I don't fully understand this question about cubic spline interpolation

[Firepanda]

If each spline is given in the form of

g_i(x) = a_i(x-x_i)³ + b_i(x-x_i)² + c_i(x-x_i) + d_i

where i = 1 to N for N+1 data points.

Then given that b₁ and b_N+1 are zero (because the second derivatives are zero at the endpoints, due to this being a natural cubic spline), then there are N-1 equations for b₂...b_N.

Since we then use the value of b_i to work out the unknowns a_i, c_i and d_i then there are four equations per i. Then since b_i and b_N+1 are zero, we can work out a_i, c_i and d_i on the last splines (3 each).

So is the answer 4(N-1) +6?

I don't understand what it means by 'Show there are enough equations..', any ideas?

Thanks

 

[I like Serena]

Hi Firepanda!

If you have N+1 datapoints, that means you have N cubic splines.
Each spline has 4 unknown parameters for a total of 4N parameters.
To solve all parameters you need as many equations as parameters.

So you need exactly 4N equations...