What is numerical analysis?

Numerical analysis is a general term for methods that approximately solve mathematical problems as concrete numerical values that computers can handle. It is used to solve problems such as differential equations and integrals, for which exact solutions cannot be obtained, to a practical level of accuracy.

What is the bisection method?

The bisection method is an algorithm that finds a root of the equation f(x)=0 by repeatedly dividing an interval [a,b] in half—where f(a) and f(b) have opposite signs—and narrowing down to the sub-interval containing the root. Based on the Intermediate Value Theorem, it is the most fundamental root-finding method, halving the interval with each step.

What types of errors arise in numerical computation?

The main types are rounding error (error from truncation or rounding when representing numbers with a finite number of digits) and truncation error (error from terminating an infinite computation after a finite number of steps). There are also pitfalls such as catastrophic cancellation (loss of significant digits when subtracting nearly equal values) and absorption error (a small number being ignored when added to a much larger number).

Numerical Analysis Introduction

Fundamentals of Numerical Computation (High School Level)

Overview

In this introduction, we develop an intuitive understanding of what numerical computation is. We learn what it means to solve mathematical problems with a computer and what kinds of errors arise in the process.

Learning Objectives

Understand the basic ideas behind numerical computation
Learn the concept of errors (rounding error, truncation error)
Be able to find the root of an equation using the bisection method
Understand the basic ideas of numerical differentiation and numerical integration
Understand that there are cases where computers cannot compute "exactly"

Chapter 1 What is Numerical Computation?
The purpose of numerical computation, the difference between analytical and numerical solutions, and the idea of approximation
Chapter 2 Fundamentals of Error
Absolute error, relative error, rounding error, truncation error
Chapter 3 Bisection Method
The most basic method for finding a root of $f(x) = 0$
Chapter 4 Introduction to Numerical Differentiation
The idea of finite difference approximation $f'(x) \approx \frac{f(x+h) - f(x)}{h}$
Chapter 5 Introduction to Numerical Integration
Riemann sums and an intuitive understanding of the trapezoidal rule
Chapter 6 Pitfalls in Computation
Catastrophic cancellation, absorption error, overflow and underflow

Visualizing the Bisection Method

The bisection method finds a root of $f(x) = 0$ by repeatedly halving the interval. For a continuous function $f$, if $f(a)$ and $f(b)$ have opposite signs, the Intermediate Value Theorem guarantees that a root exists in the interval $(a, b)$. The bisection method exploits this property by checking the sign at the midpoint $c = (a+b)/2$ and narrowing down to the sub-interval containing the root.

When f(a) and f(b) have opposite signs, a root exists in the interval. The midpoint c = (a+b)/2 halves the interval.

Prerequisites

Basic middle school mathematics (equations, graphs of functions)
High school mathematics (basic concepts of differentiation and integration) is desirable

Overview

Learning Objectives

Table of Contents

Visualizing the Bisection Method

Prerequisites