Solving Cubic Spline Interpolation with Conditions and Coefficients

[Orphen89]

Homework Statement

Consider the use of cubic splines to interpolate a set of data. Suppose at some stage in the calculation we arrive at the following spline functions for two consecutive intervals

\tilde{f_{0}} = x^{3} + ax^{2} + bx + c over the interval -1 \leq x \leq 1
\tilde{f_{1}} = 2x^{3} + x^{2} - x + 4 over the interval 1 \leq x \leq 2


a) State the conditions that should be imposed on the two functions
b) Hence, compute a, b and c

I'm having a bit of difficulty getting started with this question. I've managed to do Lagrangian and Newton interpolation okay, but the lecture notes covering cubic splines do not go into much detail, so I honestly have no idea what to do. If anyone can provide a bit of help then I'd appreciate it.

Thanks in advance.

 

[HallsofIvy]

To be a spline the to pieces must meet smoothly. That is, you must have f_0(1)= f_1(1), f_0'(1)= f_1'(1), and f_0"(1)= f_1"(1). Those three conditions give you 3 equations to solve for a, b, and c.