32. Some Planned Future Topics

Further revisions of these notes, perhaps in the planned future github book [Introduction to Numerical Methods and Their Analysis (with Python)] (https://lemesurierb.github.io/introduction-numerical-methods-analysis-python/)

  • Implicit methods for solving ODE IVPs.

  • More on multi-step methods for solving ODE IVPs: further methods, stability, and convergence.

  • Stiffness in systems of ODEs, and the use of implicit methods to solve stiff systems efficiently.

  • Solving boundary value problems for ODEs.

  • More on finding the minimum of a function, in particular for functions of several variables.

  • More on computing eigenvalues and eigenvectors.

  • More on iterative methods for solving systems of linear equations.

  • Special methods for solving systems of linear equations with matrices of special forms: tridiagonal, banded, diagonally dominant, positive definite, etc.

  • Solving systems of nonlinear equations.

  • Splines: a version of piecewise polynomial interpolation that maintains smoothness (differentiabilty etc.) at the nodes where adjacent polynomial pieces join.

31. Least-squares Fitting to Data: Appendix on The Geometrical Approach 1. Initial Value Problems for ODEs, Part 6: A Very Brief Introduction to Multistep Methods