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

MATH307 April 2012

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Question 01 (a)

Suppose you are given a set of N data points (xn, yn), with xn increasing, and you wish to interpolate these points with a spline function ƒ, where ƒ(x) is given by the cubic polynomial pn(x) on each interval (xn, xn+1) for n = 1, ..., N-1: $\displaystyle p_{n}(x)=a_{n}(x-x_{n})^{3}+b_{n}(x-x_{n})^{2}+c_{n}(x-x_{n})+d_{n}$ (a) Write down the equations required for ƒ(x) to be continuous and to pass through the data points. How many equations does this provide?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

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Hint

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Solution
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. For ƒ(x) to pass through all the data points, we must have pn(xn) = yn and for ƒ(x) to be continuous, we must have pn(xn+1) = yn+1. Each of these give us N-1 equations because there are N-1 polynomials. So in total, (N-1) + (N-1) = 2N - 2 equations.