5. Revision Notes and Plans

5.1. Recent Changes

• 2025-11-05 Added code for solving tridiagonal linear systems in section Tridiagonal system solving in the module numerical_methods.

• 2025-10-29 Error bounds for linear algebra, matrix norms, condition numbers, etc.: Added the proof of the formula (2.10) for the matrix norm \(\| A \|_\infty\) in Error bounds for linear algebra, matrix norms, condition numbers, etc..

• 2025-10-29 Added references to [Kincaid and Chenney, 1990].

• 2025-10-25 Section Solving Ax = b with LU factorization: added Algorithm 2.9 for general upper trianglar matrices and Algorithm 2.10, and expanded the notes on LU factorization workinf for both types of SDD matrix.

• 2025-10-10 Section Faster Methods for Solving Ax=b with Tridiagonal and Banded Matrices: added Algorithm 2.19 for solving tri-diagonal systems by row-reduction, and some Exercises on measuring cost by operation counts.

• 2025-10-10 Section Operation Counts for Solving Simultaneous Linear Equations: added some (references to) exercises.

• 2025-10-07 Section Errors and Error Formulas for Polynomial Collocation: added Example 3.4 on approximating an exponential function to the

• 2025-10-07 Section Approximating Derivatives by the Method of Undetermined Coefficients: added Aside on an Alternative Approach: Differentiating a Collocating Polynomial.

• 2025-09-30 Computing Eigenvalues and Eigenvectors: the Power Method and Beyond: added the proof of the Gershgorin Circle Theorem Theorem 2.16 and the Rayleigh Iteration method for computing all eigenvalues.

• 2025-09-24 Section Iterative Methods for Simultaneous Linear Equations: added code for the Jacobi method and examples with that, and modified the exercises.

• 2025-09-01: Added references to Sullivan [2021] and Dionne [2023] throughout.

• 2025-08-17/18: Updated references in Chapters 1 and 2.

• 2025-08-17: Added more exercises.

• 2025-07-10: Ongoing adoption of myST/sphinx formatting for exercises, examples, remarks (formerly asides) and such, provided by the extensions sphinx_proof and sphinx_exercise.

• 2025-07-10: Gathered the exercises in each section into a subsection “Exercises” at the end of each section, with cross-references from where each is relevant.

• 2025-03-18: Updated the Bibliography.

• 2024-11-06: Renumbered exercises to be sequential through each chapter (“chapter.item”), rather than within each section, to be consistent with the numbering of other items (definitions, theorems, examples etc.). However, the PDF version insists on “chapter.section.item” format for everything. (Done to Runge-Kutta Methods so far.)

• 2024-10-08: Added the iteration cost result in the section on Root Finding by Interval Halving

• 2024-10-08: Added the derivation of the error bound \(\displaystyle Rel(x_a) := \frac{\|x - x_a\|}{\| x \|} \leq \|A\|\|A^{-1}\|\frac{\|r\|}{\|b\|} \) in the section on Error bounds for linear algebra etc..

• 2024-10-07: Added some comments to the section on Solving Equations by Fixed Point Iteration

• 2024-09-13: Revised the function bisection1 in module numerical_methods.

• 2024-08-26/31: Added further exercises.

• 2024-06-18: Renamed, dropping the “Introduction to” and correspondingly dropping “introduction-to-” from its URLs.

5.2. Planned Content Improvements

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

• 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 — A Brief Introduction onwards.

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

5.3. 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.