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Numerical Analysis: UNIT 2 - Part 2 Unit 3

This document discusses numerical methods for interpolation and differentiation including: - Newton's forward and backward interpolation formulas for equal intervals - Newton forward and backward difference formulas and examples of using difference tables - Numerical differentiation techniques including Newton's finite difference formulas, Richardson extrapolation, and derivatives for unequally spaced data. - Higher accuracy Newton's finite difference formulas are derived by including additional terms from the Taylor series expansion. - Richardson extrapolation combines two estimates with errors of O(h^2) to obtain a more accurate approximation with error of O(h^4).

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101 views14 pages

Numerical Analysis: UNIT 2 - Part 2 Unit 3

This document discusses numerical methods for interpolation and differentiation including: - Newton's forward and backward interpolation formulas for equal intervals - Newton forward and backward difference formulas and examples of using difference tables - Numerical differentiation techniques including Newton's finite difference formulas, Richardson extrapolation, and derivatives for unequally spaced data. - Higher accuracy Newton's finite difference formulas are derived by including additional terms from the Taylor series expansion. - Richardson extrapolation combines two estimates with errors of O(h^2) to obtain a more accurate approximation with error of O(h^4).

Uploaded by

egamr
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Numerical Analysis

2019

UNIT 2 – part 2
UNIT 3
 Interpolation for equal intervals
Newton’s forward interpolation formula is
Newton’s backward interpolation formula is
 Newton forward difference formula
 Example 1

o Sol. The difference table is

o Newton Forward Interpolation formula is

Unit 2- Part 2, Unit 3 Page 1


 Example 2

o Sol. The difference table is

o Newton Forward Interpolation formula is

Unit 2- Part 2, Unit 3 Page 2


 Newton backward difference formula

 Example 1

o Sol. The difference table is

o Newton Backward Interpolation formula is

Unit 2- Part 2, Unit 3 Page 3


 Numerical Differentiation

 Newton’s Finite Difference Formulas

 Richardson Extrapolation

 Newton’s Finite Difference Formulas

Unit 2- Part 2, Unit 3 Page 4


 For both the Forward and Backward difference, the error is O(h)

 Halving the step size h approximately halves the error of the

Forward and Backward differences

 The Centered difference approximation is more accurate than the

Forward and Backward differences because the error is O(h2)

 Halving the step size h approximately quarters the error of the

Centered difference.

 Example

o Sol. With h=0.5

Unit 2- Part 2, Unit 3 Page 5


o Sol. With h=0.25

 High Accuracy Newton’s Finite Difference Formulas

 Derivation of high accuracy forward and backward

o High-accuracy divided-difference formulas can be generated by

including additional terms from the Taylor series expansion

o Inclusion of the 2nd derivative term has improved the accuracy

to O(h2).

Unit 2- Part 2, Unit 3 Page 6


 Derivation of the second derivative of the centered formula

 Forward finite-divided-difference formulas

Unit 2- Part 2, Unit 3 Page 7


 Backward finite-divided-difference formulas

 Centered finite-divided-difference formulas

Unit 2- Part 2, Unit 3 Page 8


 Example

 Richardson Extrapolation
 There are two ways to improve derivative estimates when employing
finite divided differences:
o Decrease the step size, or
o Use a higher-order formula that employs more points.
o Use Richardson extrapolation
 Richardson extrapolation
o uses two derivative estimates (with O(h2) error) to compute a
third (with O(h4) error) , more accurate approximation.
o We can derive this formula following the same steps used in the
case of the integrals:

Unit 2- Part 2, Unit 3 Page 9


 Example

o Sol

Unit 2- Part 2, Unit 3 Page 10


 Richardson Extrapolation Table

 Example

o Sol

 First Column

Unit 2- Part 2, Unit 3 Page 11


 Richardson Table

Unit 2- Part 2, Unit 3 Page 12


 Derivatives of Unequally Spaced Data
 Derivation formulas studied so far (especially the ones with O(h2)
error) require multiple points to be spaced evenly.
 For unequal intervals, Fit a Lagrange interpolating polynomial, and
then calculate the 1st derivative.
 Derivative of Lagrange interpolating polynomial

 Example

Unit 2- Part 2, Unit 3 Page 13

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