Automatic Piecewise Approximation

Question

I would like to create the following function: It should approximate a function func[x] by piecewise polynomial interpolations, splitting the approximate function in n different polynomials, from xMin to xMax.

autoPiecewiseApproximation[func[x],{xMin,xMax,n}]

Example: the square root function

For instance for two subdivisons n = 2 and xMin = 0, xMax = 1, the function would automatically generate this:

m = Piecewise[{{Fit[
     Table[{x, Sqrt[x]}, {x, 0, 0.5, .01}], {1, x, x^2, E^x}, x] , 
    0 <= x <= 0.5}, {Fit[
     Table[{x, Sqrt[x]}, {x, 0.5, 1, .01}], {1, x, x^2, E^x}, x] , 
    0.5 < x <= 1}}]

Anyone knows how to do this correctly ?

Answer 1

pW[f_, n_, m_, min_: 0, max_: 1] := Piecewise[
  {Fit[Table[{x, f @ x}, {x, ##, (#2 - #)/m}], {1, x, x^2, E^x}, x], # < x <= #2} & @@@
   Partition[Subdivide[min, max, n], 2, 1]] 

Examples:

pW[Sqrt, 4, 10] // TeXForm

$\small\begin{cases} -149.176 x^2-246.803 x+251.787 e^x-251.768 & 0<x\leq \frac{1}{4} \\ -2.882 x^2-1.65139 x+3.18162 e^x-2.99226 & \frac{1}{4}<x\leq \frac{1}{2} \\ -0.876557 x^2+0.489058 x+0.663232 e^x-0.41176 & \frac{1}{2}<x\leq \frac{3}{4} \\ -0.418328 x^2+0.738036 x+0.220335 e^x+0.0813601 & \frac{3}{4}<x\leq 1 \end{cases}$

pW[Sqrt, 5, 50, 5, 15] // TeXForm

$\small\begin{cases} -0.010714 x^2+0.328728 x+\text{9.866783884779016$\grave{ }$*${}^{\wedge}$-6} e^x+0.858866 & 5<x\leq 7 \\ -0.00658734 x^2+0.280252 x+\text{6.475622629784991$\grave{ }$*${}^{\wedge}$-7} e^x+1.00608 & 7<x\leq 9 \\ -0.00455749 x^2+0.248165 x+\text{5.00746734942773$\grave{ }$*${}^{\wedge}$-8} e^x+1.13528 & 9<x\leq 11 \\ -0.00338884 x^2+0.224974 x+\text{4.292376170350944$\grave{ }$*${}^{\wedge}$-9} e^x+1.25171 & 11<x\leq 13 \\ -0.00264528 x^2+0.207225 x+\text{3.949544062291019$\grave{ }$*${}^{\wedge}$-10} e^x+1.3585 & 13<x\leq 15 \end{cases}$