The goal of this paper is to present a script-based system for calculating Lagrange Interpolating Polynomials from data points provided in text files. This simple solution utilizes Batch for the management of text files, Python for performing calculations, and HTML for presenting the obtained results in an aesthetically pleasing manner. This straightforward approach shows how one could easily and efectively implement such mathematical concepts with an interactive interface.
image processing - for finding the unknown or missing
pixels e.g. in the process of converting an image between
Interpolation is a mathematical concept used for the con- resolutions without afecting the image quality [
Since ancient times interpolation played an impor- very often use also other methods, ie. [18] presented
tant role in calculating the planetary movement [
Nowadays it has even more widespread usage. In Batch is a language dedicated for Windows operating
medicine, it is implemented e.g. to obtain 3D images systems. It works through simple executable commands
out of 2D magnetic resonance imagining (MRI) or com- that allow for automation of repetitive tasks [21].
Recputed tomography (CT) scans [
Interpolation is also applied in digital signal processing handling and management more accessible. The
math- for the process known as upsampling (transforming the ematical calculations are performed with the help of
sampled signal into a higher sampling frequency) [
This paper is structured in the following manner: ifrstly the theory of Lagrange interpolating polynomials ture is described, then a case study is demonstrated with the use of examples, and last but not least a conclusion with possible future improvement or further develop- The program calculates Lagrange interpolating polynoment ideas listed. mials for given sets of data points extracted from a text ifle IN.txt and saves obtained results into OUT.txt file, ifnally, it displays the results in a table format on a sim2. Lagrange Interpolating ple website from Program-Report.html file. The system Polynomial consists of four files, and during operation, it creates two more files. Additionally, if the user decides to perform Lagrange interpolation is applied to estimate or find the the "backup" option, a BACKUP directory will be created polynomial that best fits with the given data points. with a copy of the appropriate files.
For given set of data points , where = 0, 1, ..., it can be obtained + 1 Lagrange interpolating polynomials : () = ∏︁
( − ) =0,̸= ( − ) For given interpolating points, (, ), where = 0, 1, ..., formula for estimating the interpolating polynomial states as follows:
() = ∑︁ ()
=0 After simplifying the result, a polynomial will be obtained for which the following mathematical relationship will be satisfied: ∀ ∈ : () = 00()+11()+...+11() =
Algorithm 1 Lagrange Interpolating Polynomial Algorithm Require: xv = list of arguments x, yv = list of values y Ensure: length(xv) == length(yv) = Symbol(’x’) unsimplified = 0 for = 0 to ℎ() do Φ = 1 for = 0 to ℎ() do if ̸= then
Φ = Φ · ((−− ) ) end if end for unsimplified = unsimplified + · Φ end for result = simplify(unsimplified)
The general file organization of the proposed system is as follows: • Fundamental files: 1. IN.txt - file that contains the input data
points for calculations 2. Batch.bat – script that controls and manages the workings of the whole system 3. Program1.py – processes the input data and creates the OUT.txt with output data 4. Program2.py – extracts data from both IN.txt and OUT.txt files to generate a report with obtained calculation results in the form of a simple website.
lations from Program1.py file 2. Program-Report.html – HTML file with the website format of the report.
• Files generated during the program execution:
Subsequent data sets for calculations should be included in the IN.txt file and meet the following requirements : • one set of data points should be in one line of the text file • numbers should be separated by the traditional comma
1, 1, 2, 2, ..., , 1, 1, 2, 2, ..., ,
...
Where numbers and state the data point number.
After the Batch.bat script file is executed and the Start option is chosen, the Program1.py will be executed. It processes the input data extracted from the IN.txt file and saves the obtained results in an OUT.txt file. Next, the Program2.py is executed and it extracts data from both IN.txt and OUT.txt to then process them and create a simple HTML website saved to Program-Report.html file.
After successful program execution, a new window will be opened in the default browser with a report presenting the results of program operation. Additionally, after selecting the "Backup" option, a copy of the IN.txt and OUT.txt files will be created and Program-Report.html, if they exist, otherwise an appropriate message will be displayed. The copy will be saved to the directory with the current date and time.
The above chart represents how files interact with each other. The core of the whole system is based on the main batch script file called Batch.bat, which manages each subsequent or chosen action performed by the program. It is expected that the user begins with the Start option, which initiates the calculations by extracting the input data from the IN.txt file, then processes them appropriately, and finally saves the results into the OUT.txt ifle and presents them in a table on an HTML site. User has also an option to get acquainted with the program workings instructions that are included in the Descrip- Figure 2: The menu window of the program. tion option. After successful calculations, the user can also save the whole program log data into a given folder by choosing the Backup option that will save there all There are 4 options to choose: related files. By choosing the Exit option the user can 1. Start ifnish the process. 2. Description
The program can be executed by opening the Batch.bat script. Then the main window will appear as follows:
4. Exit
The Start option begins the calculations and the program execution. The successful operation finishes with a new web browser window with the report being opened.
If any issues are detected the program will display an appropriate warning message.
The program is protected against incorrect input data, as shown in the example below, where both incorrect and correct values are given as subsequent sets of points.
The program is protected against incorrect data, both in the IN.txt file and during the operation of the Batch.bat script when selecting subsequent options.
The aim of this paper has been achieved. A simple scriptbased system for the calculation of Lagrange interpolating polynomials has been proposed. User-friendly scripts allow potential users to input their data points in a text ifle and then generate a Lagrange interpolating polynomial that can be used to estimate intermediate values of the given function.
Through testing, it has been found that the suggested approach is accurate and efective as it generates proper results and is capable of handling large datasets without any lag in performance. In the future, the project can be further improved by adding more functionalities such as displaying visualizations of estimated functions along with the data points on a graph next to the already presented results in the table on the website.
This project demonstrates how Lagrange interpolating polynomials can serve as a tool for the estimation of function and analysis. However, the script-based approach and text file data storage make it more accessible for entry-level users and allow for easy modification and customization.