Numerical analysis and 7th degree splines
Response to the following problem:
Given the points x={xo, x1, x2,.... xn}^T and the function values f={fo, f1, f2, ....fn}^T at those points, we want to generate a 7th degree spline, i.e. a piecewise 7th degree polynomial approximation.
a) Why would we want to do this? Why not just use Newton's Interpolatory Divided Difference formula to get an nth degree interpolating polynomial?
b) Write down the interpolant Sk(x) on the subinterval [xk, xk+1]
c) Write down the conditions required for the spline S(x) to fit the data and have 6 continuous derivatives, i.e. S(x) E C^6[x0,xn].
d) How many unknown coefficients, in total, do we have to determine?
e) How many equations do we have (in part c)?
f) Suggest additional boundary conditions, giving enough additional equations to close the system (assume we have no additional information about f).