Rational interpolation

### 17.1.3 Rational interpolation

##### Thiele interpolation.

The thiele command finds the rational interpolation given a tabulated function.

- thiele takes two or three arguments:
  - data, a matrix with two columns. The first column contains the x coordinates and the second column contains the corresponding y coordinates.

    Instead of a single matrix, data can be given as sequence containing the vector of x coordinates and the vector of y coordinates (in which case the call to thiele has three arguments).

  - v, an identifier, number or symbolic expression (default: x).
- thiele(data,v) returns R(v) where R is the Thiele rational interpolant.
- Note that the obtained interpolant may have singularities in the range [minx,maxx].

#### Examples

thiele([[1,3],[2,4],[4,5],[5,8]],x)

Warning: interpolant has singularities in [4,5]

19 x2−45 x−154
18 x−78

assume(a>1):; f:=unapply(thiele([1,2,a],[3,4,5],t),t)

t↦ |
5 a t−2 a−13 t+10
a t−3 t+2

Indeed:

simplify(apply(f,[1,2,a]))

| ⎡ ⎣ | 3,4,5 | ⎤ ⎦

In the following example, data is obtained by sampling the function f(x)=(1−x4) e1−x3 on [−1,2].

xdata:=linspace(-1.0,2.0,13):; ydata:=apply(x->(1-x^4)*exp(1-x^3),xdata):; p:=thiele(xdata,ydata,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 | ⎞ ⎠

plot(t,x=-1..2); scatterplot(xdata,ydata,color=blue+point_width_2)

##### Rational interpolation without poles.

The ratinterp command computes a family of pole-free rational functions which interpolate given data. Rational interpolation usually gives better results than the classic polynomial interpolation, which may oscillate highly in some cases.

- ratinterp takes one mandatory argument and one or two optional arguments:
  - Matrix data with 2 columns with rows corresponding to points (xk,yk), k=0,1,…,n, or the sequence of lists data_x=[x0,x1,…,xn] and data_y=[y0,y1,…,yn], where a=x0<x1<⋯<xn=b,
  - an identifier x, which may also be a number, symbolic expression or list of numbers a (by default x),
  - Optionally, d, an integer such that 0≤ d≤ n (by default ⌊ (n+1)/2⌋).
- ratinterp(data ⟨,var,d ⟩) or ratinterp(data_x,data_y ⟨,var,d ⟩) returns a rational interpolation r(a) of the given points using the method of Floater and Hormann (2006). If a is a list of numbers a1,a2,…,am, then the list [r(a1)…,r(am)] is returned. There are at most n+1 distinct interpolants which can be specified by varying the parameter d, which allows you to choose the most suitable one.

#### Examples

ratinterp([1,3,5,8],[2,-1,3,4])

− |
29 x3
168
| + |
17 x2
7
| − |
1507 x
168
| + |
61
7

ratinterp([1,3,5,8],[2,-1,3,4],x,2)

− |
29 x3
168
| + |
17 x2
7
| − |
1507 x
168
| + |
61
7

With xdata and ydata as in the third example of thiele above:

plot(ratinterp(xdata,ydata,x,0),x=-1..2,color=blue); plot(ratinterp(xdata,ydata,x,6),x=-1..2,color=red); legend(1.5+3i,"$______ @d = 0$",color=blue); legend(1.5+2.5i,"$______ @d = 6$",color=red);