Difference in cubic spline formula

[zzmanzz]

Homework Statement

Hi, this is more of a review question and I'm just looking at solutions of natural cubic spline equations and some will give the cubic spline as:

1. s(x) = a + bx + cx^2 + dx^3

on Wolfram

while other pages will give:

2. s(x) = a + b(x - t) + c(x-t)^2 + d(x-t)^3

where t is the first coordinate in the each point given. I'm not sure which is the standard version and it can make a difference when I evaluate the derivatives so any clarification is appreciated.

 

[fresh_42]

As I didn't know the term cubic spline, I looked it up on Wikipedia and found
$$
c(t)=(2p_0-2p_1+m_0+m_1)t^3 + (-3p_0+3p_1-2m_0-m_1)t^2+m_0t+p_0=(2t^3-3t^2+1)p_0+(-2t^3+3t^2)p_1+(t^3-2t^2+t)m_0+(t^3-t^2)m_1
$$
so the answer seems to be: both, depending on how you would like to interpret and group the coefficients.

 

[RPinPA]

It's all how you want to parametrize the spline. The first form may be more convenient for proofs, defining each in terms of the same $x$. But when you evaluate that, as $x$ gets larger and larger, so do $x^2$ and $x^3$ and you're going to be dealing with small differences between large numbers. It's mathematically correct and gives the right answer theoretically, but might lead to some round-off errors in actually evaluating.

The second form is in terms of the distance $(x - t)$ from the last point. It won't have the problem of numerical stability because $(x - t)$ will never get very large.

In fact I've often used a normalized parameter $(x - x_i)/(x_{i+1}-x_i)$ which ranges from 0 at the beginning of each interval to 1 at the end of the interval. I've found that makes the derivations and resulting equations extremely easy.

Since all of these things are linear in $x$, if expanded and simplified they should all lead to the same polynomials given the same constraints.