Numerical Methods Course

Lesson 1 • Algorithm Stability and Conditioning

Defines numerical stability and condition numbers for problems. Prepares students to assess algorithm reliability before implementation.

Lesson 2 • Programming Environment Setup

Configures a scientific computing environment using a high-level language. Ensures all students can implement and test algorithms from chapter two onward.

Lesson 3 • Computational Complexity and Efficiency

Introduces Big-O notation and operation counting for numerical algorithms. Enables cost comparison between competing methods.

Lesson 4 • Floating-Point Number Representation

Explains IEEE 754 standard and machine epsilon. Connects representation limits to practical precision loss in computation.

Lesson 1 • Bisection and Bracketing Methods

Covers the bisection method and its guaranteed convergence properties. Introduces the concept of bracketing as a foundation for more advanced root-finders.

Lesson 2 • Systems of Nonlinear Equations

Extends scalar root-finding to multivariate systems using Newton's method with Jacobians. Prepares students for nonlinear problems in later applied chapters.

Lesson 3 • Secant and Quasi-Newton Methods

Presents derivative-free alternatives to Newton's method. Compares superlinear convergence against the cost of derivative evaluation.

Lesson 4 • Fixed-Point Iteration

Reformulates equations as fixed-point problems and analyses convergence. Builds intuition for iterative schemes used in later chapters.

Lesson 1 • Gaussian Elimination and Back Substitution

Implements row reduction to upper triangular form and back substitution. Establishes the baseline direct solver for all subsequent matrix methods.

Lesson 2 • Iterative Methods for Linear Systems

Presents Jacobi, Gauss-Seidel, and conjugate gradient methods for large sparse systems. Analyses convergence criteria and compares cost against direct methods.

Lesson 3 • Special Matrix Structures

Exploits symmetry, positive definiteness, and sparsity to reduce computational cost. Introduces Cholesky factorisation and banded matrix solvers.

Lesson 4 • Matrix Norms and Condition Numbers

Quantifies solution sensitivity using matrix norms and condition numbers. Directly applies error analysis concepts from chapter one to linear systems.

Lesson 1 • Cubic Spline Interpolation

Constructs piecewise cubic splines with continuity conditions. Demonstrates superior smoothness over global high-degree polynomials.

Lesson 2 • Interpolation Error Analysis

Derives the interpolation error bound and identifies Runge's phenomenon. Motivates the use of Chebyshev nodes to minimise maximum error.

Lesson 3 • Hermite and Osculatory Interpolation

Extends interpolation to match derivative values at nodes. Provides the theoretical basis for cubic spline construction in the next section.

Lesson 4 • Multivariate Interpolation Techniques

Extends interpolation to two-dimensional grids using bilinear and bicubic methods. Prepares students for numerical PDE and data science applications.

Lesson 1 • Gaussian Quadrature

Selects optimal node and weight pairs to maximise polynomial exactness. Demonstrates superior accuracy over Newton-Cotes for smooth integrands.

Lesson 2 • Richardson Extrapolation and Romberg Integration

Applies Richardson extrapolation to eliminate leading error terms systematically. Builds the Romberg table for high-accuracy integration with minimal function evaluations.

Lesson 3 • Finite Difference Formulas

Derives forward, backward, and centred difference approximations from Taylor series. Establishes first- and second-order accuracy for derivative estimation.

Lesson 4 • Newton-Cotes Quadrature Rules

Develops trapezoidal, Simpson's, and higher-order Newton-Cotes rules. Derives composite forms and their error terms for practical integration.

Lesson 1 • Systems of ODEs and Higher-Order Equations

Converts higher-order ODEs to first-order systems and applies vector solvers. Prepares students for coupled PDE semi-discretisations in the next chapter.

Lesson 2 • Runge-Kutta Methods

Develops the classical RK4 method and its family through Butcher tableaux. Balances accuracy and function evaluation cost for non-stiff problems.

Lesson 3 • Multistep Methods

Presents Adams-Bashforth and Adams-Moulton predictor-corrector schemes. Compares efficiency against single-step methods for smooth long-time integration.

Lesson 4 • Stability Analysis and Stiff Systems

Defines absolute stability regions and identifies stiff ODEs. Motivates implicit solvers such as backward differentiation formulas for stiff problems.

Lesson 1 • Finite Differences for Parabolic PDEs

Applies explicit and implicit schemes to the heat equation. Derives the Crank-Nicolson method and its second-order accuracy in time and space.

Lesson 2 • Classification and Discretisation of PDEs

Classifies PDEs as elliptic, parabolic, or hyperbolic and selects appropriate discretisation strategies. Connects ODE methods from chapter six to the method of lines.

Lesson 3 • Hyperbolic PDEs and Wave Equations

Solves the advection and wave equations using upwind and Lax-Wendroff schemes. Analyses numerical diffusion and dispersion as key accuracy metrics.

Lesson 4 • Finite Element Method Introduction

Introduces the weak formulation and Galerkin finite element method for 1D boundary value problems. Provides a conceptual bridge to advanced FEM courses.

Lesson 1 • Constrained and Nonlinear Optimisation

Introduces Lagrange multipliers, penalty methods, and sequential quadratic programming. Prepares students to solve engineering design and data fitting problems with constraints.

Lesson 2 • Singular Value Decomposition

Computes the SVD and applies it to rank determination, pseudoinverse, and data compression. Provides the most robust tool for ill-conditioned least squares problems.

Lesson 3 • QR Factorisation and Orthogonal Methods

Decomposes matrices via Gram-Schmidt and Householder reflections for stable least squares. Demonstrates superior numerical stability over normal equations.

Lesson 4 • Unconstrained Optimisation Methods

Applies gradient descent, Newton's method, and quasi-Newton updates to minimise smooth functions. Analyses convergence rates and line search strategies.