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10.2.4  Rational interpolation

The thiele command finds the rational interpolation.

Instead of a single matrix data, two vectors x=(x1,x2,…,xn) and y=(y1,y2,…,yn) This method computes Thiele interpolated continued function based on the concept of reciprocal differences.

It is not guaranteed that R is continuous, i.e. it may have singularities in the shortest segment which contains all components of the x coordinates.

Examples

thiele([[1,3],[2,4],[4,5],[5,8]],x)
     
19 x2−45 x−154
18 x−78
          
thiele([1,2,a],[3,4,5],3)
     
13 a−29
a−7
          

In the following example, data is obtained by sampling the function f(x)=(1−x4e1−x3.

data_x:=[-1,-0.75,-0.5,-0.25,0,0.25,0.5,0.75,1,1.25,1.5,1.75,2]:; data_y:=[0.0,2.83341735599,2.88770329586, 2.75030303645,2.71828182846,2.66568510781, 2.24894558809,1.21863761951,0.0,-0.555711613283, -0.377871362418,-0.107135851128,-0.0136782294833]:; thiele(data_x,data_y,x)
     

1.55286115659 x6+5.87298387514 x5−5.4439152812 x4+1.68655817708 x3         
2.40784868317 x2−7.55954205222 x+9.40462512097
/
         

x6
1.24295718965 x5−1.33526268624 x4+4.03629272425 x3         
0.885419321 x2−2.77913222418 x+3.45976823393
         

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