What are the end conditions for Cubic spline?

A class of end conditions is derived for cubic spline interpolation at equally spaced knots. These conditions are in terms of function values at the knots and give rise to O(h*) spline approximations.

What is not a knot spline?

Not-a-knot Spline Without specifying any extra conditions at the end points (other than that the spline interpolates the data points there), the not-a-knot spline requires that the third derivative of the spline is continuous at x1 and xN 1.

What are the conditions used in clamped cubic spline interpolation?

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.

What are the advantages of cubic spline fitting?

Cubic spline is used as the method of interpolation because of the advantages it provides in terms of simplicity of calculation, numerical stability and smoothness of the interpolated curve.

What is natural cubic spline?

‘Natural Cubic Spline’ — is a piece-wise cubic polynomial that is twice continuously differentiable. In mathematical language, this means that the second derivative of the spline at end points are zero.

What is a cubic spline model?

A cubic spline is a piecewise cubic function that interpolates a set of data points and guarantees smoothness at the data points. Cubic spline interpolants are continuous in the zeroth through second derivatives and pass through all the data points.

How many points is a cubic spline?

A special type of spline is the Bézier curve. This is a cubic function defined by four points. The two end points are used, together with two ‘control’ points. The slope of the curve at one end is a tangent to the line between that end point and one of the control points.

Is a cubic spline continuous?

Cubic splines are popular because they are easy to implement and produce a curve that appears to be seamless. Cubic splines avoid this problem, but they are only piecewise continuous, meaning that a sufficiently high derivative (third) is discontinous.

How do you find a natural cubic spline?

it is a natural cubic spline is simply expressed as z0 = zn = 0. S (x) is a linear spline which interpolates (ti ,zi ). interpolant S (x), and then integrate that twice to obtain S(x). Si (x) = zi x − ti+1 ti − ti+1 + zi+1 x − ti ti+1 − ti .

What are the conditions for a not a knot spline?

Not-a-knot Spline Without specifying any extra conditions at the end points (other than that the spline interpolates the data points there), the not-a-knot spline requires that the third derivative of the spline is continuous at x 1 and x N

How to interpolate cubic spline with end conditions?

The function applies Lagrange end conditions to each end of the data, and matches the spline endslopes to the slope of the cubic polynomial that fits the last four data points at each end. Data values at the same site are averaged. pp = csape (x,y,conds) uses the end conditions specified by conds.

Are there other end conditions for the spline?

There exist other end conditions: “Clamped spline”, that specifies the slope at the ends of the spline, and the popular “not-a-knot spline”, that requires that the third derivative is also continuous at the x1 and xN−1 points. For the “not-a-knot” spline, the additional equations will read:

Is the Clamped spline constructed to have zero slope?

The clamped spline is constructed to have zero slope at the end points. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Natural Spline Clamped Spline Not−a−knot Spline Data Points 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 Natural Spline Clamped Spline Not−a−knot Spline Data Points