How to fit complex valued data?

• To: mathgroup at smc.vnet.net
• Subject: [mg64096] How to fit complex valued data?
• From: James McCambridge <James.McCambridge at usa.dupont.com>
• Date: Wed, 1 Feb 2006 04:34:29 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Esteemed Colleagues,

I would like to fit a complex valued data set with FindFit (I'm willing to
try other methods too, but I thought I'd start out with the basics). Is
this possible?

For example, I am interested in fitting the complex permittivity of liquid
polymers vs frequency, which has a functional form:

In[1]:=    expr[b_,x_]:=(1+I b x)^-1;

Using this form to generate data, with b = 1.0, I get:

In[2]:=    data ={{1,0.5 -0.5 I},{2.,0.2 -0.4 I},{3.,0.1 -0.3 I
},{4.,0.0588235 -0.235294 I},{5.,0.0384615 -0.192308 I},{6.,0.027027
-0.162162 I},{7.,0.02 -0.14 I},{8.,0.0153846 -0.123077 I},{9.,0.0121951
-0.109756 I},{10.,0.00990099 -0.0990099 I}};

Using FindFit, I come up against an error message.

In[3]:=   FindFit[data,expr[b,y],{{b,1.}}, y]

has the output

Out[3]:=   FindFit::nrlnum: The function value {0.+0. I, 0.+0. I, -1.38778
10^-17+0. I, <<4>>, 0.+0. I, 0.+0. I, 0.+0. I} is not a list of real
numbers with dimensions {10} at {b} = {0.}

The form expr[0.,x] doesn't look poorly behaved, so what gives?

I could separately fit the real and imaginary components, but this often
gives two somewhat different sets of parameters; I would like to obtain
the parameters which optimize the fit to BOTH the real and imaginary
parts.

Jim McCambridge

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