Spline interpolation

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In the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. That is, instead of fitting a single, high-degree polynomial to all of the values at once, spline interpolation fits low-degree polynomials to small subsets of the values: For example, fitting nine cubic polynomials between each of the pairs of ten points, instead of fitting a single degree-ten polynomial to all of them. Spline interpolation is often preferred over polynomial interpolation because the interpolation error can be made small even when using low degree polynomials for the spline.[1] Spline interpolation also avoids the problem of Runge's phenomenon, in which oscillation can occur between points when interpolating using high degree polynomials.

Introduction[edit]

Originally, spline was a term for elastic rulers that were bent to pass through a number of predefined points, or knots. These were used to make technical drawings for shipbuilding and construction by hand, as illustrated by Figure 1.

Figure 1: Interpolation with cubic splines between eight points. Hand-drawn technical drawings for shipbuilding are a historical example of spline interpolation; drawings were constructed using flexible rulers that were bent to follow pre-defined points.

We wish to model similar kinds of curves using a set of mathematical equations, with one polynomial for each pair of knots, and , where . So there will be polynomials, and knots: The first polynomial starts at , and the last polynomial ends at .

The curvature of any curve is defined as:

To make the spline take a shape that minimizes the bending (under the constraint of passing through all knots), we will define both and to be continuous everywhere, including at the knots. Thus the first and second derivatives of each successive polynomial must have identical values at the knots, which is to say that

This can only be achieved if polynomials of degree 3 or higher — cubic polynomials or higher — are used. The classical approach is to use polynomials of exactly degree 3 — cubic splines.

Algorithm to find the interpolating cubic spline[edit]

We wish to find each polynomial given the points through . To do this, we will consider just a single piece of the curve, , which will interpolate from to . This piece will have slopes and at its endpoints. Or, more precisely:

The full equation can be written in the symmetrical form

 

 

 

 

(1)

where

 

 

 

 

(2)

 

 

 

 

(3)

 

 

 

 

(4)

But what are and ? To derive these critical values, we must consider that

It then follows that

 

 

 

 

(5)

 

 

 

 

(6)

Setting t = 0 and t = 1 respectively in equations (5) and (6) one gets from (2) that indeed first derivatives q′(x1) = k1 and q′(x2) = k2 and also second derivatives

 

 

 

 

(7)

 

 

 

 

(8)

If now (xi, yi), i = 0, 1, ..., n are n + 1 points and

 

 

 

 

(9)

where i = 1, 2, ..., n and are n third degree polynomials interpolating y in the interval xi−1xxi for i = 1, ..., n such that q′i (xi) = q′i+1(xi) for i = 1, ..., n−1 then the n polynomials together define a differentiable function in the interval x0xxn and

 

 

 

 

(10)

 

 

 

 

(11)

for i = 1, ..., n where

 

 

 

 

(12)

 

 

 

 

(13)

 

 

 

 

(14)

If the sequence k0, k1, ..., kn is such that, in addition, q′′i(xi) = q′′i+1(xi) holds for i = 1, ..., n-1, then the resulting function will even have a continuous second derivative.

From (7), (8), (10) and (11) follows that this is the case if and only if

 

 

 

 

(15)

for i = 1, ..., n-1. The relations (15) are n − 1 linear equations for the n + 1 values k0, k1, ..., kn.

For the elastic rulers being the model for the spline interpolation one has that to the left of the left-most "knot" and to the right of the right-most "knot" the ruler can move freely and will therefore take the form of a straight line with q′′ = 0. As q′′ should be a continuous function of x one gets that for "Natural Splines" one in addition to the n − 1 linear equations (15) should have that

i.e. that

 

 

 

 

(16)

 

 

 

 

(17)

Eventually, (15) together with (16) and (17) constitute n + 1 linear equations that uniquely define the n + 1 parameters k0, k1, ..., kn.

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:

where .

Example[edit]

Figure 2: Interpolation with cubic "natural" splines between three points.

In case of three points the values for are found by solving the tridiagonal linear equation system

with

For the three points

,

one gets that

and from (10) and (11) that

In Figure 2, the spline function consisting of the two cubic polynomials and given by (9) is displayed.

See also[edit]

Computer code[edit]

TinySpline: Open source C-library for splines which implements cubic spline interpolation

SciPy Spline Interpolation: a Python package that implements interpolation

Cubic Interpolation: Open source C#-library for cubic spline interpolation

References[edit]

  1. ^ Hall, Charles A.; Meyer, Weston W. (1976). "Optimal Error Bounds for Cubic Spline Interpolation". Journal of Approximation Theory. 16 (2): 105–122. doi:10.1016/0021-9045(76)90040-X.

External links[edit]