README

METODA NEJMENSICH CTVERCU (Least Squares Method)

!!!! Pull requests are welcome!

Dedicated to Frantisek — whom I admired as a child. He was a great programmer and always repaired our family computer when it was broken.

This is an experiment in reviving legacy code. The original source code was photographed from a printed listing and can be found in the data/ folder. I modified the code to compile and run on Turbo Pascal 5.5 (the original was written for TP 3.x).

The original source: a printed listing, photographed.

My plan is to compare the results with a modern Python implementation to verify correctness and explore the numerical differences.

Legacy Turbo Pascal (DOS) program for computing a least-squares polynomial approximation of measured data points. Originally written for Borland Turbo Pascal and intended to run in a DOS environment.

Note: The program interface is in Czech. An English version is planned for the future.

📌 Overview

This program implements the method of least squares to approximate a set of measured points
((x_i, y_i)) using a polynomial of low degree.

The user:

1. Enters measured data points
2. Chooses the degree of the polynomial
3. The program computes polynomial coefficients
4. Displays approximation errors
5. Allows evaluation of the polynomial for arbitrary x

🧮 Mathematical Background

The program constructs a polynomial:

[ P(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n ]

Coefficients (a_i) are computed by solving the normal equations:

[ (A^T A) a = A^T y ]

Where:

• (A) is the Vandermonde matrix of input x values
• (y) is the vector of measured y values

A custom Gaussian elimination routine is used to solve the linear system.

🗂 Program Structure

Constants

• maxt = 2 — maximum polynomial degree
• max = 100 — maximum number of data points

Data Types

• pole — array of real values (input data)
• vektor — vector of polynomial coefficients
• matice — augmented matrix for linear equations

🔧 Main Components

eliminace

Performs Gaussian elimination and back-substitution to solve a linear system.

polynom

Evaluates the polynomial using Horner’s method.

mocnina

Computes integer powers of a real number.

🖥 User Interaction

• Text-based interface using the CRT unit
• Single-key input with ReadKey
• Interactive correction of entered points
• Live evaluation of polynomial values

⚠ Limitations

• Requires DOSBox + Turbo Pascal 5.5 to run
• Limited polynomial degree (maxt = 2)
• Numerical robustness — see Numerical Stability Considerations below

▶ How to Run

Requirements

• DOSBox - DOS emulator
  bashCopy

  sudo apt install dosbox

First Time Setup

After cloning, extract Turbo Pascal from the included disk images:

cd turbo_pascal
./setup_turbopascal.sh

Run the Program

cd turbo_pascal
./run_mnc.sh

This opens DOSBox, compiles and runs the program.

More Info

See turbo_pascal/README.md for:

• Turbo Pascal IDE usage
• Code changes from original TP3 to TP5.5
• Troubleshooting

🔢 Numerical Stability Considerations

The implementation of the least squares method in this program uses normal equations combined with a hand-written Gaussian elimination routine.

While this approach is simple and historically common, it has known numerical drawbacks:

• Forming normal equations squares the condition number of the system
• No pivoting is used during Gaussian elimination
• Rows are not normalized during elimination
• Intermediate values can grow rapidly

As a result, the algorithm may suffer from:

• Loss of precision for closely spaced x values
• Large rounding errors for higher polynomial degrees
• Numerical instability for larger datasets

Practical Impact

In its current configuration, the program is reasonably safe only for:

• Very low polynomial degrees (maxt = 2)
• Small numbers of data points
• Well-scaled input values

Future Improvements

If this code is ever modernized, recommended improvements include:

• Using QR decomposition instead of normal equations
• Adding partial or full pivoting to Gaussian elimination
• Scaling or normalizing input data
• Using numerically stable linear algebra libraries

🔄 Future Plan (Optional)

Phase 2 — Porting & Modernization

Potential future improvements:

• Port to Free Pascal or modern Pascal
• Remove CRT dependencies
• Numerical fixes — see Future Improvements above
• Increase supported polynomial degree
• Add file-based input/output

📁 Files

????????? README.MD                        # This document
????????? data/                            # Photos of original printed code
????????? turbo_pascal/
    ????????? README.md                    # Technical details & setup
    ????????? MNC.PAS                      # Working TP5.5 version
    ????????? MNC_ORIGINAL.PAS             # Original TP3 source (historical)
    ????????? installed/                   # Turbo Pascal 5.5 installation (created by setup_turbopascal.sh)
    ????????? *.sh                         # Run scripts

🕰 Historical Note

This code reflects classic 1980s numerical programming practices:

• Manual matrix handling
• Custom solvers
• Text-mode UI

Preserved for educational, historical, and experimental purposes.

✍ Author / Origin

• Original program: DALSAN soft
• Turbo Pascal program lister © 1983 Borland International
• Documentation & execution plan added later