How to implement matrices in C# efficiently

[Proektsoftbg]

The most straightforward way to program matrices in C-like languages is to use two-dimensional arrays [ , ]. But it is also very inefficient in respect to engineering analysis.

We can easily implement efficient data structures for different types of matrices through OOP, using inheritance and polymorphism. All matrices will be represented by vector of row vectors. For square symmetric matrices, we will store only the elements in the triangle above and along the main diagonal. So, each row will have different size. Also, we will reuse most of the available vector functionality right away.

By overriding the indexer's getter and setter for each type, we will make all matrix types behave in the same way externally, although they are quite different internally. The huge savings in both memory and time by not storing and processing part of the elements are worth the little overhead introduced by polymorphism.

And the LargeVector class is the same structure we used for the sparse vectors we discussed a few days ago. Initially, it does not reserve any memory until we assign some values. After that, it stores only the non-zero elements, so practically, in this way we get a sparse symmetric matrix.

[alishrih]

But, why not implement Math.NET?
Even if it is not as "fast", a somewhat feature-complete product can have a larger reach, especially between engineers.

[Proektsoftbg]

Hi! Thank you this question. Yes, there are a lot of math libraries available today. For example, we could also use Boost with C#:
https://beta.boost.org/doc/libs/1_82_0/libs/math/doc/html/math_toolkit/stat_tut/weg/c_sharp.html

But as you can probably see, Calcpad is not a just a UI wrapper of some external math library. If we wanted to be, we would have started it like this years ago. But we are building another project that is different and standalone.

And it is not aimed to be another heavy, feature-rich math system. There are plenty of those already. Just the opposite. We want a simple, lightweight, fast and easy to learn tool for engineers to write everyday calculations. Most engineers wouldn't use more than 10% of all this advanced math. But most want to start working with 10 min of training.

And I just do not want to make Calcpad dependent on other external math libraries. This option has a lot of limitations. For example, I cannot find units of measurement in Math.NET. All their classes are built on single, double and complex unitless scalars. It is not possible to use them with our unit-aware values, which are the building blocks of the Calcpad math. Or if possible, it would require a lot of work to implement and may not be seamless.

Also, I believe that our root finding and integration procedures are better. They do not have Anderson-Bjorks method, which significantly outperforms all other bracketing methods with a little modification:
https://www.researchgate.net/publication/368554880_Modified_Anderson-Bjork's_method_for_solving_non-linear_equations_in_structural_mechanics

https://github.com/Proektsoftbg/Numerical/blob/main/Numerical.Benchmark/doc/Root/Root-test-results.pdf
Also, no adaptive Gauss-Lobatto integration, etc.

So, there are already some important things that Calcpad is ahead of Math.NET. On the other hand, Math.NET have other advanced math features that we would hardly ever need. Both are just too different and created for different purposes from the very beginning.

Anyway, vectors are 95% ready, matrices are about at 50% now. All is custom made for the Calcpad needs, for maximum performance and efficiency. I think we can do better than Math.NET. We should note that Calcpad is actually a much newer project. We just started later, so we need some time to catch up. ;)

[alishrih]

That's alright, Ned. I mentioned it because I like the project. I truly do hope it becomes popular.
When I recommended it to others, the lack of vectors were one of the issues that came up. At any rate, best of luck to you.

[Proektsoftbg]

Thank you!

Just a little more time please and we will have vectors and matrices soon. :)