Science:Math Exam Resources/Courses/MATH307/April 2012/Question 01 (b)
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Question 01 (b) 

Suppose you are given a set of N data points (x_{n}, y_{n}), with x_{n} increasing, and you wish to interpolate these points with a spline function ƒ, where ƒ(x) is given by the cubic polynomial p_{n}(x) on each interval (x_{n}, x_{n+1}) for n = 1, ..., N1: (b) Write down the equations required for ƒ(x) to have continuous first and second derivatives. How many equations does this provide? 
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Hint 

Science:Math Exam Resources/Courses/MATH307/April 2012/Question 01 (b)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. For to have continuous first derivatives, we must have . This gives us N2 equations because we cannot fit the endpoints to anything. For to have continuous second derivatives, we must have . This gives us N2 equations because again, we cannot fit the endpoints to anything. So in total, equations 