Is Cubic Spline continuous?
As we have seen, a straight polynomial interpolation of evenly spaced data tends to build in distortions near the edges of the table. Cubic splines avoid this problem, but they are only piecewise continuous, meaning that a sufficiently high derivative (third) is discontinous.
Why do we use cubic splines?
Cubic spline interpolation is a special case for Spline interpolation that is used very often to avoid the problem of Runge’s phenomenon. This method gives an interpolating polynomial that is smoother and has smaller error than some other interpolating polynomials such as Lagrange polynomial and Newton polynomial.
What is the order of cubic spline?
Abstract: Bickley [5] had suggested the use of cubic splines for the solution of general linear two-point boundary-value problems. It is well known since then that this method gives only order h2 uniformly convergent approximations. But cubic spline interpolation itself is a fourth-order process.
How is the interpolating function for the cubic spline derived?
The interpolating function takes on specified values at the knots. Given the set of values {xi,f(xi)}it is desired to pass a smooth curve through those points, and the curve shall have continuous first and second derivatives at the knots. The formulas for the cubic spline are derived as follows for one region.
Which is the most common type of spline?
Cubic splines are most common. In this case the function is represented by a cubic polynomial within each interval and has continuous first and second derivatives at the knots. Two more conditions can be specified arbitrarily.
How to calculate the spline of a function?
The following code supplies a vector y(x), fits those points to a natural spline [pp = spline(x,y)], evaluates the spline at a set of points xx [v=ppval(pp,xx);], and then plots the spline (in blue) as well as the knots (in red). Since the original function is a cubic function, the spline interpolation is exact. »x = [0 1 2 3 4];»y = [0 1 8 27 64];
Which is the correct method for cubic interpolation?
For the cubic interpolation, we solve the following equation and get . The errors of these three methods are , , and , respectively. Obviously the higher the degree of the interpolating polynomial, the higher the accuracy.