Advanced Numerical Methods with Matlab 1

Function Approximation and System Resolution

Bouchaib Radi, Abdelkhalak El Hami

• Publisher: Wiley-ISTE
• Year: 2018
• ISBN:
  9781119516552
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

Most physical problems can be written in the form of mathematical equations (differential, integral, etc.). Mathematicians have always sought to find analytical solutions to the equations encountered in the different sciences of the engineer (mechanics, physics, biology, etc.). These equations are sometimes complicated and much effort is required to simplify them. In the middle of the 20th century, the arrival of the first computers gave birth to new methods of resolution that will be described by numerical methods. They allow solving numerically as precisely as possible the equations encountered (resulting from the modeling of course) and to approach the solution of the problems posed. The approximate solution is usually computed on a computer by means of a suitable algorithm.

The objective of this book is to introduce and study the basic numerical methods and those advanced to be able to do scientific computation. The latter refers to the implementation of approaches adapted to the treatment of a scientific problem arising from physics (meteorology, pollution, etc.) or engineering (structural mechanics, fluid mechanics, signal processing, etc.).

PART 1
Introduction

1
Review of Linear Algebra

We will denote the fields of real and complex numbers by ℝ and ℂ respectively. If there is no need to distinguish between them, we will instead simply refer to the field 𝕂 of scalars. The set E, equipped with the two operations of addition and scalar multiplication, denotes a vector space over 𝕂 (or a 𝕂-vector space).

1.1. Vector spaces

1.1.1. General definitions

DEFINITION.– A vector space over the field 𝕂 is a set E equipped with the two following operations:

• – addition, which equips E with the structure of a commutative group;
• – an “outer” product of an element of E by an element of 𝕂, satisfying the following properties:
  • - ∀λ, μ ∈ 𝕂, ∀x ∈ E : (λμ)x = λ(μx);
  • - ∀λ, μ ∈ 𝕂, ∀x ∈ E : (λ + μ)x = λx + μx;
  • - ∀λ ∈ 𝕂, λ(x + y) = λx + λy;
  • - ∀x ∈ E : 1x = x (where 1 is the identity element of 𝕂).

The elements of E are called vectors.

DEFINITION.– Let F be a subset of the vector space E. F is a vector subspace of E if it is closed under the operations of E. In other words:

• – ∀x, y ∈ F, x + y ∈ F;
• – ∀λ ∈ 𝕂, ∀x ∈ F, λx ∈ F.

DEFINITION.– Let E be a 𝕂-vector space, and suppose that F and G are two vector subspaces of ...

Table of contents

• Cover
• Table of Contents
• Title
• Copyright
• Preface
• PART 1: Introduction
• PART 2: Approximating Functions
• PART 3: Solving Linear Systems
• PART 4: Appendices
• Bibliography
• Index
• End User License Agreement