S.Baskar

1 introduction to numerical AnalysisS. Baskar2 General InstructionsCourse Number :SI 507 Course Title : numerical AnalysisCourse Preliminaries:Continuity of a Function and Intermediate Value Theorem; Mean ValueTheorem for Differentiation and Integration; Taylor s Theorem (1 and 2 dimensions). Analysis:Floating-Point Approximation of a Number; Loss of Significance and Error Propagation;Stability in numerical Systems:Gaussian Elimination; Pivoting Strategy; LU factorization; Residual Corrector Method;Solution by Iteration; Conjugate Gradient Method; Ill-Conditioned Matrices, Matrix Norms; Eigenvalue prob-lem - Power Method; Gershgorin s Equations:Bisection Method; Fixed-Point Iteration Method; Secant Method; Newton Method;Rate of Convergences; Solution of a System of Nonlinear Equations; Unconstrained by Polynomials:Lagrange Interpolation; Newton Interpolation and DividedDifferences;Hermite Interpolation; Error of the Interpolating Polynomials; Piecewise Linear and Cubic Spline Interpola-tion; Trigonometric Interpolation; Data Fitting and Least-Squares Approximation and Integration:Difference formulae.

2 Some Basic Rules of Integration; Adaptive Quadra-tures; Gaussian Rules; Composite Rules; Error Equations:Euler Method; Runge-Kutta Methods; Multi-Step Formulae; Predictor-CorrectorMethods; Stability and Convergence; Two Point Boundary Value K. E. Atkinson,An introduction to numerical Analysis(2nd edition), Wiley-India, S. D. Conte and Carl de Boor,Elementary numerical Analysis - An Algorithmic Approach(3rd edition),McGraw-Hill, Rules1. Attendance in lectures as well as tutorials is compulsory. Students not fulfilling the 80% attendance require-ment may be awarded the XX Attendance will be recorded through an attendance sheet that will be circulated in the class. Each studentis expected to sign against his/her name only. Students who are found indulging in proxy attendance or anyform of cheating will be severely Plan1. There will be two quizzes (dates will be announced later),each of weightage 10% and one hour The Mid-Semester Examination scheduled during 11-18 September 2010 will be of 30% The End-Semester Examination scheduled during 16-28 November will be of 40% Lab assignments will be given through out the semester andthe students are expected to complete theassignment and produce all the outputs asked at the end of thesemester.

3 A oral viva will be conducted toeach student. The weightage will be of 10%.5. To pass the course (DD), one needs to score minimum of 40% ofthe maximum mark scored in the class. Forinstance, if the maximum mark scored is 80% at the end of the semester, then the passing mark will be 32%.Higher grades will be based on the over all performance of Page: Course related materials will be uploaded addition to the references provided above, class notes will be distributed in the class as a typedmaterial. These notes are meant only for SI 507 in Autumn 2010 as a supplementary material and cannotbe considered as a text book. Students are requested to refer the text books listed under course syllabusfor more details. These notes may have errors of all kind and the author request the readers to take careof such error while going through the material. The author will be grateful to those who brings to hisnotice any kind of.

4 71 Mathematical Preliminaries.. Continuity of a Function .. Differentiation of a Function .. Integration of a Function .. Taylor s Formula .. 12 Exercise 1 .. 142 Error Analysis.. Floating-Point Form of Numbers .. Chopping and Rounding a Number .. Different Type of Errors .. Loss of Significant Digits .. Propagation of Error .. 21 Exercise 2 .. 253 Linear Systems.. Gaussian Elimination .. LU Factorization Method .. Error in Solving Linear Systems .. Matrix Norm .. Iterative Methods .. Eigenvalue Problem: The Power Method.. Gerschgorin s Theorem .. 43 Exercise 3 .. 454 Nonlinear Equations.. Fixed-Point Iteration Method .. Bisection Method.. Secant Method .. Newton-Raphson Method.. System of Nonlinear Equations .. Unconstrained Optimization .. 626 ContentsExercise 4 .. 645 Interpolation by Polynomials.

5 Lagrange Interpolation .. Newton Interpolation and Divide Differences .. Error in Polynomial Interpolation .. Piecewise Linear and Cubic Spline Interpolation .. 75 Exercise 5 .. 776 numerical Differentiation and Integration.. numerical Differentiation .. numerical Integration .. 83 Exercise 6 .. 917 numerical Ordinary Differential Equations.. Review on Theory .. Discretization .. Euler s Method.. Runge-Kutta Method .. An Implicit Methods .. Multistep Methods: Predictor and Corrector .. 101 Exercise 7 .. 103 References.. 105 Index.. 107 IntroductionNumerical analysis is a branch of Mathematics that deals with devisingefficient methods for obtainingnumerical solutions to difficult Mathematical of the Mathematical problems that arise in science and engineering are very hard and sometimeimpossible to solve exactly. Thus, an approximation to a difficult Mathematical problem is very impor-tant to make it more easy to solve.

6 Due to the immense development inthe computational technology, numerical approximation has become more popular and a modern tool for scientists and engineers. As aresult many scientific softwares are developed (for instance, Matlab, Mathematica, Maple etc.) to handlemore difficult problems in an efficient and easy way. These softwares contain functions that uses standardnumerical methods, where a user can pass the required parameters and get the results just by a singlecommand without knowing the details of the numerical method. Thus, one may ask why we need tounderstand numerical methods when such softwares are at our hands?In fact, there is no need of a deeper knowledge of numerical methods and their analysis in most of thecases in order to use some standard softwares as an end user. However, there are at least three reasonsto gain a basic understanding of the theoretical background of numerical Learning different numerical methods and their analysis will make aperson more familiar with thetechnique of developing new numerical methods.

7 This is important when the available methods arenot enough or not efficient for a specific problem to be In many circumstances, one has more methods for a given problem. Hence, choosing an appropriatemethod is important for producing an accurate result in lesser With a sound background, one can use methods properly (especially when a method has its ownlimitations and/or disadvantages in some specific cases) and, most importantly, one can understandwhat is going wrong when results are not as analysis include three parts. The first part of the subject is about the development of amethod to a problem. The second part deals with the analysis of the method, which includes the erroranalysis and the efficiency analysis. Error analysis gives us the understanding of how accurate the resultwill be if we use the method and the efficiency analysis tells us how fast we can compute the third part of the subject is the development of an efficient algorithm to implement the method asa computer code.

8 A complete knowledge of the subject includes familiarity in all these three parts. Thiscourse is designed to meet this first course in Calculus is indispensable for numerical analysis. The first chapter of these lecturenotes quickly reviews all the essential calculus for following this course. Few theorems that are repeatedlyused in the course are collected and presented with an outline of their 2 introduces the concept of errors. One may be surprised to see errors at the initial stageof the course when no methods are introduced. Of course, thereare two types of errors involved in amethod, namely,1. the error involved in approximating a problem and2. the error due to first type of error is purely mathematical and often known astruncation error. The second one isdue to the floating-point approximation of a number. This error is committed by computer due to theirlimited memory capacity.

9 For instance, the number 1/3= hasinfinitely many digits and since acomputer can deal with a number with finite number of digits, this number has to be approximated tothe number with finite number of digits (depending on the memory capacity of the computer).Such an approximation is called thefloating-point approximation. Chapter 2 is devoted mainly tothe floating-point error and related methods to solve linear systems and computation of eigenvalues and eigen vectors are thesubject of the chapter 2. In this chapter, we discuss direct methods which gives exact solution to thesystems mathematically. However, when we implement these direct methods on a computer we will getan approximate solution as the computed solution involves floating-point error. The chapter then discusssome iterative methods for solving linear systems. After a brief discussion of matrix analysis, the chapterends with power method for computing a eigenvalue and the corresponding eigen vector for a given all eivenvalues can be computed using this method and also not allmatrices can be applicable tothis method.

10 Gershgorin s theorem may be used to decide whether power method can be used for a givenmatrix. We state this theorem without proof and discuss its application to power 4 introduces various iterative methods for a nonlinear equation and their convergence anal-ysis. The methods are further extended to system of nonlinear equations. Unconditioned optimization isdiscussed at the end of the chapter. Interpolation by polynomials,data fitting and least-square approx-imation are the subject of Chapter 5. Chapter 6 introduced numerical differentiation and notes end with some basic methods for solving ordinary differential PreliminariesThis chapter reviews some of the results from calculus that are frequently used in this course. Onlydefinitions and important theorems with outline of a proof are provided. However, the readers are assumedto be familiar with a first course in 1 defines continuity of a function and proves intermediate value theorem.