# hw5 - B 2 ,i ( t ) and determine its values and the values...

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Homework 5 Foundations of Computational Math 2 Spring 2011 Solutions will be posted Monday, 2/14/11 Problem 5.1 Recall that we have derived diferent sets oF linear equations For the coe±cients oF an inter- polating cubic spline. Assume that f ( x ) = x 3 and analyze the equations and boundary conditions that de²ne s ( x ) in the Forms below and determine what can be said about the relationship between s ( x ) and f ( x ). 5.1.a . s ( x ) is determined by Ts ′′ = d where s ′′ is a vector containing s ′′ i 1 i n - 1 and boundary conditions s ′′ 0 = f ′′ ( x 0 ) and s ′′ n = f ′′ ( x n ). 5.1.b . s ( x ) is determined by ˜ Ts = ˜ d where s is a vector containing s i 1 i n - 1 and boundary conditions s 0 = f ( x 0 ) and s n = f ( x n ). Problem 5.2 Assuming that the nodes are uniFormly spaced, we have derived the Form oF the cubic B- spline B 3 ,i ( t ) and determined its values and the values oF B 3 ,i ( t ) and B ′′ 3 ,i ( t ) at the nodes t i 2 , t i 1 , t i 1 , t i , t i +1 , and t i +2 . We also derived B 1 ,i ( t ) and saw that it was the Familiar hat Function. 5.2.a . Derive the Formula oF the quadratic B-spline

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Unformatted text preview: B 2 ,i ( t ) and determine its values and the values oF B 2 ,i ( t ) and B 2 ,i ( t ) at the appropriate nodes. 5.2.b . Derive the Formula oF the quintic B-spline B 5 ,i ( t ) and determine its values and the values oF B 5 ,i ( t ) and B 5 ,i ( t ) at the appropriate nodes. Problem 5.3 Consider a set oF equidistant mesh points, x k = x + kh , 0 k m . 5.3.a . Determine a cubic spline b i ( x ) that satises the Following conditions: b i ( x j ) = b iF j < i-1 or j > i + 1 1 iF j = i b i ( x ) = b i ( x ) = 0 For x = x i 2 and x = x i +2 (or simplicity, you may assume that 2 < i < m-2.) 1 5.3.b . Show that b i ( x ) = 0 when | x-x i | 2 h . 5.3.c . Show that b i ( x ) > 0 when | x-x i | < 2 h . 5.3.d . What is the relationship between the spline, b i ( x ), and a B-spline? 2...
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## This note was uploaded on 11/10/2011 for the course MAD 5404 taught by Professor Gallivan during the Spring '11 term at FSU.

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hw5 - B 2 ,i ( t ) and determine its values and the values...

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