Template Class Hermite

• Defined in File opoly.hh

Inheritance Relationships

Base Type

• public opoly::poly< INT, REAL > (Template Class poly)

Class Documentation

template<typename INT, typename REAL>
class opoly::Hermite : public opoly::poly<INT, REAL>

Class defining the Hermite orthogonal polynomial

• INT is a integer type, can be also unsigned

• REAL is a floating point type, can be float or a high precision number class

Public Types

typedef INT int_type

typedef int_type *int_pointer

typedef int_type const *int_const_pointer

typedef int_type int_reference

typedef int_type const &int_const_reference

typedef REAL real_type

typedef real_type *pointer

typedef real_type const *const_pointer

typedef real_type &reference

typedef real_type const &const_reference

Public Functions

inline Hermite()

inline virtual real_type operator()(int_type n, const_reference x) const override

Evaluate the polynomial

Parameters

• n – [in] the degree op the polynomial

• x – [in] the value at which the polynomial is evaluated

Returns

the value \( p(x) \)

inline virtual int_type svar(int_type n, const_reference x) const override

Evaluate the sign variation of the corresponding sturm sequence.

Parameters

• n – [in] the degree op the polynomial

• x – [in] the value at which the polynomial is evaluated

Returns

the number of sign variations

inline virtual int_type eval(int_type n, const_reference x, reference p) const override

Evaluate the polynomial and the sign variation of the corresponding sturm sequence.

Parameters

• n – [in] the degree op the polynomial

• x – [in] the value at which the polynomial is evaluated

• p – [out] the value \( p(x) \)

Returns

the number of sign variations

inline virtual int_type eval(int_type n, const_reference x, reference p, reference dp) const override

Evaluate the polynomial and its derivative. Moreover return the sign variation of the corresponding sturm sequence.

Parameters

• n – [in] the degree op the polynomial

• x – [in] the value at which the polynomial is evaluated

• p – [out] the value \( p(x) \)

• dp – [out] the value \( p'(x) \)

Returns

the number of sign variations

inline virtual int_type sequence(int_type n, const_reference x, pointer p) const override

Evaluate the Sturm sequence

Parameters

• n – [in] the degree of the polynomial

• x – [in] the value at which the polynomial is evaluated

• pvec – [out] the sequence \( p_0(x) \), \( p_1(x) \), …, \( p_n(x) \) of the 3 term recurrence

Returns

the number of sign variations

inline virtual real_type weight(const_reference x) const override

Weight function of the orthogonal polynomial

Parameters

x – [in] the value at which the weight is evaluated