alfi::util::poly Namespace Reference

Namespace with utility functions for polynomial operations. More...

Functions
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
void	normalize (Container< Number > &p, const Number &epsilon=std::numeric_limits< Number >::epsilon())
	Normalizes a polynomial by removing leading zero coefficients.
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
Container< Number >	mul (const Container< Number > &p1, const Container< Number > &p2)
	Multiplies two polynomials.
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
std::pair< Container< Number >, Container< Number > >	div (const Container< Number > &dividend, const Container< Number > &divisor, const Number &epsilon=std::numeric_limits< Number >::epsilon())
	Divides one polynomial by another.

Detailed Description

Namespace with utility functions for polynomial operations.

This namespace provides a set of functions for manipulating polynomials, such as normalization, multiplication, and division. All polynomials are represented as containers of coefficients in descending degree order.

Function Documentation

normalize()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

void alfi::util::poly::normalize	(	Container< Number > &	p,
		const Number &	epsilon = std::numeric_limits<Number>::epsilon() )

Normalizes a polynomial by removing leading zero coefficients.

Removes leading coefficients that are considered zero within a given tolerance epsilon. If all coefficients are close to zero, the last one is preserved. If the container is empty, a single zero coefficient is inserted.

Parameters

p	the polynomial to normalize (a container of coefficients in descending degree order)
epsilon	the tolerance used to determine whether a coefficient is considered zero (default is machine epsilon)

mul()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

Container< Number > alfi::util::poly::mul	(	const Container< Number > &	p1,
		const Container< Number > &	p2 )

Multiplies two polynomials.

Given two polynomials \(p_1\) and \(p_2\) represented as containers of coefficients, this function computes their product using the convolution formula:

\[ (p_1 \cdot p_2)[k] = \sum_{i+j=k}{p_1[i] \cdot p_2[j]} \]

If either polynomial is empty, the function returns an empty container.

Parameters

p1	the first polynomial
p2	the second polynomial

Returns

the product polynomial (either empty or of size p1.size() + p2.size() - 1)

div()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

std::pair< Container< Number >, Container< Number > > alfi::util::poly::div	(	const Container< Number > &	dividend,
		const Container< Number > &	divisor,
		const Number &	epsilon = std::numeric_limits<Number>::epsilon() )

Divides one polynomial by another.

This function divides the dividend polynomial \(A\) by the divisor polynomial \(B\), and returns the quotient \(Q\) and the remainder \(R\) such that:

\[ A(x) = B(X) \cdot Q(x) + R(x) \]

with either \(R\) being effectively zero or the degree of \(R\) being lower than the degree of \(B\).

The division is performed using a tolerance epsilon to determine when a coefficient is considered zero.

If the divisor is effectively zero or if the dividend has a lower degree than the divisor, the function returns an empty quotient and the dividend as the remainder.

Parameters

dividend	the polynomial to be divided
divisor	the polynomial to divide by
epsilon	the tolerance used to determine whether a coefficient is considered zero (default is machine epsilon)

Returns

a pair {quotient, remainder}