Polynomial Interpolation

Functions
Vector	NewtonPolynomial (const Vector &x, const Vector &y)
	Build a polynomial P(z) by Newton's divided differences and return the coefficient vector a. More...
Real	NewtonPolynomialEvaluation (const Vector &a, const Vector &x, const Real z0)
	Evaluate P(z0), i.e. the value of P(z) at z=z0, where P(z) is a Newton polynomial defined by a and x. More...

Detailed Description

Function Documentation

NewtonPolynomial()

Vector NewtonPolynomial	(	const Vector &	x,
		const Vector &	y
	)

Build a polynomial P(z) by Newton's divided differences and return the coefficient vector a.

A Newton polynomial is in the form P(z) = a(1) + a(2)*(z-x(1))

• a(3)*(z-x(1))*(z-x(2)) + ... + a(n)*(z-x(1))*...*(z-x(n-1)), such that P(x(i)) = y(i) for i=1,...,n.

  Parameters

  x	is the input vector of the domain values.
  y	is the input vector of the range values.

  Returns

  The vector a of coefficients.

  All vectors a, x, and y are of the same length n.

  Remarks

  If x(i)==x(j) for some i j, then it is assumed that the derivative of P(z) is to be zero at x(i) and the Hermite polynomial interpolation is applied.

  In general, if there are k x(i)'s with the same value x0, then the jth order derivative of P(z) is zero at z=x0 for j=1,...,k-1.

NewtonPolynomialEvaluation()

Real NewtonPolynomialEvaluation	(	const Vector &	a,
		const Vector &	x,
		const Real	z0
	)

Evaluate P(z0), i.e. the value of P(z) at z=z0, where P(z) is a Newton polynomial defined by a and x.

The newton polynomial is P(z) = a(1) + a(2)*(z-x(1))

• a(3)*(z-x(1))*(z-x(2)) + ... + a(n)*(z-x(1))*...*(z-x(n-1)).

Remarks

This routine also works for evluating the function value of a Hermite interpolating polynomial, which is in the same form as a Newton polynomial.

Returns

The evaluated P(z0), i.e. the value of P(z) at z=z0.