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Electronic Transactions on Numerical Analysis

EBook NameElectronic Transactions on Numerical Analysis
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EBook Description: its book of mathematics on numerical analysis.

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This book was begun around January 1991, as a response to a need I perceived in the classroom. Students who have not had calculus but intend to pursue the mathematical sciences are not ready for a formalistic proof-laden treatment of the subject.

1 Relations

This book starts with one of its most abstract topics, so don't let the abstract nature deter you. Relations are quite simple but like virtually all simple mathematical concepts they have their subtle aspects. Relations are defined on collections (or sets) of objects.

2 Graphs

We are concerned with graphs. Every graph contains objects that are known as nodes or vertices depending on the mood of the speaker. A graph always contains at least one node. A graph usually also contains lines that go between nodes. These lines are known as arcs or edges, again depending on the mood of the speaker.

3 Integer Functions

The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple functions. The first (and most important) of these is the floor function. We denote this function by lxm although it can be denoted by floor(x) It is also widely known as the greatest integer function and is found in some computer languages as the integer function (sometimes denoted INT(x)).

4 Algorithms

An algorithm is the abstract version of a computer program. In essence a computer program is the codification of an algorithm, and every algorithm can be coded as a computer program.

5 Recursive Algorithms

Recursion is a form of definition and of algorithms that is very important in computer science theory as well as in practice. Recursive algorithms can be inefficient or efficient.1 A recursive definition or a recursive algorithm is characterized by self-reference.

6 Induction

The principle of induction is a one of the most fundamental laws of mathematics. Students come across it, as here, as a method of proof. However, it is a basic property of the natural numbers (i.e. the positive integers) and can be considered as part of the definition of the natural numbers. Since we do not define the natural numbers here, the principle will be simply given as an axiom.1

7 Graph Searching

Algorithms for searching graphs go back at least to the nineteenth century when they were used for mazes. Until the advent of computers they were an extremely obscure topic (as was graph theory). We are going to look at two algorithms which represent the two main approaches to searching graphs.

8 Horner's Algorithm

Horner's algorithm is an excellent example of an algorithm. It is a simple and useful algorithm, but it does not have much to do with discrete mathematics.

9 Shortest Paths

A fundamental problem in graphs is finding the shortest path from vertex A to vertex B. Fortunately there are several simple (and efficient algorithms for doing this). We will look at the three best of these: Dijkstra’s algorithm, Floyd’s algorithm, and the Bellman-Ford algorithm. First we need to discuss different types of shortest path problems and various conventions that we will use in solving them.

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