Fitting data using the roots of two coupled transcendental equations

Question

I am trying to use FindRoot to solve two simultaneous transcendental equations for a variety of temperatures and then use the resulting curves to fit a data set. The actual formulas I am using are somewhat complicated, so I worked up a simpler example but am still having trouble getting even that to work.

I generated a set of noisy test data:

 tdata = Table[{x, 
    FindRoot[{a == 2 + x + 1, b == 3*1^2}, {a, 1}, {b, 1}][[1, 2]] + 
     RandomReal[{-0.1, 0.1}]}, {x, 0, 10, 0.5}];

And am trying to simply get FindFit to reproduce the data:

    FindFit[tdata, 
 FindRoot[{a == 2 + x + q, b == 3 q^2}, {{a, 1}, {b, 1}}][[1, 
  2]], {{q, 1}}, x]

Here, I replaced the "1" in tdata with a parameter "q" and I'm trying to extract that with FindFit. x is the independent variable. The other code I am writing does essentially the same thing, but there are more constants in the equation. When I run it, I get the following error:

FindRoot::nlnum: The function value {-1.-1. q-1. x,1.-3. q^2} is not \
a list of numbers with dimensions {2} at {a,b} = {1.,1.}.
FindFit::nrlnum: The function value \
{-2.93229+(b==3.),-3.49817+(b==3.),-3.9952+(b==3.),-4.42837+(b==3.),<<\
3>>,-6.4968+(b==3.),-7.08838+(b==3.),-7.4854+(b==3.),<<11>>} is not a \
list of real numbers with dimensions {21} at {q} = {1.}.

What's interesting is that I CAN get this to work with only one transcendental equation. (Well, it also throws a similar error message but actually returns a number that is off from the correct value by about 0.01):

FindFit[tdata, FindRoot[a == 2 + x + q, {a, 1}][[1, 2]], {{q, 1}},
  x]

I would be grateful for any help anyone can give me.

Answer 1

FindFit does only work with numeric arguments of the fitted function.

tdata=Table[{x,FindRoot[{a==2+x+1,b==3*1^2},{a,1},{b,1}][[1,2]]+RandomReal[{-0.5,0.5}]},{x,0,10,0.5}];
f[q_?NumericQ,x_?NumericQ]:=FindRoot[{a==2+x+q,b==3 q^2},{{a,1},{b,1}}][[1,2]]
nlmf=NonlinearModelFit[tdata,f[q,x],{{q,1}},x];
nlmf["ParameterTable"]
Show[ListPlot@tdata,Plot[nlmf[x],{x,0,10},PlotStyle->Red]]