clone.

LagrangeInterpolation() [4/4]

template<typename TEuclideanRing>

DGtal::LagrangeInterpolation< TEuclideanRing >::LagrangeInterpolation ( LagrangeInterpolation< TEuclideanRing > && other )

default

Move constructor.

Parameters

other

the object to clone.

Member Function Documentation

basis()

template<typename TEuclideanRing>

Polynomial DGtal::LagrangeInterpolation< TEuclideanRing >::basis ( Size i ) const

inline

Parameters

i	any value between 0 included and size() excluded.

Returns

the numerator of i-th Lagrange basis polynomial (divide by denominator to get the exact Lagrange basis polynomial).

Definition at line 199 of file LagrangeInterpolation.h.

    {
      ASSERT( i < size() );
      return myLagrangeBasis[ i ];
    }

Referenced by DGtal::LagrangeInterpolation< Integer >::selfDisplay().

BOOST_CONCEPT_ASSERT()

template<typename TEuclideanRing>

DGtal::LagrangeInterpolation< TEuclideanRing >::BOOST_CONCEPT_ASSERT

(

(concepts::CEuclideanRing< Ring >)

)

degree()

template<typename TEuclideanRing>

Size DGtal::LagrangeInterpolation< TEuclideanRing >::degree ( ) const

inline

Returns

the degree of Lagrange polynomials (i.e. size()-1)

Definition at line 132 of file LagrangeInterpolation.h.

    {
      return size() - 1;
    }

denominator()

template<typename TEuclideanRing>

Ring DGtal::LagrangeInterpolation< TEuclideanRing >::denominator ( ) const

inline

Returns

the common denominator to Lagrange bases and Lagrange polynomial.

Definition at line 138 of file LagrangeInterpolation.h.

    {
      return myDeterminant;
    }

init()

template<typename TEuclideanRing>

void DGtal::LagrangeInterpolation< TEuclideanRing >::init ( const std::vector< Ring > & xvalues )

inline

Initializes the Lagrange interpolation object with its x-values, and computes the corresponding Lagrange Basis.

Parameters

xvalues

the vector of X-values, corresponding to the abscissa where the interpolation will take place.

Definition at line 148 of file LagrangeInterpolation.h.

    {
      myX           = xvalues;
      // Compute Vandermonde determinant
      myDeterminant = NumberTraits< Ring >::ONE;
      for ( Dimension i = 0; (i+1) < size(); ++i )
        for ( Dimension j = i + 1; j < size(); ++j )
          myDeterminant *= myX[ j ] - myX[ i ];
      // compute Lagrange polynomial basis.
      myLagrangeBasis.clear();
      myLagrangeBasis.reserve( size() );
      for ( Dimension j = 0; j < size(); ++j )
        {
          Ring c = myDeterminant;
          for ( Dimension m = 0; m < size(); ++m )
            if ( m != j ) 
              c /= myX[ j ] - myX[ m ];
          Polynomial P = c; // constant
          for ( Dimension m = 0; m < size(); ++m )
            if ( m != j ) 
              P *= mmonomial<Ring>( 1 ) - myX[ m ] * mmonomial<Ring>( 0 );
          myLagrangeBasis.push_back( P );
        }
    }

Referenced by DGtal::LagrangeInterpolation< Integer >::LagrangeInterpolation().

OK()

template<typename TEuclideanRing>

bool DGtal::LagrangeInterpolation< TEuclideanRing >::OK ( ) const

inline

Checks the validity/consistency of the object.

Returns

'true' if the object is valid, 'false' otherwise.

Definition at line 224 of file LagrangeInterpolation.h.

    {
      return myDeterminant != NumberTraits< Ring >::ZERO;
    }

operator=() [1/2]

template<typename TEuclideanRing>

LagrangeInterpolation & DGtal::LagrangeInterpolation< TEuclideanRing >::operator= ( const LagrangeInterpolation< TEuclideanRing > & other )

default

Assignment.

Parameters

other

the object to copy.

Returns

a reference on 'this'.

operator=() [2/2]

template<typename TEuclideanRing>

LagrangeInterpolation & DGtal::LagrangeInterpolation< TEuclideanRing >::operator= ( LagrangeInterpolation< TEuclideanRing > && other )

default

Move assignment.

Parameters

other

the object to move.

Returns

a reference on 'this'.

polynomial()

template<typename TEuclideanRing>

Polynomial DGtal::LagrangeInterpolation< TEuclideanRing >::polynomial ( const std::vector< Ring > & yvalues )

inline

Parameters

[in]

yvalues

the y-values corresponding to the current x-values, in the same order.

Returns

the numerator of the Lagrange polynomial that interpolates all given values (divide by denominator to get the exact Lagrange polynomial).

Note

if yvalues has not the correct size, then the zero polynomial is returned.

Definition at line 182 of file LagrangeInterpolation.h.

    {
      Polynomial P;
      if ( yvalues.size() != size() ) return P;
      for ( Dimension j = 0; j < size(); ++j )
        P += yvalues[ j ] * myLagrangeBasis[ j ];
      return P;
    }

selfDisplay()

template<typename TEuclideanRing>

void DGtal::LagrangeInterpolation< TEuclideanRing >::selfDisplay ( std::ostream & that_stream ) const

inline

Writes/Displays the object on an output stream.

Parameters

that_stream

the output stream where the object is written.

Definition at line 212 of file LagrangeInterpolation.h.

    {
      that_stream << "[LagrangeInterpolation det=" << myDeterminant << std::endl;
      for ( Size i = 0; i < size(); i++ )
        that_stream << "l_" << i << "=" << basis( i ) << std::endl;
      that_stream << "]";
    }

Referenced by DGtal::operator<<().

size()

template<typename TEuclideanRing>

Size DGtal::LagrangeInterpolation< TEuclideanRing >::size ( ) const

inline

Returns

the number of interpolation points (or number of x-values)

Definition at line 126 of file LagrangeInterpolation.h.

    {
      return myX.size();
    }

Referenced by DGtal::LagrangeInterpolation< Integer >::basis(), DGtal::LagrangeInterpolation< Integer >::degree(), DGtal::LagrangeInterpolation< Integer >::init(), DGtal::LagrangeInterpolation< Integer >::polynomial(), and DGtal::LagrangeInterpolation< Integer >::selfDisplay().

Field Documentation

myDeterminant

template<typename TEuclideanRing>

Ring DGtal::LagrangeInterpolation< TEuclideanRing >::myDeterminant

protected

The determinant of the Vandermonde matrix corresponding to X-values.

Definition at line 235 of file LagrangeInterpolation.h.

myLagrangeBasis

template<typename TEuclideanRing>

std::vector<Polynomial> DGtal::LagrangeInterpolation< TEuclideanRing >::myLagrangeBasis

protected

The Lagrange polynomial basis corresponding to X-values.

Definition at line 237 of file LagrangeInterpolation.h.

myX

template<typename TEuclideanRing>

std::vector< Ring > DGtal::LagrangeInterpolation< TEuclideanRing >::myX

protected

The vector of X-values (abscissa)

Definition at line 233 of file LagrangeInterpolation.h.

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

• LagrangeInterpolation.h