Topic 10. Interpolation. Parabolic interpolation. Spline interpolation

Output data:

• the value of the unknown function f (x) at two or more points: f ( _xо) = y _o ; f (x ₁ ) = y ₁ ; ...; f (x _n ) = y _n ;
• the order of the interpolating polynomial (n-1).

The essence of the method. As an interpolating polynomial, we will take the form of a polynomial:

 ,

and the values ​​of P _n (x) in the interpolation nodes must coincide with the values ​​of the given function f (x), that is: P _n (x) = f (x _i ) = y _i (i = 0,1,2, ..., n).

This condition allows you to go to the system with (n + 1) linear equations. Since the interpolation nodes are different, then this system of linear equations has only one solution. And hence, the interpolation polynomial P _n (x) exists and will be unique. We will derive this polynomial in the following way. First, let's consider the auxiliary polynomial F (x), which takes the value of F ₀ (x ₀ ) = 1 at the interpolation node x ₀ , and in all other nodes x _i (i = 1,2, ..., n), the value of this polynomial is equal zero: F ₀ (x ₁ ) = F ₀ (x ₂ ) = ... = F ₀ (x _n ) = 0. Such a polynomial will look like:

                                             .

The interpolation nodes x ₁ , x ₂ , ..., x _n are the roots of the polynomial F ₀ (x ₀ ), and at the point x = x _{0 the} numerator is equal to the denominator, and hence, F ₀ (x ₀ ) = 1. A similar polynomial will be constructed for a node x = x ₁ , whose form:

                                             .

The same polynomials can be constructed for all other nodal points of interpolation. In the general form, the polynomials F _i (x), (i = 0,1,2, ..., n) can be written as:

                                 .

Then the desired interpolation polynomial will look like:

                            .

The product F _i (x) y _i (i = 0,1,2, ..., n) is zero in all interpolation nodes, except node xi, where they are equal to yi. Moreover, the order of the polynomial P _n (x) is equal to n, since each term of the sum of F _i (x) y _i also has the order n.

The well-defined form P _n (x) is called the Lagrange interpolation polynomial. In turn, the Lagrange interpolation formula for determining the values ​​of functions in the intermediate points xÎ] x _o ; x _n [, x χ _i (i = 0,1, ..., n) has the form:

                               .

In the partial case, when there are two interpolation nodes, this formula represents the formula of linear interpolation, for the three nodes the quadratic interpolation formula.

Since the polynomial P _n (x) is a parabolic curve of the nth order, such an interpolation is called parabolic.

The error of the parabolic interpolation depends on: 1) the polynomial; 2) the errors of the interpolation nodes, and also 3) the error of the calculation.