MINISTRY OF EDUCATION AND SCIENCE OF UKRAINE

KHARKIV NATIONAL UNIVERSITY OF RADIO ELECTRONICS

Department of ST

Report

on the implementation of laboratory work No. 4

"Interpolations"

In the discipline "Numerical methods"

Complied by student KNT-23-1: Roman, Kravchenko Kharkiv 2024

Accepted: P.E. Sytnikova,

Purpose of the work: acquisition of practical skills of interpolation of functions in modeling problems of complex objects and systems. Comparative analysis of methods and ways of solving problems of interpolation of functions.

Variant 11:

1. The graph of the function .

2. Independently set equidistant interpolation nodes.

   1. Constructed interpolation polynomials using Lagrange's interpolation formula, Newton's first interpolation formula, Newton's second interpolation formula, cubic spline interpolation formula.

• Lagrange's interpolation formula:

1. The formula:

2. Write down the Lagrange’s formula for our case of 4 nodes:

3. Construct a table of 4 nodes.

x
y = f(x)

4. Substitute the table values into the formula.

5. The obtained function is

• Newton's first interpolation formula:

1. The formula:

Where

2. Write the Newton’s first interpolation formula for the case of 4 nodes:

3. Generate the finite difference table. Where

and

x	y

4. Substitute the values into the formula:

4. The obtained function is

• Newton's second interpolation formula.

1. The formula:

2. Write the Newton’s second interpolation formula for the case of 4 nodes:

3. Generate the finite difference table.

   x	y

4. Substitute the values into the formula:

4. The obtained function is

4. After this step we can notice that using Lagrange’s and Newton’s formulas we obtain the same exact function.

• The formula for cubic spline.

1. If then cubic spline can be represented by the formula on this interval:

2. Table of the required values:

   x	1	2	3	4
   y	1.6	4.9	9.1	13.9
   y`	2.8	3.8	4.5	5

3. Substitute values into the formula:

4. We now have got three cubic polynomials for each interval: