Lecture 20: Overview, review and looking forward

Lecture 20: Overview, review and looking forward#

Review#

Floating point numbers#

\[(\beta, t, L, U)\]

  • Representations and rounding

  • \(eps\), machine precision

Matrices and vectors#

\[\begin{split} A = \begin{pmatrix} 4 & 5 \\ 3 & -1 \end{pmatrix} \end{split}\]

  • Multiplication

  • Inner product

  • Euclidean norms

Systems of linear equations#

\[ A \vec{x} = \vec{b} \]

  • Translating between matrices and systems of linear equations

  • Deriving simple linear systems of equations

  • Solving triangular systems of equations

  • Gaussian Elimination

  • LU factorisation

  • Jacobi iteration

  • Gauss Seidel iteration

  • Initial guesses, convergence and stopping criteria

Dynamic problems#

\[ y'(t) = f(t, y) \quad \text{subject to} \quad y(t_0) = y_0 \]

  • Derivatives, gradients/slopes and rates of change

  • Reading and drawing simple graphs

  • Deriving models for simple dynamic problems

  • Initial conditions

  • Euler’s method

  • Midpoint scheme

Nonlinear equations#

\[ f(x) = 0 \]

  • Roots of an equation

  • Bisection algorithm

  • Newton’s method

  • Quasi Newton methods

  • Hybrid methods

Data fitting#

  • How to form a system of linear equations to find a simple curve of best fit

  • How to find a best fit solution

  • When to choose which curve (from a simple choice)

  • Problems with our approach

What next….?#

  • You will apply ideas from this course in many areas of computer science including

    • graphics;

    • artificial intelligence/machine learning.

  • There are plenty of opportunities for Final Year Projects on topics in numerical computation. If you are interested talk to me (or Yongxing Wang).

  • There are also funded places for further study in numerical computation….

Thank you for your attention!

Good luck with your exam and future studies

I hope you have a good Christmas break!

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