We have seen how it is possible to smoothly interpolate between a set of points ( , with the

We have also seen how extrapolating such polynomials beyond the first and last nodes can yield less than satisfactory results, which we fixed by specifying the first and last gradients and then adding new first and last nodes to ensure that the first and last polynomials would represent straight lines.

Now we shall see how cubic spline interpolation can break down rather more dramatically and how we might fix it.

*x*,

_{i}*y*)

_{i}*x*known as nodes and the_{i}*y*as values, by specifying the gradients_{i}*g*at the nodes and calculating values between adjacent pairs using the uniquely defined cubic polynomials that match the values and gradients at them._{i}We have also seen how extrapolating such polynomials beyond the first and last nodes can yield less than satisfactory results, which we fixed by specifying the first and last gradients and then adding new first and last nodes to ensure that the first and last polynomials would represent straight lines.

Now we shall see how cubic spline interpolation can break down rather more dramatically and how we might fix it.

Full text...