Q9_PrecisionAnalysis.m

% =========================================================================================
% FILE NAME: Q9_PrecisionAnalysis.m
% AUTHOR: [Amey Thakur](https://github.com/Amey-Thakur)
% COURSE REPO: https://github.com/Amey-Thakur/COMPUTATIONAL-METHODS-AND-MODELING-FOR-ENGINEERING-APPLICATIONS
% RELEASE DATE: September 08, 2023
% LICENSE: Creative Commons Attribution 4.0 International (CC BY 4.0)
%
% DESCRIPTION:
% This script analyzes IEEE 754 floating-point precision limits in MATLAB using 
% realmax and realmin to detect overflow and underflow conditions during computation.
%
% PROBLEM STATEMENT (Q9):
% The functions realmax and realmin give the largest and smallest possible numbers 
% that can be handled by MATLAB. Calculations generating numbers that are too large 
% or too small result in overflow and underflow. 
% a. Use MATLAB to calculate c = ab for a = 3e150, b = 5e200.
% b. Suppose d = 5e-200, use MATLAB to calculate f = d/a.
% c. Use MATLAB to calculate the product x = abd two ways:
%    i) calculating the product directly as x = a*b*d
%    ii) by splitting up the calculation as y = b*d and then x = a*y.
%
% Reference: MATLAB for Engineering Applications, William J. Palm, Chapter 1, Q9.
%
% TECHNOLOGY STACK:
% - Programming Language: MATLAB (R2023a+)
% =========================================================================================

% --- Environment Initialization ---
clc;
clear;

% --- IEEE 754 Boundary Identification ---
upper_limit = realmax; 
lower_limit = realmin;

fprintf('--- MATLAB Precision Boundaries ---\n');
fprintf('RealMax: %e\n', upper_limit);
fprintf('RealMin: %e\n\n', lower_limit);

% --- Variable Initialization ---
a = 3 * 10^150;
b = 5 * 10^200;
d = 5 * 10^-200;

% --- Computational Scenarios ---

% a. Calculation of c = a * b
% Analyzing potential Overflow
c = a * b;

% b. Calculation of f = d / a
% Analyzing potential Underflow
f = d / a;

% c. Product x = a * b * d
% Comparing direct calculation vs. sequential splitting
x_direct = a * b * d;

y_temp = b * d;
x_split = a * y_temp;

% --- Results Visualization ---
fprintf('--- Overflow/Underflow Simulation ---\n');

% Overflow detection logic
status_c = 'Finite';
if isinf(c)
    status_c = 'OVERFLOW (Inf)';
end
fprintf('c = a * b: %e (Status: %s)\n', c, status_c);

% Underflow detection logic
status_f = 'Finite';
if f == 0
    status_f = 'UNDERFLOW (0)';
end
fprintf('f = d / a: %e (Status: %s)\n', f, status_f);

fprintf('\n--- Calculation Sequence Dependency ---\n');
fprintf('Direct (a*b*d): %e\n', x_direct);
fprintf('Split (a*(b*d)): %e\n', x_split);

% Professional Scholarly Footer
fprintf('\nScientific Insight: Numerical stability is highly dependent on calculation order. Sequential splitting (as in step c) prevents intermediate overflow that occurs in direct multiplication.\n');