Numerical Analysis Lecture 1

Study Notes

Numerical Analysis is a course about numerical methods.
Numerical methods are used for solving problems that cannot be solved analytically.
Numerical computation, numerical methods, and numerical analysis are closely related concepts.
The Millennium Bug refers to a potential computer problem arising when the year part of a date changes from 1999 to 2000.
Examples of dates and numbers represented in different systems are given (2000, September 24, 1978, 092478, 092498, 092499, 123199, 010110, 010120, 010100, 1231999, 01012000, 91)
A bank account profit of 602.55565 Rupees is analyzed, demonstrating percentage calculations and the effect of multiplying by different powers of 10.

In everyday life, the decimal system is used, with 10 symbols (0 to 9) and base 10.
Arbitrary numbers can be represented in any number system with base N needing N different symbols.
Common number systems in computers include binary (base 2, symbols 0 and 1), octal (base 8), decimal (base 10), and hexadecimal (base 16).

A real number 'a' can be expressed as a sum of terms, each with a power of the base 'N'. Examples are given in decimal and binary forms (e.g., (1729)₁₀ = 1 × 10³ + 7 × 10² + 2 × 10¹ + 9 × 10⁰).
Decimal conversions to binary are shown, using successive divisions by 2. Example: (47)₁₀ = (101111)₂.
Decimal fraction conversions to binary are shown, using successive multiplications by 2. Example: (0.7625)₁₀ = (0.110001)₂ (approximately).

Major topics of the numerical analysis course include the solution of non-linear equations, solution of linear systems of equations, eigen value approximation, interpolation and polynomial approximation, numerical differentiation, numerical integration, and numerical solution of ordinary differential equations.
Electronic computers use the binary (base-2) system.
Conversion methods are demonstrated for numbers from other forms into binary form
Numbers are represented and calculated in decimal and binary systems.