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# spline

Cubic spline data interpolation

## Syntax

yy = spline(x,Y,xx)
pp = spline(x,Y)

## Description

yy = spline(x,Y,xx) uses a cubic spline interpolation to find yy, the values of the underlying function Y at the values of the interpolant xx. For the interpolation, the independent variable is assumed to be the final dimension of Y with the breakpoints defined by x.

The sizes of xx and yy are related as follows:

• If Y is a scalar or vector, yy has the same size as xx.

• If Y is an array that is not a vector,

• If xx is a scalar or vector, size(yy) equals [d1, d2, ..., dk, length(xx)].

• If xx is an array of size [m1,m2,...,mj], size(yy) equals [d1,d2,...,dk,m1,m2,...,mj].

pp = spline(x,Y) returns the piecewise polynomial form of the cubic spline interpolant for later use with ppval and the spline utility unmkpp. x must be a vector. Y can be a scalar, a vector, or an array of any dimension, subject to the following conditions:

• If x and Y are vectors of the same size, the not-a-knot end conditions are used.

• If x or Y is a scalar, it is expanded to have the same length as the other and the not-a-knot end conditions are used. (See Exceptions (1) below).

• If Y is a vector that contains two more values than x has entries, the first and last value in Y are used as the endslopes for the cubic spline. (See Exceptions (2) below.)

### Exceptions

1. If Y is a vector that contains two more values than x has entries, the first and last value in Y are used as the endslopes for the cubic spline. If Y is a vector, this means

• f(x) = Y(2:end-1)

• df(min(x)) = Y(1)

• df(max(x)) = Y(end)

2. If Y is a matrix or an N-dimensional array with size(Y,N) equal to length(x)+2, the following hold:

• f(x(j)) matches the value Y(:,...,:,j+1) for j=1:length(x)

• Df(min(x)) matches Y(:,:,...:,1)

• Df(max(x)) matches Y(:,:,...:,end)

 Note   You can also perform spline interpolation using the interp1 function with the command interp1(x,y,xx,'spline'). Note that while spline performs interpolation on rows of an input matrix, interp1 performs interpolation on columns of an input matrix.

## Examples

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### Spline Interpolation of Sine Data

This generates a sine curve, then samples the spline over a finer mesh.

```x = 0:10;
y = sin(x);
xx = 0:.25:10;
yy = spline(x,y,xx);
plot(x,y,'o',xx,yy)
```

### Spline Interpolation of Distribution and Specify Endpoint Slopes

This illustrates the use of clamped or complete spline interpolation where end slopes are prescribed. Zero slopes at the ends of an interpolant to the values of a certain distribution are enforced.

```x = -4:4;
y = [0 .15 1.12 2.36 2.36 1.46 .49 .06 0];
cs = spline(x,[0 y 0]);
xx = linspace(-4,4,101);
plot(x,y,'o',xx,ppval(cs,xx),'-');
```

### Extrapolation Using Cubic Spline

The two vectors

```t = 1900:10:1990;
p = [ 75.995  91.972  105.711  123.203  131.669 ...
150.697 179.323  203.212  226.505  249.633 ];
```

represent the census years from 1900 to 1990 and the corresponding United States population in millions of people. The expression

```spline(t,p,2000)
```

uses the cubic spline to extrapolate and predict the population in the year 2000. The result is

```ans =

270.6060

```

### Spline Interpolation of Angular Data

The statements

```x = pi*[0:.5:2];
y = [0  1  0 -1  0  1  0;
1  0  1  0 -1  0  1];
pp = spline(x,y);
yy = ppval(pp, linspace(0,2*pi,101));
plot(yy(1,:),yy(2,:),'-b',y(1,2:5),y(2,2:5),'or'), axis equal
```

generate the plot of a circle, with the five data points y(:,2),...,y(:,6) marked with o's. Note that this y contains two more values (i.e., two more columns) than does x, hence y(:,1) and y(:,end) are used as endslopes.

### Spline Interpolation of Sine and Cosine Data

The following code generates sine and cosine curves, then samples the splines over a finer mesh.

```x = 0:.25:1;
Y = [sin(x); cos(x)];
xx = 0:.1:1;
YY = spline(x,Y,xx);
plot(x,Y(1,:),'o',xx,YY(1,:),'-'); hold on;
plot(x,Y(2,:),'o',xx,YY(2,:),':'); hold off;
```

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### Algorithms

A tridiagonal linear system (with, possibly, several right sides) is being solved for the information needed to describe the coefficients of the various cubic polynomials which make up the interpolating spline. spline uses the functions ppval, mkpp, and unmkpp. These routines form a small suite of functions for working with piecewise polynomials. For access to more advanced features, see the interp1 reference page, the command-line help for these functions, and the Curve Fitting Toolbox™ spline functions.

## References

[1] de Boor, C., A Practical Guide to Splines, Springer-Verlag, 1978.