What is Cubic Spline formula?

Use CubicSpline to plot the cubic spline interpolation of the dataset x = [0, 1, 2] and y = [1, 3, 2] for 0 ≤ x ≤ 2 . To determine the coefficients of each cubic function, we write down the constraints explicitly as a system of linear equations with 4 ( n − 1 ) unknowns.

Are cubic spline simple to compute?

This produces a so-called “natural” cubic spline and leads to a simple tridiagonal system which can be solved easily to give the coefficients of the polynomials. 3 in An Introduction to Splines for Use in Computer Graphics and Geometric Modelling.

What is cubic spline used for?

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 a clamped cubic spline?

The clamped cubic spline gives more accurate approximation to the function f(x), but requires knowledge of the derivative at the endpoints. Condition 1 gives 2N relations. Conditions 2, 3 and 4 each gives N − 1 relations.

How many knots do I need for a cubic spline?

A restricted cubic spline has the additional property that the curve is linear before the first knot and after the last knot. The number of knots used in the spline is determined by the user, but in practice we have found that generally five or fewer knots are sufficient.

How does cubic spline interpolation work?

Cubic spline interpolation is a mathematical method commonly used to construct new points within the boundaries of a set of known points. These new points are function values of an interpolation function (referred to as spline), which itself consists of multiple cubic piecewise polynomials.

How to get clamped cubic spline coefficients in Mathematica?

A similar notebook cspline.nb contains a Mathematica command which produces the clamped cubic spline coefficients for a set of 2D data points and an example of its use. Neither command currently has error checking capabilities and assumes that the list of data points or other input arguments are correct.

What are the end conditions for a cubic spline?

End conditions for the spline, specified as ‘complete’ or ‘clamped’, ‘not-a-knot’, ‘periodic’, ‘second’, ‘variational’, or as a 1-by-2 matrix. The predefined options for conds impose identical end conditions at each end of the data.

How to find a cubic spline interpolant in math?

Task: Find S(x) such that it is a cubic spline interpolant. •The requirement that it is to be a cubic spline gives us 3(n −1) equations. •In addition we require that S(x i) = y i, i = 0,··· ,n which gives n +1 equations. •This means we have 4n −2 equations in total. •We have 4n degrees of freedom (a i,b ic id i) n−1 i=0.

Which is the best description of a spline curve?

A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces. The general approach is that the user enters a sequence of points, and a curve is constructed whose shape closely follows this sequence.