Lab 2

Task 1

Newton polynomial

Newton interpolation polynomial:

Where — a divided difference.

Input data:

• Table of y(x);
• Degree of polynomial;
• Values of X.

Is necessary to obtain:

• Interpolation function output;
• Exact value;
• Interpolation error. (As it turned out, it is not necessary)

To calculate the error of a function, which is represented by Newton polynomial:

Estimation of the polynomial interpolation error for the above given formula is called a priori.

It allows you to identify factors that determine the accuracy: steepness of function and configuration of the nodes (In the center - more precisely).

Task 2

Inverse Interpolation

Description:

We use inverse interpolation to solve a system of transcendental equations with two variables.
Nobody does this. Generally use Taylor series to solve similar problems. 

Main idea:

We rearrange the columns of x and y values and interpolate transformed table using Newton polynomial.

Task 3

Spline interpolation

A cubic spline is a curve consisting of "docked" polynomials of the third degree . At the docking points, values and derivatives of two neighboring polynomials are equal.

A system of equations for determination of :

The above system of linear algebraic equations is solved by the sweep method. We find an array .

With found, the remaining coefficients are determined by the formulas:

Main idea

1. Building a table;
2. Set value of x;
3. Find the k-th interval, which includes x;
4. 
5. Output the result y.

About The Sweep Method

It is based on use of canonical form of a system of linear algebraic equations with a tridiagonal matrix.

The sweep method is performed in 2 stages:

1. Trace: by the means of the formulae for given initial coefficients , we find all the sweep coefficients

2. Backtrace: with a known value of we determine all - back stroke by formulae:

Task 4

TODO

Prepared by @gofixyourself