Who cares about how computers do matrix math? All of you, probably.

Data science is largely about manipulating matrices since almost all data can be represented as a matrix: time-series, structured data, anything that fits in a spreadsheet or SQL database, images, and language often represented as word embeddings. A typical first linear algebra course focuses on how to solve matrix problems by hand, for instance, spending time using Gaussian Elimination with pencil and paper to solve a small system of equations manually.

## Computational Algebra and Number Theory

However, it turns out that the methods and concerns for solving larger matrix problems via a computer are often drastically different :. Accuracy: Computers represent numbers which are continuous and infinite in a discrete and finite way, which means that they have limited accuracy. Rounding errors can add up, particularly if you are iterating! Furthermore, some math problems are not very stable, meaning that if you vary the input a little, you get a vastly different output.

Memory use: for matrices that have lots of zero entries, or that have a particular structure, there are more efficient ways to store these in memory.

### Related Content

Basically, we considered anything that could speed up your matrix computations as fair game to cover here! This course uses the same top down, code first, application centered teaching method as we used in our Practical Deep Learning for Coders course, and which I describe in more detail in this blog post and this talk I gave at the San Francisco Machine Learning meetup.

Basically, our goal is to get students working on interesting applications as quickly as possible, and then to dig into the underlying details later as time goes on. This approach is very different from how most math courses operate: typically, math courses first introduce all the separate components you will be using, and then you gradually build them into more complex structures.

## Ontario Research Centre for Computer Algebra - ORCCA

I love math I even have a math PhD! In the area of computational algebra and algebraic geometry, such problems as finding improved methods for computing primary decompositions, finding versal deformations, computing with inverse systems of ideals, and calculating Hilbert functions will be investigated as well as developing parallel algorithms for finding Groebner bases, applying subalgebra membership determination to finding polynomial functional decomposition and improving subalgebra algorithms for computing with subfields of rational function fields.

A generalization of multivariate resultants to general commutative rings will be investigated. The computer algebra system Macaulay will be extended to handle splines, primary decompositions, ideal radicals, inhomogeneous Groebner bases, polynomial factorization, rational arithmetic and floats. Macaulay will be improved with regard to root finding, Hilbert function calculation, data structures representing monomial ideals, and reducing virtual memory thrashing.

In the area of real and algebraically closed fields and algebraic cell decomposition, the Ben-Or, Kozen, Reif decision procedure will be improved and implemented and polynomial root finding techniques will be used to give an improved algorithm for computing full adjacency relations in algebraic cell decomposition. In the area of integration and root finding techniques, the problem of lifting positive characteristic integration back to characteristic zero will be investigated and the Ben-Or, Feig, Kozen, Tiwari algorithm for approximating all real roots of a polynomial with real coefficients will be implemented.

Finally, in the area of flows and dynamical systems, Groebner basis techniques will be applied to the problem of calculating a bound on the number of limit cycles of a polynomial flow and general utilities for the implementation of several related perturbation methods will be developed and implemented.

Please report errors in award information by writing to: awardsearch nsf. This leads to many applications. For example, here are some important problems of various fields that will be studied using matroids in this project.

## Other Free Computer Algebra Systems

A graph can be viewed as an analogue of an algebraic curve. A divisor on a graph is an assignments of integers to its vertices. We define a game in which at each step a vertex is chosen and lends a token to each of its neighbours or borrows one from each. Two divisors are equivalent if there is a sequence of moves taking one to the other. The mathematical structures arising from this process are very rich. In this setting there is an analogue of the classical Riemann-Roch theorem, and a canonical algebraic object a variety encoding equivalences of divisors on it which is closely related to the graphic matroid.

I plan to solve several important problems, such as establishing algebraic properties of this variety, by studying the Tutte polynomial. Consider a network in which every edge has an associated probability of being operational. One can think of the vertices as a set of people who pass messages among themselves and edges as communication links among them.

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Computing various reliability notions of messaging in such a network has many applications. A well-studied case arises when a person is fixed as a source and multiple people as targets, and the objective is to find the probability of the source being able to communicate with the targets. The network reliability is obtained by plugging special values in the Tutte polynomial of the associated matroid.