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# Numerical Analysis Pages (one page at a time)

Notation used in the course

Basic Programming Constructs

Some Programming Principles

Basic Introduction to Programming in Matlab

Matlab function m-file

On programming assignment 1

## Computer Arithmetic

Comparing numbers

Floating point representation

Pictures of a (toy) Floating point system

Floating point arithmetic

Swamping and Cancellation

The set of floats

Comparing reals and comparing floats

Error Analysis

What is a Solution?

Condition analysis

Three perspectives on cancellation

On evaluating a function

## Polynomials

Polynomials

Polynomial Evaluation

The Taylor Polynomial

A Taylor Polynomial Picture

Rate of Convergence - the 'Big O' notation

Polynomials are Vectors

Fundamental Theorem of Algebra

## Equations in One Variable

Solving equations in one variable

Method of bisection

Newton's method

The secant method

Order of Convergence

Newton's method convergence analysis

Secant method convergence analysis

The Geometry of Newton & Secant

Compare Bisect, Newton & Secant

Hybrid methods for root finding

Roots of a polynomial (Mueller's method and deflation)

Example of a bad polynomial

## Interpolation and Approximation

Polynomial Interpolation

Evaluating the Interpolator (Barycentric weights)

Interpolation Nodes

Lagrange interpolation example

Hermite interpolation

Hermite interpolation on 2 nodes

Kissing (osculating) polynomials

FFT Tease

Interpolating splines

Least squares

## Differentiation and Quadrature

Numerical differentiation is easy

Numerical differentiation is hard

Quadrature (Newton-Cotes)

Composite Newton-Cotes

Optimal Quadrature (Gaussian Quad)

Basic Monte Carlo

Rejected (rejection sampling)

Smart numerical integration (adaptive)

Adaptive quadrature: recursive m-file

## Ordinary Differential Equations

The IVP and Euler's method

A Picture Euler's method

Can we solve this IVP? (well-posed)

Systems of IVP's

Example System of IVP's

Local Truncation Error

Runge-Kutta methods

Runge-Kutta methods order 2

Adaptive Runge-Kutta methods

Multi-step methods

Adaptive Multistep Methods

Stiff problems