Interpolating a function of two variables

Question

I am trying to create the interpolating function for a function of two variables, over a finite area. Just for consistency we can think of a function:

MyFunc[a_,b_]:=Sin[a*b]/Sqrt[1+a^2+b^2]

I've read the documentation and, in order to get the interpolating function I use:

MyTable=Table[MyFunc[a,b],{a, -5, 5, 0.1}, {b, -5, 5, 0.1}]
MyApproximateFunc = ListInterpolation[MyTable]

this seem to work, but when I try to plot MyFunc and MyApproximateFunc they look very different: I must have missed some detail...

Plot3D[MyFunc[a, b], {a, -5, 5}, {b, -5, 5}]
Plot3D[MyApproximateFunc[a, b], {a, -5, 5}, {b, -5, 5}]

Thanks in advance for your kind help!

Answer 1

OK then after Sjoerd C. de Vries request ;-), you can either use (less thinking)

MyTable = Table[{a, b, MyFunc[a, b]}, {a, -5, 5, 0.1}, {b, -5, 5, 0.1}] // Flatten[#, 1] &; 
MyApproximateFunc = Interpolation[MyTable];

or (from @J.M.'s comment, less memory)

 MyTable = Table[MyFunc[a, b], {a, -5, 5, 0.1}, {b, -5, 5, 0.1}] // Flatten[#, 1] &; 
 MyApproximateFunc = ListInterpolation[MyTable, {{-5, 5}, {-5, 5}}];

to get

  GraphicsRow[{Plot3D[MyFunc[a, b], {a, -5, 5}, {b, -5, 5}],
               Plot3D[MyApproximateFunc[a, b], {a, -5, 5}, {b, -5, 5}]}]

Answer 2

Another possibility is FunctionInterpolation. Straight from the Documentation:

   MyApproximateFunc = 
    FunctionInterpolation[
     Evaluate[Table[D[MyFunc[a, b], {{a, b}, k}], {k, 0, 2}]], {a, -5, 
      5}, {b, -5, 5}]

We have to provide derivatives to get a better interpolation.

GraphicsRow[{Plot3D[MyFunc[a, b], {a, -5, 5}, {b, -5, 5}], 
  Plot3D[MyApproximateFunc[a, b], {a, -5, 5}, {b, -5, 5}]}]