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In themathematical field ofnumerical analysis,spline interpolation is a form ofinterpolation where the interpolant is a special type ofpiecewisepolynomial called aspline. 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,^{[clarification needed]} instead of fitting a single degree-nine polynomial to all of them. Spline interpolation is often preferred overpolynomial interpolation because theinterpolation error can be made small even when using low-degree polynomials for the spline.^[1] Spline interpolation also avoids the problem ofRunge's phenomenon, in which oscillation can occur between points when interpolating using high-degree polynomials.

Originally,spline was a term for elastic rulers that were bent to pass through a number of predefined points, orknots. These were used to maketechnical drawings forshipbuilding and construction by hand, as illustrated in the figure.

We wish to model similar kinds of curves using a set of mathematical equations. Assume we have a sequence of${\displaystyle n+1}$ knots,${\displaystyle (x_0,y_0)}$ through${\displaystyle (x_n,y_n)}$. There will be a cubic polynomial${\displaystyle q_i(x)=y}$ between each successive pair of knots${\displaystyle (x_{i-1},y_{i-1})}$ and${\displaystyle (x_i,y_i)}$ connecting to both of them, where${\displaystyle i=1,2,\dots ,n}$. So there will be${\displaystyle n}$ polynomials, with the first polynomial starting at${\displaystyle (x_0,y_0)}$, and the last polynomial ending at${\displaystyle (x_n,y_n)}$.

Thecurvature of any curve${\displaystyle y=y(x)}$ is defined as

${\displaystyle \kappa =\frac{y^{″}}{(1+y^{\prime 2}{)}^{3/2}},}$

where${\displaystyle y^{\prime }}$ and${\displaystyle y^{″}}$ are the first and second derivatives of${\displaystyle y(x)}$ with respect to${\displaystyle x}$.To make the spline take a shape that minimizes the bending (under the constraint of passing through all knots), we will define both${\displaystyle y^{\prime }}$ and${\displaystyle y^{″}}$ to be continuous everywhere, including at the knots. Each successive polynomial must have equal values (which are equal to the y-value of the corresponding datapoint), derivatives, and second derivatives at their joining knots, which is to say that

${\displaystyle \{\begin{array}{l}{q}_i(x_i)=q_{i+1}(x_i)=y_i\\ q_i^{\prime }(x_i)=q_{i+1}^{\prime }(x_i)\\ q_i^{″}(x_i)=q_{i+1}^{″}(x_i)\end{array}1\le i\le n-1.}$

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

In addition to the three conditions above, anatural cubic spline has the condition that${\displaystyle q_1^{″}(x_0)=q_n^{″}(x_n)=0}$.

In addition to the three main conditions above, aclamped cubic spline has the conditions that${\displaystyle q_1^{\prime }(x_0)=f^{\prime }(x_0)}$ and${\displaystyle q_n^{\prime }(x_n)=f^{\prime }(x_n)}$ where${\displaystyle f^{\prime }(x)}$ is the derivative of the interpolated function.

In addition to the three main conditions above, anot-a-knot spline has the conditions that${\displaystyle q_1^{‴}(x_1)=q_2^{‴}(x_1)}$ and${\displaystyle q_{n-1}^{‴}(x_{n-1})=q_n^{‴}(x_{n-1})}$.^[2]

We wish to find each polynomial${\displaystyle q_i(x)}$ given the points${\displaystyle (x_0,y_0)}$ through${\displaystyle (x_n,y_n)}$. To do this, we will consider just a single piece of the curve,${\displaystyle q(x)}$, which will interpolate from${\displaystyle (x_1,y_1)}$ to${\displaystyle (x_2,y_2)}$. This piece will have slopes${\displaystyle k_1}$ and${\displaystyle k_2}$ at its endpoints. Or, more precisely,

${\displaystyle q(x_1)=y_1,}$

${\displaystyle q(x_2)=y_2,}$

${\displaystyle q^{\prime }(x_1)=k_1,}$

${\displaystyle q^{\prime }(x_2)=k_2.}$

The full equation${\displaystyle q(x)}$ can be written in the symmetrical form

${\displaystyle q(x)=(1-t(x))y_1+t(x)y_2+t(x)(1-t(x))((1-t(x))a+t(x)b),}$

1

where

${\displaystyle t(x)=\frac{x-x_1}{x_2-x_1},}$

2

${\displaystyle a=k_1(x_2-x_1)-(y_2-y_1),}$

3

${\displaystyle b=-k_2(x_2-x_1)+(y_2-y_1).}$

4

But what are${\displaystyle k_1}$ and${\displaystyle k_2}$? To derive these critical values, we must consider that

${\displaystyle q^{\prime }=\frac{dq}{dx}=\frac{dq}{dt}\frac{dt}{dx}=\frac{dq}{dt}\frac{1}{x_2-x_1}.}$

It then follows that

${\displaystyle q^{\prime }=\frac{y_2-y_1}{x_2-x_1}+(1-2t)\frac{a(1-t)+bt}{x_2-x_1}+t(1-t)\frac{b-a}{x_2-x_1},}$

5

${\displaystyle q^{″}=2\frac{b-2a+(a-b)3t}{{(x_2-x_1)}^2}.}$

6

Settingt =0 andt =1 respectively in equations (5) and (6), one gets from (2) that indeed first derivativesq′(x₁) =k₁ andq′(x₂) =k₂, and also second derivatives

${\displaystyle q^{″}(x_1)=2\frac{b-2a}{{(x_2-x_1)}^2},}$

7

${\displaystyle q^{″}(x_2)=2\frac{a-2b}{{(x_2-x_1)}^2}.}$

8

If now(x_i,y_i),i = 0, 1, ...,n aren + 1 points, and

${\displaystyle q_i=(1-t)y_{i-1}+t{y}_i+t(1-t)((1-t)a_i+t{b}_i),}$

9

wherei = 1, 2, ...,n, and${\displaystyle t=\frac{x-x_{i-1}}{x_i-x_{i-1}}}$ aren third-degree polynomials interpolatingy in the intervalx_i−1 ≤x ≤x_i fori = 1, ...,n such thatq′_i (x_i) =q′_i+1(x_i) fori = 1, ...,n − 1, then then polynomials together define adifferentiable function in the intervalx₀ ≤x ≤x_n, and

${\displaystyle a_i=k_{i-1}(x_i-x_{i-1})-(y_i-y_{i-1}),}$

10

${\displaystyle b_i=-k_i(x_i-x_{i-1})+(y_i-y_{i-1})}$

11

fori = 1, ...,n, where

${\displaystyle k_0=q_1^{\prime }(x_0),}$

12

${\displaystyle k_i=q_i^{\prime }(x_i)=q_{i+1}^{\prime }(x_i),i=1,\dots ,n-1,}$

13

${\displaystyle k_n=q_n^{\prime }(x_n).}$

14

If the sequencek₀,k₁, ...,k_n is such that, in addition,q′′_i(x_i) =q′′_i+1(x_i) holds fori = 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

${\displaystyle \frac{k_{i-1}}{x_i-x_{i-1}}+\left(\frac{1}{x_i-x_{i-1}}+\frac{1}{x_{i+1}-x_i}\right)2k_i+\frac{k_{i+1}}{x_{i+1}-x_i}=3\left(\frac{y_i-y_{i-1}}{{(x_i-x_{i-1})}^2}+\frac{y_{i+1}-y_i}{{(x_{i+1}-x_i)}^2}\right)}$

15

fori = 1, ...,n − 1. The relations (15) aren − 1 linear equations for then + 1 valuesk₀,k₁, ...,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 withq′′ = 0. Asq′′ should be acontinuous function ofx, "natural splines" in addition to then − 1 linear equations (15) should have

${\displaystyle q_1^{″}(x_0)=2\frac{3(y_1-y_0)-(k_1+2k_0)(x_1-x_0)}{{(x_1-x_0)}^2}=0,}$

${\displaystyle q_n^{″}(x_n)=-2\frac{3(y_n-y_{n-1})-(2k_n+k_{n-1})(x_n-x_{n-1})}{{(x_n-x_{n-1})}^2}=0,}$

i.e. that

${\displaystyle \frac{2}{x_1-x_0}{k}_0+\frac{1}{x_1-x_0}{k}_1=3\frac{y_1-y_0}{(x_1-x_0{)}^2},}$

16

${\displaystyle \frac{1}{x_n-x_{n-1}}{k}_{n-1}+\frac{2}{x_n-x_{n-1}}{k}_n=3\frac{y_n-y_{n-1}}{(x_n-x_{n-1}{)}^2}.}$

17

Eventually, (15) together with (16) and (17) constituten + 1 linear equations that uniquely define then + 1 parametersk₀,k₁, ...,k_n.

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 thethird derivative is also continuous at thex₁ andx_n−1 points.For the "not-a-knot" spline, the additional equations will read:

${\displaystyle q_1^{‴}(x_1)=q_2^{‴}(x_1)\Rightarrow \frac{1}{\mathrm{\Delta }{x}_1^2}{k}_0+\left(\frac{1}{\mathrm{\Delta }{x}_1^2}-\frac{1}{\mathrm{\Delta }{x}_2^2}\right)k_1-\frac{1}{\mathrm{\Delta }{x}_2^2}{k}_2=2\left(\frac{\mathrm{\Delta }{y}_1}{\mathrm{\Delta }{x}_1^3}-\frac{\mathrm{\Delta }{y}_2}{\mathrm{\Delta }{x}_2^3}\right),}$

${\displaystyle q_{n-1}^{‴}(x_{n-1})=q_n^{‴}(x_{n-1})\Rightarrow \frac{1}{\mathrm{\Delta }{x}_{n-1}^2}{k}_{n-2}+\left(\frac{1}{\mathrm{\Delta }{x}_{n-1}^2}-\frac{1}{\mathrm{\Delta }{x}_n^2}\right)k_{n-1}-\frac{1}{\mathrm{\Delta }{x}_n^2}{k}_n=2\left(\frac{\mathrm{\Delta }{y}_{n-1}}{\mathrm{\Delta }{x}_{n-1}^3}-\frac{\mathrm{\Delta }{y}_n}{\mathrm{\Delta }{x}_n^3}\right),}$

where${\displaystyle \mathrm{\Delta }{x}_i=x_i-x_{i-1},\mathrm{\Delta }{y}_i=y_i-y_{i-1}}$.

In case of three points the values for${\displaystyle k_0,k_1,k_2}$ are found by solving thetridiagonal linear equation system

with

${\displaystyle a_{11}=\frac{2}{x_1-x_0},}$

${\displaystyle a_{12}=\frac{1}{x_1-x_0},}$

${\displaystyle a_{21}=\frac{1}{x_1-x_0},}$

${\displaystyle a_{22}=2\left(\frac{1}{x_1-x_0}+\frac{1}{x_2-x_1}\right),}$

${\displaystyle a_{23}=\frac{1}{x_2-x_1},}$

${\displaystyle a_{32}=\frac{1}{x_2-x_1},}$

${\displaystyle a_{33}=\frac{2}{x_2-x_1},}$

${\displaystyle b_1=3\frac{y_1-y_0}{(x_1-x_0{)}^2},}$

${\displaystyle b_2=3\left(\frac{y_1-y_0}{{(x_1-x_0)}^2}+\frac{y_2-y_1}{{(x_2-x_1)}^2}\right),}$

${\displaystyle b_3=3\frac{y_2-y_1}{(x_2-x_1{)}^2}.}$

For the three points

${\displaystyle (-1,0.5),(0,0),(3,3),}$

one gets that

${\displaystyle k_0=-0.6875,k_1=-0.1250,k_2=1.5625,}$

and from (10) and (11) that

${\displaystyle a_1=k_0(x_1-x_0)-(y_1-y_0)=-0.1875,}$

${\displaystyle b_1=-k_1(x_1-x_0)+(y_1-y_0)=-0.3750,}$

${\displaystyle a_2=k_1(x_2-x_1)-(y_2-y_1)=-3.3750,}$

${\displaystyle b_2=-k_2(x_2-x_1)+(y_2-y_1)=-\mathrm{1.6875.}}$

In the figure, the spline function consisting of the two cubic polynomials${\displaystyle q_1(x)}$ and${\displaystyle q_2(x)}$ given by (9) is displayed.

• Akima spline
• Circular interpolation
• Cubic Hermite spline
• Centripetal Catmull–Rom spline
• Discrete spline interpolation
• Monotone cubic interpolation
• Non-uniform rational B-spline
• Multivariate interpolation
• Polynomial interpolation
• Smoothing spline
• Spline wavelet
• Thin plate spline
• Polyharmonic spline

• 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.

• Cubic Spline Interpolation Online Calculation and Visualization Tool (with JavaScript source code)
• "Spline interpolation",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Dynamic cubic splines with JSXGraph
• Lectures on the theory and practice of spline interpolation
• Paper which explains step by step how cubic spline interpolation is done, but only for equidistant knots.
• Numerical Recipes in C, Go to Chapter 3 Section 3-3
• A note on cubic splines
• Information about spline interpolation (including code in Fortran 77)
• 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

• Splines (mathematics)
• Interpolation

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