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Newton’s Backward Difference Interpolation Formla-Numerical Analysis-Lecture Handouts

Lecture Notes, Numerical Analysis

Post: August 5th, 2012
Description
This course contains solution of non linear equations and linear system of equations, approximation of eigen values, interpolation and polynomial approximation, numerical differentiation, integration, numerical solution of ordinary differential equations. This lecture includes: Newton, Backward, Difference, Interpolation, Formula, Function, Derivation, Equisapced, Shift, Operator, Binomial, Expansion
This course contains solution of non linear equations and linear system of equations, approximation of eigen values, interpolation and polynomial approximation, numerical differentiation, integration, numerical solution of ordinary differential equations. This lecture includes: Newton, Backward, Difference, Interpolation, Formula, Function, Derivation, Equisapced, Shift, Operator, Binomial, Expansion
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Contents
Numerical Analysis –MTH603 VU For interpolating the value of the function y = f (x) near the end of table of values, and to extrapolate value of the function a short distance forward from y , Newton’s backward n interpolation formula is used Let y = f (x) be a function which takes on values f (x ), f (x -h), f (x -2h), …, f (x ) corresponding to equispaced values x , x -h, x -2h, n n n 0 nn n …, x . Suppose, we wish to evaluate the function f (x) at (x + ph), where p is any real 0 n number, then we have the shift operator E, such that f ( xn + ph) = E p f ( xn ) = ( E −1 ) − p f ( xn ) = (1 − ∇) − p f ( xn ) Binomial expansion yields, p( p + 1) 2 p( p + 1)( p + 2) 3  f ( xn + ph) = 1 + p∇ + ∇+ ∇+ 2! 3!  NEWTON’S BACKWARD DIFFERENCE INTERPOLATION FORMULA Derivation + That is p( p + 1)( p + 2) n! ( p + n − 1)  ∇ n + Error  f ( xn )  f ( xn + ph) = f ( xn ) + p∇f ( xn ) + + p( p + 1) 2 p( p + 1)( p + 2) 3 ∇ f ( xn ) + ∇ f ( x..

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