Computational algebra

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Such an exact representation implies that, even when the size of the output is small, the intermediate data generated during a computation may grow in an unpredictable way. This behavior is called expression swell. To obviate this problem, various methods are used in the representation of the data, as well as in the algorithms that manipulate them. The usual numbers systems used in numerical computation are floating point numbers and integers of a fixed bounded size.

None of these is convenient for computer algebra, due to expression swell. Therefore, the basic numbers used in computer algebra are the integers of the mathematicians, commonly represented by an unbounded signed sequence of digits in some base of numeration , usually the largest base allowed by the machine word. These integers allow to define the rational numbers , which are irreducible fractions of two integers. Programming an efficient implementation of the arithmetic operations is a hard task. Therefore, most free computer algebra systems and some commercial ones such as Mathematica and Maple software , use the GMP library , which is thus a de facto standard.

Except for numbers and variables , every mathematical expression may be viewed as the symbol of an operator followed by a sequence of operands. In computer algebra software, the expressions are usually represented in this way. This representation is very flexible, and many things, that seem not to be mathematical expressions at first glance, may be represented and manipulated as such.

Conversely, any mathematical expression may be viewed as a program. Executing this program consists in evaluating the expression for given values of a and b ; if they do not have any value—that is they are indeterminates—, the result of the evaluation is simply its input. This process of delayed evaluation is fundamental in computer algebra. As the size of the operands of an expression is unpredictable and may change during a working session, the sequence of the operands is usually represented as a sequence of either pointers like in Macsyma or entries in a hash table like in Maple.

Such a complicated expression is clearly not acceptable, and a procedure of simplification is needed as soon as one works with general expressions. This simplification is normally done through rewriting rules. There are several classes of rewriting rules that have to be considered. They are systematically applied in the computer algebra systems.

Course Overview

The first difficulty occurs with associative operations like addition and multiplication. In other words, in the internal representation of the expressions, there is no subtraction nor division nor unary minus, outside the representation of the numbers. A second difficulty occurs with the commutativity of addition and multiplication.

The problem is to recognize quickly the like terms in order to combine or cancel them. In fact, the method for finding like terms, consisting of testing every pair of terms, is too costly for being practicable with very long sums and products. For solving this problem, Macsyma sorts the operands of sums and products with a function of comparison that is designed in order that like terms are in consecutive places, and thus easily detected. In Maple , the hash function is designed for generating collisions when like terms are entered, allowing to combine them as soon as they are introduced.

This design of the hash function allows also to recognize immediately the expressions or subexpressions that appear several times in a computation and to store them only once. This allows not only to save some memory space, but also to speed up computation, by avoiding repetition of the same operations on several identical expressions.

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Some rewriting rules sometimes increase and sometimes decrease the size of the expressions to which they are applied. This is the case of distributivity or trigonometric identities. For the distributivity, the computer function that applies this rewriting rule is generally called "expand". The reverse rewriting rule, called "factor", requires a non-trivial algorithm, which is thus a key function in computer algebra systems see Polynomial factorization.

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In this section we consider some fundamental mathematical questions that arise as soon as one wants to manipulate mathematical expressions in a computer. We consider mainly the case of the multivariate rational fractions. This is not a real restriction, because, as soon as the irrational functions appearing in an expression are simplified, they are usually considered as new indeterminates. For example,. There are two notions of equality for mathematical expressions. The syntactic equality is the equality of the expressions which means that they are written or represented in a computer in the same way.

Being trivial, the syntactic equality is rarely considered by mathematicians, although it is the only equality that is easy to test with a program. The semantic equality is when two expressions represent the same mathematical object, like in. It is known from Richardson's theorem that there may not exist an algorithm that decides if two expressions representing numbers are semantically equal, if exponentials and logarithms are allowed in the expressions.

Therefore, semantical equality may be tested only on some classes of expressions such as the polynomials and rational fractions. To test the equality of two expressions, instead of designing specific algorithms, it is usual to put expressions in some canonical form or to put their difference in a normal form , and to test the syntactic equality of the result. Unlike in usual mathematics, "canonical form" and "normal form" are not synonymous in computer algebra.

In other words, zero has a unique representation by expressions in normal form. Normal forms are usually preferred in computer algebra for several reasons. Firstly, canonical forms may be more costly to compute than normal forms.

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For example, to put a polynomial in canonical form, one has to expand by distributivity every product, while it is not necessary with a normal form see below. Secondly, It may be the case, like for expressions involving radicals, that a canonical form, if it exists, depends on some arbitrary choices and that these choices may be different for two expressions that have been computed independently.

This may make impracticable the use of a canonical form. At the beginning of computer algebra, circa , when the long-known algorithms were first put on computers, they turned out to be highly inefficient. A typical example of this kind of work is the computation of polynomial greatest common divisors , which is required to simplify fractions.

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Surprisingly, the classical Euclid's algorithm turned out to be inefficient for polynomials over infinite fields, and thus new algorithms needed to be developed. The same was also true for the classical algorithms from linear algebra. From Wikipedia, the free encyclopedia. Anyone can view the notebooks online by clicking on the links in the readme Table of Contents.

However, to really learn the material, you need to interactively run the code, which requires installing Anaconda on your computer or an equivalent set up of the Python scientific libraries and you will need to be able to clone or download the git repo. Accompanying the notebooks is a playlist of lecture videos, available on YouTube. If you are ever confused by a lecture or it goes too quickly, check out the beginning of the next video, where I review concepts from the previous lecture, often explaining things from a new perspective or with different illustrations. You can ask questions or share your thoughts and resources using the Computational Linear Algebra category on our fast.

This course assumes more background than our Practical Deep Learning for Coders course.

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If you are new to linear algebra, I recommend you watch the beautiful 3Blue 1Brown Essence of Linear Algebra video series as preparation. There is no official registration for the course; rather it is up to you to work through the materials at your own pace. You can get started now, by watching the 1st video:.

Knowing how these algorithms are implemented will allow you to better combine and utilize them, and will make it possible for you to customize them if needed. Several of the topics we cover are areas of active research, and there is recent research that has not yet been added to existing libraries. New fast. What is numerical linear algebra? However, it turns out that the methods and concerns for solving larger matrix problems via a computer are often drastically different : Speed: when you have big matrices, matrix computations can get very slow. There are different ways of addressing this: Different algorithms: there may be a less intuitive way to solve the problem that still gets the right answer, and does it faster.

Vectorizing or parallelizing your code. Locality: traditional runtime computations focus on Big O , the number of operations computed. However, for modern computing, moving data around in memory can be very time-consuming, and you need ways to minimize how much memory movement you do. Scalability: in many cases you are interested in working with more data than you have space in memory for.

Teaching Methodology 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. The Details The primary resource for this course is the free online textbook of Jupyter Notebooks, available on Github.

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Computational algebra Computational algebra
Computational algebra Computational algebra
Computational algebra Computational algebra
Computational algebra Computational algebra
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