NumericalAnalysis-683-2010aug

# NumericalAnalysis-683-2010aug - 1 MAT 683 Numerical...

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 MAT 683 Numerical Analysis Qualifying Exam Syracuse University August 24, Fall 2010 Ex 1. Let x < x 1 be two distinct real numbers. Let be such that 0 < < x 1- x . Find a polynomial p of degree 3 such that p ( x ) = p ( x + ) = 1 p ( x 1 ) = p ( x 1 + ) = 0 Let p ( x ) = lim → p ( x ). Find p ( x ), p ( x 1 ), p ( x ), p ( x 1 ) and p ( x 1 ). What is the Hermite interpolation problem that the polynomial p satisfies? Ex 2. Let α , 0 < α < 1 be a given real number and let x 1 =- 1, x 2 =- α , x 3 = α , x 4 = 1 and let w 1 ,w 2 ,w 3 ,w 4 be real numbers. We consider a quadrature Q ( f ) = 4 X j =1 w j f ( x j ) for any function f continuous on [- 1 , 1]. (a) Find w 1 ,w 2 ,w 3 ,w 4 as functions of α so that Q ( p ) = R 1- 1 p ( x ) d x for every polynomial p of degree ≤ 3. (b) Does there exist an α such that Q ( p ) = R 1- 1 p ( x ) d x for every polynomial p of degree r for r > 3 ? If yes what is the maximal value of r and what are the corresponding values values of α and w...
View Full Document

## This note was uploaded on 06/19/2011 for the course MATH 684 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.

### Page1 / 2

NumericalAnalysis-683-2010aug - 1 MAT 683 Numerical...

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

View Full Document
Ask a homework question - tutors are online