Spline interpolation

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.

IntroductionEdit

 
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.

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 in the figure.

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 splineEdit

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, "natural splines" in addition to the n − 1 linear equations (15) should have

 
 

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", which specifies the slope at the ends of the spline, and the popular "not-a-knot spline", which 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  .

ExampleEdit

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

See alsoEdit

Computer codeEdit

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

ReferencesEdit

  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 linksEdit