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. 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 avoids the problem of Runge's phenomenon, in which oscillation can occur between points when interpolating using high degree polynomials.

IntroductionEdit

Originally, spline was a term for elastic rulers that were bent to pass through a number of predefined points ("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 were made for shipbuilding etc. using flexible rulers that were bent to follow pre-defined points

The approach to mathematically model the shape of such elastic rulers fixed by n + 1 knots   is to interpolate between all the pairs of knots   and   with polynomials  .

The curvature of a curve   is given by:

 

As the spline will take a shape that minimizes the bending (under the constraint of passing through all knots) both   and   will be continuous everywhere and at the knots. To achieve this one must have that

 

This can only be achieved if polynomials of degree 5 or higher are used. The classical approach is to use polynomials of degree 3, called cubic splines, which can achieve the continuity of the first derivative, but not that of second derivative.

Algorithm to find the interpolating cubic splineEdit

A third-order polynomial   for which

 
 
 
 

can be written in the symmetrical form

 

 

 

 

 

(1)

where

 

 

 

 

 

(2)

 

 

 

 

 

(3)

 

 

 

 

 

(4)

As

 

one gets that:

 

 

 

 

 

(5)

 

 

 

 

 

(6)

Setting x = x1 and x = x2 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  .

ExampleEdit

 
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 alsoEdit

Computer codeEdit

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.
  • Schoenberg, Isaac J. (1946). "Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions: Part A.—On the Problem of Smoothing or Graduation. A First Class of Analytic Approximation Formulae". Quarterly of Applied Mathematics. 4 (2): 45–99. doi:10.1090/qam/15914.
  • Schoenberg, Isaac J. (1946). "Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions: Part B.—On the Problem of Osculatory Interpolation. A Second Class of Analytic Approximation Formulae". Quarterly of Applied Mathematics. 4 (2): 112–141. doi:10.1090/qam/16705.

External linksEdit