How to create InterpolatingFunction from data?

Question

If I have a function at a number of non-uniformly spaced data points:

youtReal[7.8]=2.0332;, youtReal[8.54]=6.24352354;..................

......... and the imaginary parts:

youtImag[7.8]=2.54345345; youtImag[8.54]=-5.2434;.................

Is there an easy way to generate an InterpolatingFunction from this data consisting of complex values i.e. so I have

yout[7.8]=2.0332+2.54345345I;

and can also just get an interpolated value for say yout[8.12] and so on.

To make this more concrete, the initial data is read in from a text file that looks like

youtReal[39.983881559265192985] =0.003175121157662356963;
 youtImag[39.983881559265192985] =-0.025570345562818877981;
 youtReal[39.945803445544929806] =0.0030816430773355882379;
 youtImag[39.945803445544929806] =-0.025607948113924885628;
 youtReal[39.907725331824666628] =0.0029879382430246325535;
 youtImag[39.907725331824666628] =-0.025645261479980185081;
 youtReal[39.869647218104403449] =0.0028940073645911109315;
 youtImag[39.869647218104403449] =-0.025682284691974913792;
  youtReal[39.831569104384140271] =0.0027998511548903118804;
  youtImag[39.831569104384140271] =-0.025719016782491712295;
  youtReal[39.793490990663877092] =0.0027054703297639490629;
 youtImag[39.793490990663877092] =-0.025755456785713930289;
 youtReal[39.755412876943613914] =0.0026108656080328711776;
 youtImag[39.755412876943613914] =-0.025791603737433792661;
youtReal[39.717334763223350736] =0.00251603771148972405;
 youtImag[39.717334763223350736] =-0.025827456675060525123;
 .
 .
 .

and so on all the way down to around youtReal[~29]. So I read this in then have a function with quite arbitrarily defined value points, and lots and lots of them. I don't have a nice list of points I can just feed into ListInterpolation ab initio, I need to somehow generate this.

Answer 1

You can use DownValues to get a list of function definitions:

real = ReleaseHold[List @@@ DownValues[youtReal] /. youtReal[t_] -> t];
imag = ReleaseHold[List @@@ DownValues[youtImag] /. youtImag[t_] -> t];

Then assuming you get the real and imaginary parts for a particular point defined in order, you can now generate your list of points:

Thread[{real, imag}] /. {{t_, a_}, {t_, b_}} -> {t, a + b I}

Answer 2

InterpolatingFunction works fine with complex data. For instance, modifying the first example in the help file to make it complex-valued and unevenly spaced

 points = {{0, 0 + I}, {2, 3 + 2 I}, {3, 4}, {4, 3 - 2 I}, {5, 0}};
 ifun = Interpolation[points]

allows you to find interpolated values at points such as

 ifun[2.3]