Math128_hw3 - Math 128A Spring 2007 Homework 3 Solution 4.1.12(a L0(x L1(x L2(x L3(x is the unique polynomial of degree 3 interpolating the data(x0

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 128A, Spring 2007 Homework 3 Solution 4.1.12 (a) L 0 ( x ) + L 1 ( x ) + L 2 ( x ) + L 3 ( x ) is the unique polynomial of degree 3 interpolating the data ( x 0 , 1) , ( x 1 , 1) , ( x 2 , 1) , ( x 3 , 1). Since the polynomial 1 also has degree 3 and interpolates the data, we conclude L 0 ( x ) + L 1 ( x ) + L 2 ( x ) + L 3 ( x ) = 1 . (b) In general, L 0 ( x ) + ··· L n ( x ) and 1 are both degree n interpolants of ( x 0 , 1) , . . . , ( x n , 1), and therefore must be equal. 4.1.13 For j 3, note that x j 0 L 0 ( x )+ x j 1 L 1 ( x )+ x j 2 L 2 ( x )+ x j 3 L 3 ( x ) and x j are both polynomials of degree 3 interpolating ( x 0 , x j 0 ) , ( x 1 , x j 1 ) , ( x 2 , x j 2 ) , ( x 3 , x j 3 ). Since such interpolants are unique, the two polynomials must be equal. For j > 3, note that each L i ( x ) has degree 3, so a linear combination of them can’t have the same degree as x j . 4.1.15 Note: this is the complicated solution. In analogy to the Lagrange basis { L i ( x ) } , we want to find a basis { H i ( x ) } for the (vector space of) polynomials of degree 3 so that the coordinates of P ( x ) with respect to
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 05/07/2008 for the course MATH 128A taught by Professor Rieffel during the Spring '08 term at University of California, Berkeley.

Page1 / 4

Math128_hw3 - Math 128A Spring 2007 Homework 3 Solution 4.1.12(a L0(x L1(x L2(x L3(x is the unique polynomial of degree 3 interpolating the data(x0

This preview shows document pages 1 - 2. Sign up to view the full document.

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