Revision Notes and Plans

6. Revision Notes and Plans#

6.1. Recent Changes#

6.2. Planned Content Improvements and Additions#

  • Add a section on “direct” methods for second order equations \(u'' = f(t, u, u')\), and in particular for conservative and Hamiltonian systems.

  • Add an introduction to methods for PDE IVPs.

  • Add brief introductions to background facts about existence and uniqueness of solutions, and well-posedness for DE’s: ODE IVP, ODE BVP, PDE IVP, etc.

  • Expand/complete the chapter on Optimization (Minimization).

  • Complete the section on root-finding for functins of several variables.

  • Add notes on linear multi-step methods: zero-stability, th eDahlquist theorem and absolute stability regions.

  • Add a brief chapter on ODE BVP’s, with:

    • an introduction to the Galerkin/finite element method for ODE BVP’s.

    • an introduction to then finite difference method for ODE BVP’s.

  • ODE IVP’s: add reference and citatons to the OER materials from TU Delft, in particular “Vuik et al - c”.

  • ODE IVP’s: Stability regions and “consistency plus stabilty equals convergence” results.

  • ODE IVP’s: In the notes on consistency/stabilty/canvergence in multitep methods, add or change to the less obviously wrong example of Sauer’s Eq. 6.81 in Section 6.7.2.

  • Move the results on “when does naive row reduction work?” to a subsequent (no-Python?) section, which introduces the various main cases of “well-behaved” matrices like DD, SDD and positive definite.

  • Continue revising Initial Value Problems for Ordinary Differential Equations from Error Control and Variable Step Sizes on.

  • Add definition of positive definite matrices in Row Reduction (Gaussian Elimination), and then add more results for this case in various places.

  • Improve the presentation of error control methods such as Runge-Kutta-Fehlberg in Error Control and Variable Step Sizes.

  • Add more references to [Gear, 1971] in Initial Value Problems for Ordinary Differential Equations

  • Add Python versions of most or all algorithms presented in the text (typically as pseudo-code); in general, only in the module numerical_methods but with some “didactic” examples in the main text, to illustrate Python features such as import, array creation and manipulation, and so on. Currently (2025-11-05) this is being done in Initial Value Problems for Ordinary Differential Equations and some of Linear Algebra and Simultaneous Equations.

  • Continue adding some more advanced exercises.

  • Continue revising the notes on cubic splines in Piecewise Polynomial Approximating Functions and Spline Interpolation.

  • Add more exercises at the end of each section, gathering from assignments, the current separate sections on exercises, and other sources.

  • Add notes on min-max (Chebychev) approximation of functions by polynomials.

  • Expand some sections that are only stubs or very brief; in particular those on

    1. Optimization (Minimization)

    2. Multi-step methods for ODEs and determining their stability.

  • Continue to merge updates to and from the Julia version, mostly from Finding the Minimum of a Function of One Variable Without Using Derivatives onwards.

  • About document “python-tutorial-introduction”: either omit it or add notes on speed/cost measurement, such as with module time.

6.3. Planned Format Revisions#

  • Continue adoption of advanced MyST/Sphinx features, such as converting most aside’s and note’s to an enumerated “remark” or “observation” via the extension sphinx_proof; convert some from “remark” to “observation” (in progress).

  • Add more subsections, so that the in-page index at right is more useful.

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