# 3. Revision notes and plans#

## Recent changes#

**2024-11-06:**Gathered exercises at the end of each section, with references from their previous locations. (To Runge-Kutta Methods so far.)**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 othe items (definitions, theorems, examples etc.) However, the PDF version insists on “chapter.section.item” format for everything. (To Runge-Kutta Methods so far.)**2024-10-14:**Revised the graphs and added some remarks in the section on*Error Formulas for Polynomial Collocation***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`numericalMethods`

.**2024-08-26/31:**Added further exercises.**2024-06-18:**Renamed, dropping the “Introduction to” and correspondingly dropping “introduction-to-” from its URLs.

## To Do#

### Content improvements#

Add references to other free/online resources.

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 derivation of the formula for the matrix norm \(\| A \|_\infty\)

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

Optimization (Minimization)

Multi-step methods for ODEs and determining their stability.

Continue to port 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`

.

### Formating, layout and style improvements#

**camelCase**everything (e.g. change from*numerical_methods*to*numericalMethods*), except where an underscore is used to indicate a subscript. (For one thing that harmonises with Julia style.)When using

`<br>`

, it should appear on its own line, not at end-of-line. (For PDF output; HTML output is more forgiving.) This is relevant to some pseudo-code appearence in the PDF; e.g in Basic Concepts and Euler’s Method:

### MyST markdown tips#

Move to MyST Markdown notation

`{doc}`

,`{ref}`

,`{eq}`

and so on as appropriate.However the enhanced mark-down style

`[text](label)`

is often preferable to uisng`{ref}`

becauseit renders better when viewing a section as a Jupyter notebook, and

it allows substituting an abbreviation or variant of the title as the link text.

To number equations for referencing, use MyST-Markdown-augmented

`$$...$$`

notation, as with`$$2+2=4$$ (eq-obvious)`

If the top-level section atop a file is labelled as with

(section-label)= then the usage`{ref}<section-label>`

can be used instead of`{doc}<file-base-name>`

This is useful for example if there are variant files distinguished by suffixes like “-python”/”-julia” but I want to use the same cross-reference text.