What is the error bound in cubic spline approximation?

[Amer]

Find the error bound of approximation of f using the cubic spline

want to find a cubic spline for f on the interval [a,b]
suppose we have n nodes with n-1 different intervals
I tried to find it using the Taylor expansion around any nodes say $$x_i \in [a,b]$$

$$f(x) - S(x) = f(x_i)-S(x_i) + (x-x_i)(f'(x_i)-S'(x_i))+ \frac{(x-x_i)^2(f''(x_i)-S''(x_i))}{2} + \frac{(x-x_i)^3(f'''(x_i)-S'''(x_i))}{3!} + \frac{(x-x_i)^4f^{(4)}(c)}{4!}$$

i considered that the first three terms are zeros so
instead of x i sub $$x_{i+1}$$ since it is most sutiable x in the subinterval and we shall take the bound of the forth derivative of the function
ending with this
$$\mid f(x) - S(x) \mid = \frac{(x_{i+1}-x_i)^4 f^{(4)}(c)}{4!}$$

is it right ?

 

[Sudharaka]

Find the error bound of approximation of f using the cubic spline

want to find a cubic spline for f on the interval [a,b]
suppose we have n nodes with n-1 different intervals
I tried to find it using the Taylor expansion around any nodes say $$x_i \in [a,b]$$

$$f(x) - S(x) = f(x_i)-S(x_i) + (x-x_i)(f'(x_i)-S'(x_i))+ \frac{(x-x_i)^2(f''(x_i)-S''(x_i))}{2} + \frac{(x-x_i)^3(f'''(x_i)-S'''(x_i))}{3!} + \frac{(x-x_i)^4f^{(4)}(c)}{4!}$$

i considered that the first three terms are zeros so
instead of x i sub $$x_{i+1}$$ since it is most sutiable x in the subinterval and we shall take the bound of the forth derivative of the function
ending with this
$$\mid f(x) - S(x) \mid = \frac{(x_{i+1}-x_i)^4 f^{(4)}(c)}{4!}$$

is it right ?

Hi Amer, :)

A more tight error bound for the cubic spline with clamped boundary conditions can be found here.

Kind Regards,
Sudharaka.