NATIONAL OPEN UNIVERSITY OF NIGERIA

SCHOOL OF SCIENCE AND TECHNOLOGY

COURSE CODE: MTH 213

COURSE TITLE: NUMERICAL ANALYSIS 1

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Course Code MTH 213 Course Title NUMERICAL ANALYSIS 1 Course Developer Dr. Ajibola S. O.

National Open University of Nigeria Lagos

Content Editor Dr. ABIOLA. Bankole

National Open University of Nigeria Lagos

Course Coordinator Dr. Ajibola S. O.

National Open University of Nigeria Lagos

Programme Leader Dr. ABIOLA.Bankole

National Open University of Nigeria Lagos

NATIONAL OPEN UNIVERSITY OF NIGERIA

COURSE GUIDE MTH213

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National Open University of Nigeria Headquarters 14/16 Ahmadu Bello Way Victoria Island Lagos Abuja Office 5, Dar Es Salaam Street Off Aminu Kano Crescent Wuse II, Abuja Nigeria. e-mail: centralinfo@nou.edu.ng URL: www.nou.edu.ng National Open University of Nigeria 2006 First Printed 2008 ISBN: All Rights Reserved Printed by: For National Open University of Nigeria

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CONTENT PAGE Module 1 Interpolation 1 Unit 1 Interpolation (Lagranges Form).. 1 Unit 2 Newtons Form of the Interpolating Polynomial 15 Unit 3 Interpolation at Equally Spaced Points 31 Module 2 Solution of Linear Algebraic Equations 55 Unit 1 Direct Method 55 Unit 2 Inverse of A Square Matrix 91 Unit 3 Iterative Methods 112 Unit 4 Eigen-Values and Eigen-Vectors 135 Module 3 Solution of Non-Linear Equations in

one Varibale . 159 Unit 1 Review of Calculus 159 Unit 2 Iteration Methods for Locating Root. 189 Unit 3 Chord Methods for Finding Root... 208 Unit 4 Approximate Root of Polynomial Equation... 235

COURSE GUIDE MTH213

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MTH 213 NUMERICAL ANALYSIS 1

Course Developer Dr. Ajibola S. O.

National Open University of Nigeria Lagos

Content Editor Dr. ABIOLA. Bankole

National Open University of Nigeria Lagos

Course Coordinator Dr. Ajibola S. O.

National Open University of Nigeria Lagos

Programme Leader Dr. ABIOLA. Bankole

National Open University of Nigeria Lagos

NATIONAL OPEN UNIVERSITY OF NIGERIA

COURSE GUIDE

vi

National Open University of Nigeria Headquarters 14/16 Ahmadu Bello Way Victoria Island Lagos Abuja Office 5, Dar Es Salaam Street Off Aminu Kano Crescent Wuse II, Abuja Nigeria. e-mail: centralinfo@nou.edu.ng URL: www.nou.edu.ng National Open University of Nigeria 2008 First Printed 2008 ISBN: All Rights Reserved Printed by .. For National Open University of Nigeria

COURSE GUIDE MTH213

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CONTENTS PAGE Introduction. 1 The Course ...... 1 Course Aims & Objectives .. 2 Working through the course 2 Course materials.. 2 Study Units.. 3 Textbooks.... 4 Assessment... 5 Tutor-Marked Assignments. 5 End of Course Examination. 5 Summary. 5

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Introduction MTH 213: Discussion of Lagranges form for; The technique of determining an approximate value of f(x) for a non-tabular value of x which lies in the internal [a, b] is called interpolation. The process of determining the value of f(x) for a value of x lying outside the interval [a, b] is called extrapolation. The Lagranges form of the interpolating polynomial derived above has same draw backs compared to Newtons form of interpolating polynomial. Before deriving Newtons general form of interpolating polynomial. We introduce the concept of divided difference and the tabular representation of divided differences. Numerical solution of systems of linear algebraic equations play a prominent role in boundary value problems, for ordinary and partial differential equations, statistical influence, optimization theory, least square fittings of data etc. Numerical methods for solving linear algebraic system may be divided into two types, direct and iterative. To understand the numerical methods for solving linear system of equations, it is necessary to have some knowledge of the properties of matrices. The prerequisite to the course shall be linear Algebra courses. The Course As a 3-credit unit course, 11 study units grouped into 3 modules of 3 units in module 1, 4 units in module 2 and 4 units in module 3. This course guide gives a brief summary of the total contents contained in the course material. The fundamental theorem of algebra and its useful calories, inverse interpolation and errors. Newtons form of the interpolating polynomial features divided differences and interpolating polynomial error types. Likewise interpolating at equally spaced points, here we talked about differences. For equally spaced nodes, we shall deal with three types of differences, namely forward, backward and central and discuss their representation in the form of a table. Also discussed her are some direct and iterative methods for finding the solution of system of linear algebraic equations. Lastly, we discussed three fundamental theorems, namely; intermediate value theorem, Rolles theorem and Lagranges mean value theorem. All these theorems give properties of continues functions defined on a

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closed interval [a, b]. Although the theorems are not proved but their utility was illustrated with examples. Course Aim & Objectives On the completion of this course, you are expected to: find the Lagranges form of interpolating polynomial complete the approximate value of f at a non-tabular point. Complete the error omitted in interpolation, if the function is

known at a non-tabular point of interest. Find an upper bound in the magnitude of the error. Write forward, backward and central differences in terms of

function values from a table of either difference and locate a difference of given order at given point.

Obtain the interpolating polynomial of f(x) for a given data by applying any one of the interpolating formulae.

Obtain the solution of systems of linear algebraic equations by using the direct methods such as Cramers rule, Gauss elimination method Lu decomposition method.

Working through the Course This course involves that you would be required to spend lot of time to read. The content of this material is very dense and require you spending great time to study it. This accounts for the great effort put into its development in the attempt to make it very readable and comprehensible. Nevertheless, the effort required of you is still tremendous. I would advice that you avail yourself the opportunity of attending the tutorial sessions where you would have the opportunity of comparing knowledge with your peers. The Course Material You will be provided with the following materials: Course Guide Study Units In addition, the course comes with a list of recommended textbooks, which through are not compulsory for you to acquire or indeed read, are necessary as supplements to the course material.

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Study Units The following are the study units contained in this course. The units are arranged into 3 identifiable but readable modules. Module 1 Unit 1 Interpolation (Lagranges Form) This unit takes one through the definition of interpolation, inverse interpolation and error. Unit 2 Newtons Form of the Interpolating Polynomial This unit is sub-divided into divided difference Newtons General Form of interpolating polynomial, and the error of the interpolating polynomial. Divided difference and derivative of the functions and further results on interpolations error. Unit 3 Interpolation at Equally Spaced Points This unit takes about the three types of differences i.e. forward, backward and central differences. Difference formulae which encompasses: Newtons ForwardDifference formula and Newtons Backward-Difference formula. Module 2 Solution of Linear Algebraic Equations. Unit 1 Direct Method This unit entails the preliminaries, Cramers rule, direct methods for special matrices. Gauss elimination methods and LU decomposition method. Unit 2 Inverse of A Square Matrix This unit is sub-divided into method of adjoints, the Gauss-Jordan reduction method and LU decomposition method. Unit 3 Iterative Methods This unit consists of the general iterative methods. The Jaccobis iteration methods and the Gauss-Seidel iteration method.

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Unit 4 Eigen-Values and Eigen-Vectors. This unit focused on the Eigen value problem. The power method and the inverse power method. Module 3 Unit 1 Review of Calculus Here, the three fundamental theorems, Taylor's theorem, error (round off and truncation errors) are discussed. Unit 2 Iteration Methods for Locating Root. This unit discussed: The initial approximation to a root (tabulation and graphical methods). Bisection method and fixed point iteration method. Unit 3 Chord Methods for Finding Root This entails Repuler-Falsi method, Newton Raphson method and convergence criterion. Unit 4 Approximate Root of Polynomial Equation. It can be sub-divided into some results on roots of polynomial equation. Birge-Vieta method and Graeffes Root squaring method