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Working with functions defined by Interpolation in Mathematica

  1. Oct 13, 2011 #1
    Working with functions defined by "Interpolation" in Mathematica

    Hello,

    Just perhaps a simple question for a Mathematica expert:

    I have a function of two variables f(a,b) defined using Interpolation option
    in Mathematica. I am wondering how to determine the value of one of the
    variable if I know the value of the other variable and the value of the function.
    Many thanks!

    Best regards,
    Evgeniy
     
  2. jcsd
  3. Oct 13, 2011 #2
    Re: Working with functions defined by "Interpolation" in Mathematica

    So you know f[x,y]==z for some y and z
    Then to find x you need to use a numerical method to find when f[x, y] - z == 0.
    Note that, depending on the function, this solution won't necessarily be unique.

    Anyway, here's a test function:

    data = Flatten[Table[{{x, y}, E^(x + y)}, {x, 4}, {y, 4}], 1]

    f = Interpolation[data]

    which you can visualize using

    ContourPlot[f[x, y], {x, 1, 4}, {y, 1, 4}]

    Then if, e.g., y=3, z=140, what does x = ?

    In[]:= FindRoot[f[x, 3] == 140, {x, 2.5}]
    Out[]= {x -> 1.93157}

    Check:
    In[]:= Exp[4.93157]
    Out[]= 138.597

    It's not exactly right, but pretty good considering how few points were used for the interpolation.

    In general you can construct a function that gives the solution using InverseFunction:

    In[26]:= Solve[g[x, y] == z, x]
    Out[26]= {{x -> InverseFunction[g, 1, 2][z, y]}}

    So

    In[28]:= InverseFunction[f, 1, 2][140, 3] // N
    Out[28]= 1.93157

    which matches the FindRoot approach (and probably uses the same or similar algorithm internally).
     
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