Mathematical Tripos II Numerical Analysis (Michaelmas 2022)

Lecture notes

• Lecture 1: Poisson's equation, finite differences, and the five-point formula
• Lecture 2: Five-point formula (continued)
• Lecture 3: Convergence + Fast Poisson solver
• Lecture 4: Fast Fourier Transform + Diffusion equation
• Lecture 5: Stability
• Lecture 6: Stability (examples)
• Lecture 7: Stability using Fourier analysis
• Lecture 8: Stability using Fourier analysis cont'd: wave equations
• Lecture 9: Diffusion equation in 2D
• Lecture 10: Splitting
• Lecture 11: Spectral methods - Fourier approximation of analytic functions
• Lecture 12: Spectral methods - Fourier series for ODEs and PDEs
• Lecture 13: Spectral methods - Chebyshev expansions
• Lecture 14: Spectral methods - Evolution equations
• Lecture 15: Iterative methods for linear systems - Jacobi and Gauss-Seidel methods
• Lecture 16: Iterative methods for linear systems - Convergence of Jacobi and Gauss-Seidel
• Lecture 17: Iterative methods for linear systems - Multigrid methods
• Lecture 18: Iterative methods for linear systems - Steepest descent
• Lecture 19: Iterative methods for linear systems - Conjugate gradient method
• Lecture 20: Iterative methods for linear systems - Convergence of CG
• Lecture 21: Eigenvalues and eigenvectors - Power iteration, inverse iteration, Rayleigh quotient iteration
• Lecture 22: Eigenvalues and eigenvectors - Simultaneous/QR iterations
• Lecture 23: Eigenvalues and eigenvectors - Shifted QR iterations
• Lecture 24: Eigenvalues and eigenvectors - Reduction to tridiagonal form, nonsymmetric eigenvalue problem

Example sheets

• Example sheet 1
• Example sheet 2
• Example sheet 3
• Example sheet 4

Other resources

• IB Numerical Analysis course
• Matlab demos
• A first course in the numerical analysis of differential equations, A. Iserles, Cambridge University Press [The course mostly follows this book]
• Iterative methods for sparse linear systems, Y. Saad
• A multigrid tutorial, W. L. Briggs, V. E. Henson, S. F. McCormick
• The definition of numerical analysis by L. N. Trefethen
• Top ten algorithms of the 20th century, by Barry Cipra