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10.2.3  Natural interpolation

The spline command finds the natural spline.

Examples

Find the natural spline of degree 3, crossing through the points x0=0,y0=1, x1=1,y1=3 and x2=2, y2=0.

spline([0,1,2],[1,3,0],x,3)
     



5
4
 x3+
13
4
 x+1,
5
4
 
x−1
3
15
4
 
x−1
2
x−1
2
+3


          

Where the first polynomial, −5/4 x3+13/4 x+1, is defined on the interval [0,1] (the first interval defined by the list [0,1,2]) and the second polynomial 5/4 (x−1)3−15/4 (x−1)2x−1/2+3 is defined on the interval [1,2], the second interval defined by the list [0,1,2].

Find the natural spline of degree 4, crossing through the points x0=0,y0=1, x1=1,y1=3, x2=2, y2=0 and x3=3, y3=−1.

spline([0,1,2,3],[1,3,0,-1],x,4)
     




62
121
 x4+
304
121
 x+1,
         
 
201
121

x−1
4
248
121

x−1
3
372
121

x−1
2+
56
121

x−1
+3,
         
139
121
 
x−2
4+
556
121
 
x−2
3+
90
121
 
x−2
2
628
121
 
x−2




         

Output is a list of three polynomial functions of x, defined respectively on the intervals [0,1], [1,2] and [2,3].

Find the natural spline interpolation of cos on [0,π/2,3π/2].

spline([0,pi/2,3*pi/2],cos([0,pi/2,3*pi/2]),x,3)
     




x3
3 π 3
x
3 π 
+1,
         



x
π 
2



3



 
3 π 3
+
2


x
π 
2



2



 
π 2



x
π
2



3 π 




         

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