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.

Introduction

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.

We wish to model similar kinds of curves using a set of mathematical equations, with one polynomial $y=q_{i}(x)$ for each pair of knots, $(x_{i-1},y_{i-1})$ and $(x_{i},y_{i})$ , where $i=1,2,\cdots ,n$ . So there will be $n$ polynomials, and $n+1$ knots: The first polynomial starts at $(x_{0},y_{0})$ , and the last polynomial ends at $(x_{n},y_{n})$ .

The curvature of any curve $y=f(x)$ is defined as:

\kappa ={\frac {y''}{(1+y'^{2})^{3/2}}}

To make the spline take a shape that minimizes the bending (under the constraint of passing through all knots), we will define both $y'$ and $y''$ 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

{\begin{cases}q'_{i}(x_{i})=q'_{i+1}(x_{i})\\q''_{i}(x_{i})=q''_{i+1}(x_{i})\end{cases}}\qquad 1\leq i\leq n-1

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

We wish to find each polynomial $q_{i}(x)$ given the points $(x_{0},y_{0})$ through $(x_{n},y_{n})$ . To do this, we will consider just a single piece of the curve, $q(x)$ , which will interpolate from $(x_{1},y_{1})$ to $(x_{2},y_{2})$ . This piece will have slopes $k_{1}$ and $k_{2}$ at its endpoints. Or, more precisely:

q(x_{1})=y_{1}

q(x_{2})=y_{2}

q'(x_{1})=k_{1}

q'(x_{2})=k_{2}

The full equation $q(x)$ can be written in the symmetrical form

q(x)={\big (}1-t(x){\big )}\,y_{1}+t(x)\,y_{2}+t(x){\big (}1-t(x){\big )}{\Big (}{\big (}1-t(x){\big )}\,a+t(x)\,b{\Big )}

1

where

t(x)={\frac {x-x_{1}}{x_{2}-x_{1}}},

2

a=k_{1}(x_{2}-x_{1})-(y_{2}-y_{1}),

3

b=-k_{2}(x_{2}-x_{1})+(y_{2}-y_{1}).

4

But what are $k_{1}$ and $k_{2}$ ? To derive these critical values, we must consider that

q'={\frac {dq}{dx}}={\frac {dq}{dt}}{\frac {dt}{dx}}={\frac {dq}{dt}}{\frac {1}{x_{2}-x_{1}}}

It then follows that

q'={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}+(1-2t){\frac {a(1-t)+bt}{x_{2}-x_{1}}}+t(1-t){\frac {b-a}{x_{2}-x_{1}}},

5

q''=2{\frac {b-2a+(a-b)3t}{{(x_{2}-x_{1})}^{2}}}.

6

Setting $t = 0$ and $t = 1$ respectively in equations (5) and (6) one gets from (2) that indeed first derivatives $q' (x 1) = k 1$ and $q' (x 2) = k 2$ and also second derivatives

q''(x_{1})=2{\frac {b-2a}{{(x_{2}-x_{1})}^{2}}}

7

q''(x_{2})=2{\frac {a-2b}{{(x_{2}-x_{1})}^{2}}}

8

If now $(x i, y i), i = 0, 1, ..., n$ are $n + 1$ points and

q_{i}=(1-t)\,y_{i-1}+t\,y_{i}+t(1-t){\big (}(1-t)\,a_{i}+t\,b_{i}{\big )}

9

where i = 1, 2, ..., n and $t={\tfrac {x-x_{i-1}}{x_{i}-x_{i-1}}}$ are n third degree polynomials interpolating $y$ in the interval $x i -1 \leq x \leq x i$ for i = 1, ..., n such that $q' i (x i) = q' i +1 (x i)$ for i = 1, ..., n−1 then the n polynomials together define a differentiable function in the interval $x 0 \leq x \leq x n$ and

a_{i}=k_{i-1}(x_{i}-x_{i-1})-(y_{i}-y_{i-1})

10

b_{i}=-k_{i}(x_{i}-x_{i-1})+(y_{i}-y_{i-1})

11

for i = 1, ..., n where

k_{0}=q_{1}'(x_{0})

12

k_{i}=q_{i}'(x_{i})=q_{i+1}'(x_{i})\qquad i=1,\dotsc ,n-1

13

k_{n}=q_{n}'(x_{n})

14

If the sequence $k 0, k 1, ..., k n$ is such that, in addition, $q'' i (x i) = q'' i +1 (x i)$ 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

{\frac {k_{i-1}}{x_{i}-x_{i-1}}}+\left({\frac {1}{x_{i}-x_{i-1}}}+{\frac {1}{x_{i+1}-x_{i}}}\right)2k_{i}+{\frac {k_{i+1}}{x_{i+1}-x_{i}}}=3\left({\frac {y_{i}-y_{i-1}}{{(x_{i}-x_{i-1})}^{2}}}+{\frac {y_{i+1}-y_{i}}{{(x_{i+1}-x_{i})}^{2}}}\right)

15

for i = 1, ..., n-1. The relations (15) are $n - 1$ linear equations for the $n + 1$ values $k 0, k 1, ..., k n$ .

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

q''_{1}(x_{0})=2{\frac {3(y_{1}-y_{0})-(k_{1}+2k_{0})(x_{1}-x_{0})}{{(x_{1}-x_{0})}^{2}}}=0,

q''_{n}(x_{n})=-2{\frac {3(y_{n}-y_{n-1})-(2k_{n}+k_{n-1})(x_{n}-x_{n-1})}{{(x_{n}-x_{n-1})}^{2}}}=0,

i.e. that

{\frac {2}{x_{1}-x_{0}}}k_{0}+{\frac {1}{x_{1}-x_{0}}}k_{1}=3{\frac {y_{1}-y_{0}}{(x_{1}-x_{0})^{2}}},

16

{\frac {1}{x_{n}-x_{n-1}}}k_{n-1}+{\frac {2}{x_{n}-x_{n-1}}}k_{n}=3{\frac {y_{n}-y_{n-1}}{(x_{n}-x_{n-1})^{2}}}.

17

Eventually, (15) together with (16) and (17) constitute $n + 1$ linear equations that uniquely define the $n + 1$ parameters $k 0, k 1, ..., k n$ .

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 $x 1$ and $x N -1$ points. For the "not-a-knot" spline, the additional equations will read:

q'''_{1}(x_{1})=q'''_{2}(x_{1})\Rightarrow {\frac {1}{\Delta x_{1}^{2}}}k_{0}+\left({\frac {1}{\Delta x_{1}^{2}}}-{\frac {1}{\Delta x_{2}^{2}}}\right)k_{1}-{\frac {1}{\Delta x_{2}^{2}}}k_{2}=2\left({\frac {\Delta y_{1}}{\Delta x_{1}^{3}}}-{\frac {\Delta y_{2}}{\Delta x_{2}^{3}}}\right)

q'''_{n-1}(x_{n-1})=q'''_{n}(x_{n-1})\Rightarrow {\frac {1}{\Delta x_{n-1}^{2}}}k_{n-2}+\left({\frac {1}{\Delta x_{n-1}^{2}}}-{\frac {1}{\Delta x_{n}^{2}}}\right)k_{n-1}-{\frac {1}{\Delta x_{n}^{2}}}k_{n}=2\left({\frac {\Delta y_{n-1}}{\Delta x_{n-1}^{3}}}-{\frac {\Delta y_{n}}{\Delta x_{n}^{3}}}\right)

where $\Delta x_{i}=x_{i}-x_{i-1},\Delta y_{i}=y_{i}-y_{i-1}$ .

Example

In case of three points the values for $k_{0},k_{1},k_{2}$ are found by solving the tridiagonal linear equation system

{\begin{bmatrix}a_{11}&a_{12}&0\\a_{21}&a_{22}&a_{23}\\0&a_{32}&a_{33}\\\end{bmatrix}}{\begin{bmatrix}k_{0}\\k_{1}\\k_{2}\\\end{bmatrix}}={\begin{bmatrix}b_{1}\\b_{2}\\b_{3}\\\end{bmatrix}}

with

a_{11}={\frac {2}{x_{1}-x_{0}}}

a_{12}={\frac {1}{x_{1}-x_{0}}}

a_{21}={\frac {1}{x_{1}-x_{0}}}

a_{22}=2\ \left({\frac {1}{x_{1}-x_{0}}}+{\frac {1}{x_{2}-x_{1}}}\right)

a_{23}={\frac {1}{x_{2}-x_{1}}}

a_{32}={\frac {1}{x_{2}-x_{1}}}

a_{33}={\frac {2}{x_{2}-x_{1}}}

b_{1}=3\ {\frac {y_{1}-y_{0}}{(x_{1}-x_{0})^{2}}}

b_{2}=3\ \left({\frac {y_{1}-y_{0}}{{(x_{1}-x_{0})}^{2}}}+{\frac {y_{2}-y_{1}}{{(x_{2}-x_{1})}^{2}}}\right)

b_{3}=3\ {\frac {y_{2}-y_{1}}{(x_{2}-x_{1})^{2}}}

For the three points

(-1,0.5)\ ,\ (0,0)\ ,\ (3,3)

,

one gets that

k_{0}=-0.6875\ ,\ k_{1}=-0.1250\ ,\ k_{2}=1.5625

and from (10) and (11) that

a_{1}=k_{0}(x_{1}-x_{0})-(y_{1}-y_{0})=-0.1875

b_{1}=-k_{1}(x_{1}-x_{0})+(y_{1}-y_{0})=-0.3750

a_{2}=k_{1}(x_{2}-x_{1})-(y_{2}-y_{1})=-3.3750

b_{2}=-k_{2}(x_{2}-x_{1})+(y_{2}-y_{1})=-1.6875

In Figure 2, the spline function consisting of the two cubic polynomials $q_{1}(x)$ and $q_{2}(x)$ given by (9) is displayed.

Computer code

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

^ 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. {{cite journal}}: Invalid |ref=harv (help)
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. {{cite journal}}: Invalid |ref=harv (help)

External links

[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.

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