Chapter One: Introduction
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1 Chapter One: Introduction Objectives 1. Understand the need for numerical methods 2. Go through the stages (mathematical modeling, solving and implementation) of solving a particular physical problem. Pre-Requisites 1. Be able to find integrals of a function. 2. Be able to differentiate functions Numerical Analysis Numerical Analysis is concerned with the Mathematical derivation, description and analysis of methods of obtaining numerical solution of Mathematical problems, many computed problem (Math., Stat., Physics, ) cannot be solved, but we can solved by numerical analysis. Errors One of the most important aspects of numerical analysis is the error analysis. Errors may occur at any stage of the process of solving a problem. By the error we mean the difference between the true value and the approximate value. Types of Errors i. Inherent Errors. Errors = True value Approximate value These are the errors involved in the statement of a problem. When the problem is first presented to the numerical analysis it may contain certain data or parameters. ii. Analytic Error. These are the errors introduced due to transforming a physical or mathematical problem into a computational problem. iii. Rounding and Chopping Errors Luckson Phiri Page 1
2 For example The most widely and important errors caused by applying numerical methods are Errors caused by chopping and rounding: 1. 1/3= where 1/3= e= where e= where iv. Formulation Errors When solving the problem using mathematical method, usually a simple model would be used to describe the source problem, there for some of the factors will be put away which means simplifying the problems which cause some error and this error is called Formulation Error. For example m m 0 v 1 ( ) c 2 The second law Newton F= m.a Where m is a mass of a particle, a is acceleration In the fact m 0 is initial mass of particle V is velocity C is velocity of light Since V < C so V/C 0 And m=m 0 v. Absolute Errors E x = X-x The absolute error of an approximate number x is the absolute value of the difference between the corresponding exact number X and the number x. It denoted by E x thus: Luckson Phiri Page 2
3 vi. Relative Errors E R =E x / X. The relative error of an approximate number x is the ratio of the absolute error the number to the absolute value of the corresponding the exact number X. Where X 0, it is denoted by E R. vii. viii. Define Truncation Errors? Error in Arithmetic Operations: a. Addition Operations: E x+y =E x +E y b. Subtraction Operations: E x-y =E x -E y c. Multiplication Operations: d. Deviation Operations: Mathematical models are an integral part in solving engineering problems. Many times, these mathematical models are derived from engineering and science principles, while at other times the models may be obtained from experimental data. Mathematical models generally result in need of using mathematical procedures that include but are not limited to a) differentiation, b) nonlinear equations, c) simultaneous linear equations, d) curve fitting by interpolation or regression, e) integration, and f) Differential equations. Systems of Linear Equations The method of elimination, existed for centuries, was put into systematic order by Karl Friedrich Gauss ( ) and by Camille Jordan ( ). This method led to the matrix method (Gauss Jordan method) that is now used for solving large systems by computer. Luckson Phiri Page 3
4 In general, a system of linear equations is a collection of two or more linear equations, each containing one or more variables. A solution of a system of linear equations consists of values of the variables that are solutions of each equation of the system. To solve a system of linear equations means to find all solutions of the system. If a system of equations has at least one solution, it is said to be consistent; if it has no solution, it is said to be inconsistent. If a consistent system of equations has exactly one solution, the equations of the system are said to be independent; if it has an infinite number of solutions, the equations are called dependent. Two Linear Equations Containing Two Variables 1. If the lines are parallel, then the system of equations has no solution, because the lines never intersect. The system is inconsistent. 2. If the lines intersect, then the system of equations has one solution, given by the point of intersection. The system is consistent and the equations are independent. 3. If the lines are coincident, then the system of equations has infinitely many solutions, represented by the totality of points on the line. The system is consistent and the equations are dependent. A system of equations is either I. Inconsistent with no solution II. Consistent with a. One solution (equations are independent) Luckson Phiri Page 4
5 b. Infinitely many solutions (equations are dependent) Finding solutions Methods Used: Graphing Substitution Elimination (Linear Combination) Luckson Phiri Page 5
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