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A Natural Cubic Spline S on $[0,2]$ is defined by:

S(x)= $S_0(x)=1+2x-x^3 \to 0 \leq x < 1 $ $S_1(x)=2+b(x-1)+c(x-1)^2+d(x-1)^3 \to 1 \leq x \leq 2$

Find b,c and d

This question seems simple I am just not sure how to go about it. Can someone please guide me as to how to find the solution

1 Answers 1

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Spline of order $k$ should have continuous derivatives up to order $k-1$. In this case,

$S_0(1) = S_1(1)$ $S_0'(1) = S_1'(1)$ $S_0''(1) = S_1''(1)$

Since this is a natural cubic spline, the second derivative at the first node $(x=0)$ and at the last node $(x=2)$ should be $0$.

and then you can solve for $b, c, d$.

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    Ok I found my blunder. Th$a$nks @Patrick Li2012-09-28