Opre - Numerical Analysis Time 3:00–3:50 Location LT-1 Instructor Subramania Pillai I office CC 103 phone 0832-2580442 e-mail

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Numerical Analysis Time: 3:00–3:50 Location: LT-1 Instructor: Subramania Pillai. I office: CC 103, phone 0832-2580442 e-mail: [email protected] 6-Feb-2009 (1) I.S.Pillai, CC 103 1 The Basic Problem Given a set of tabular values ( available from an experiment) of a function f . X x x 1 x 2 ... x n f ( X ) f ( x ) f ( x 1 ) f ( x 3 ) ... f ( x n ) Where the explicit nature of the function f is not known . 6-Feb-2009 (1) I.S.Pillai, CC 103 2 The problem is .. To find a function which fits the given data. To find a function φ such that φ ( x i ) = f ( x i ) for all ≤ i ≤ n. Such a function φ is called an interpolating function . 6-Feb-2009 (1) I.S.Pillai, CC 103 3 Geometrically The problem is : To find a function / polynomial whose graph passes through the given set of ( n + 1)- points, ( x ,f ( x )) , ( x 1 ,f ( x 1 )) , ··· , ( x n ,f ( x n )) . 6-Feb-2009 (1) I.S.Pillai, CC 103 4 Polynomial interpolation If φ is a polynomial then the process is called polynomial interpolation and φ is called an interpolating polynomial. 6-Feb-2009 (1) I.S.Pillai, CC 103 5 Some types of interpolation • Polynomial interpolation • Piecewise Polynomial( Spline ) interpolation • Rational interpolation • Trigonometric interpolation • Exponential interpolation We study : Polynomial and Piecewise polynomial interpolations. 6-Feb-2009 (1) I.S.Pillai, CC 103 6 Example Find a polynomial which fits the following data. X x x 1 x 2 f ( X ) f ( x ) f ( x 1 ) f ( x 2 ) 6-Feb-2009 (1) I.S.Pillai, CC 103 7 Example Solution: P 3 ( x ) = ( x- x 1 )( x- x 2 ) ( x- x 1 )( x- x 2 ) f ( x ) + ( x- x )( x- x 2 ) ( x 1- x )( x 1- x 2 ) f ( x 1 ) + ( x- x )( x- x 1 ) ( x 2- x )( x 2- x 1 ) f ( x 2 ) . This is a polynomial of degree 2 . • Called Lagrange’s interpolation formula. 6-Feb-2009 (1) I.S.Pillai, CC 103 8 Lagrange’s Interpolation formula Similarly, p n ( x ) = ( x- x 1 )( x- x 2 ) ··· ( x- x n ) ( x- x 1 )( x- x 2 ) ··· ( x- x n ) f ( x ) + ( x- x 2 )( x- x 3 ) ··· ( x- x n ) ( x 1- x )( x 1- x 2 ) ··· ( x 1- x n ) f ( x 1 ) + ··· + ( x- x )( x- x 1 ) ··· ( x- x n- 1 ) ( x n- x )( x n- x 1 ) ··· ( x 1- x n- 1 ) f ( x n ) is the “ unique polynomial of degree ≤ n ” which interpolates f at the n + 1 points x ,x 1 , ··· ,x n . 6-Feb-2009 (1) I.S.Pillai, CC 103 9 Lagrange’s Interpolation formula If we write l i ( x ) = ( x- x )( x- x 1 ) ··· ( x- x i- 1 )( x- x i +1 ) ··· ( x- x n ) ( x i- x )( x i- x 1 ) ··· ( x i- x i- 1 )( x i- x i +1 ) ··· ( x i- x n ) then p n ( x ) = ∑ n i =0 l i ( x ) f ( x i ) is the “ unique(how?) polynomial of degree ≤ n ” which interpolates f at the n + 1 points x ,x 1 , ··· ,x n ....
View Full Document

This note was uploaded on 02/23/2011 for the course MATH 122 taught by Professor Ritadubey during the Spring '11 term at Amity University.

Page1 / 186

Opre - Numerical Analysis Time 3:00–3:50 Location LT-1 Instructor Subramania Pillai I office CC 103 phone 0832-2580442 e-mail

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online