Aims & Learning Objectives:
Aims: To extend the real analysis of implicitly defined functions into the numerical analysis of iterative methods for computing such functions and to teach an awareness of practical issues involved in applying such methods.
The students should be able to solve a variety of nonlinear equations in many variables and should be able to assess the performance of their solution methods using appropriate mathematical analysis.
Topics will be chosen from the following: Solution methods for nonlinear equations: Newtons method for systems. Quasi-Newton Methods. Eigenvalue problems.
Theoretical Tools: Local Convergence of Newton's Method. Implicit Function Theorem. Bifurcation from the trivial solution.
Applications: Exothermic reaction and buckling problems. Continuous and discrete models.
Analysis of parameter-dependent two-point boundary value problems using the shooting method. Practical use of the shooting method.
The Lyapunov-Schmidt Reduction. Application to analysis of discretised boundary value problems.
Computation of solution paths for systems of nonlinear algebraic equations. Pseudo-arclength continuation. Homotopy methods. Computation of turning points. Bordered systems and their solution. Exploitation of symmetry. Hopf bifurcation.
Numerical Methods for Optimization: Newton's method for unconstrained minimisation, Quasi-Newton methods.