Lesson 10

In practice, it is necessary to solve two mutually inverse problems:

• function tabulation - on the analytic task of the function to get meaning for its specific arguments (compile a table of values ​​of the function at its points);
• interpolation - to get an intermediate value according to the given table value (to determine the value of the function for the given argument on the analytical expression).

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 some function that would coincide with this function f (x _i ) - 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.

Interpolation (or interpolation) is also called an approximation of the function y = f (x) to the segment [a, b] of one of the functions y = P (x), so that the function y = P (x) at the points x _i , i = 0 ..n assuming the same values as the function y = f (x), ie P (x _i ) = y _i i = 0..n.

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.

An interpolation polynomial is built in cases where:

• 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).

When interpolation is solved the following tasks:

• the choice of the most satisfactory way of constructing an interpolation polynomial of a given function for each particular case;
• estimate of the error when replacing P _n (x) »f (x) for xе] 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.