Numerical Methods T 264 Unit III.pdf - Lakireddy Bali Reddy...

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Lakireddy Bali Reddy College of Engineering, Mylavaram (Autonomous) B.Tech I Year (II-Semester) May/ June 2014 T 264- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagendram Interpolation: Introduction Errors in polynomial Interpolation Finite differences Forward Differences Backward Differences Central Differences Symbolic relations and separation of symbols Differences of a polynomial Newton’s formulae for interpolation – Lagrange’s Interpo lation formula.
Lakireddy Bali Reddy College of Engineering, Mylavaram (Autonomous) B.Tech I Year (II-Semester) May/ June 2014. T 264- Numerical Methods ECE - B Section UNIT III INTERPOLATION Faculty Name: N V Nagendram Planned Topics Lectures 3.1 Introduction 3.2 Errors in polynomial Interpolation 3.3 Finite Differences (i) Introduction (ii) Forward Differences (iii) Forward Difference Table (iv) Backward Differences (v) Backward Difference Table (vi) Central Differences (vii) Central Difference Table 3.4 Symbolic relations and Separation of symbols 3.5 Relationship between and E ; operators D and E and some more relations 3.6 Differences of a polynomial 3.7 Interpolation 3.8 Errors in polynomial Interpolation 3.9 Newton‟s Formula Interpolation Formula 3.10 Newton‟s Backward Interpolation Formula 3.11 Formula for Error in polynomial Interpolation 3.12 Central Difference Interpolation 3.13 Gauss‟s Forward Interpolation formula 3 .14 Gauss‟s Ba ckward Interpolation formula 3.15 Interpolation with unevenly spaced points 3 .16 Lagrange‟s Interpolation Formula
Lakireddy Bali Reddy College of Engineering, Mylavaram (Autonomous) B.Tech I Year (II-Semester) May/ June 2014. T 264- Numerical Methods UNIT III INTERPOLATION Faculty Name: N V Nagendram Lecture-1 Introduction: If we consider the statement y = f( x ), x 0 x x n we understand that we can find the value of y, corresponding to every value of x in the range x 0 x x n . Definition: If the function f( x ) is single valued and continuous and is known as explicitly then the values of f( x ) for certain values of x like x 0 , x 1 , x 2 , ...... , x n can be calculated. The problem now is if we are given the set of tabular values X x 0 x 1 x 2 .... x n-2 x n-1 x n Y y 0 y 1 y 2 .... y n-2 y n-1 y n Satisfying the relation y = f( x ) and the explicit definition of f( x ) is not known, is it possible to find a simple function say ( x ) such that f( x ) and ( x ) agree at the set of tabulated points. This process of finding ( x ) is called Interpolation . Definition: If ( x ) is a polynomial then the process is called “ polynomial interpolation and ( x ) is called “ interpolating polynomial . Note: Throughout this chapter we study polynomial interpolation. 3.2 Errors in polynomial Interpolation Types of Errors: Let x be the value and x * be an approximation to the value x. Then the difference x x * is called an “Error” in x * .

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