Topic 10. Interpolation. Parabolic interpolation. Spline interpolation

When solving many practical problems it is necessary to solve two mutually inverse problems: 1) on the analytical task of the function to get meaning for its specific arguments (to compile a table of values ​​of the function at some of its points) - task tabulation function; 2) get the intermediate value (according to some analytical expressions, to determine the value of the function for a given argument) according to the given table value - the interpolation problem.

Let some function y = f (x) be given tabularly - for the given values ​​of the argument x _{i the} values ​​of the function y _i = f (x _i ), i = 0, .. n) are given. It is necessary to find the analytic expression of a function that would coincide with this function f (x _i ), namely at the points x _i took the value y _i , i = 0..n. From the geometric point of view, the problem of interpolation is reduced to finding the equation of the curve y = P (x), which passes through the given points (x _i , y _i ), i = 0 ..n.

An approximation of the function y = f (x) to the segment [a, b] is one of the functions y = P (x), so that the function y = P (x) at the points x _i , i = 0..n acquires the same values , that the function y = f (x), ie P (x _i ) = y _i i = 0..n is called interpolation (or interpolation). The points x _i , i = 0..n are called interpolation nodes, the function y = P (x) is an interpolating function, and the formula f (x) »P (x) is an interpolation formula.

Interpolation polygon is built in cases:

• the function is given tabularly for some arguments of the argument, but you need to find its value for an argument that is not in the table;
• the function is given graphically, but it is necessary to find its approximate analytic expression;
• the function is given by a complex analytical expression that is not convenient for integration, differentiation.

Depending on the type of function y = P (x), the methods of interpolation are divided into:

• parabolic (algebraic polynomials) - linear, quadratic, etc.
• transcendental (trigonometric).

As a rule, under interpolation understand the following tasks:

• the choice of the most satisfactory way of constructing an interpolation polynomial of a given function for each particular case;
• estimation of the error when replacing P _n (x) f (x) for xÎ] a; b [;
• optimal selection of interpolation nodes for less error.

The task of extrapolation is to find the value of a function for an argument that lies outside the function definition table.