How to solve this FindFit::nrlnum:

Question

the following code show error message

FindFit::nrlnum: "The function value {6.71759 -9.36268*10^-8\ I,6.51797 -8.42641*10^-8\ I,6.31789 -7.49014*10^-8\ I,6.1179 -6.55387*10^-8\ I,5.91794 -5.61761*10^-8\ I,5.71797 -4.68134*10^-8\ I,<<5>>,4.53457 +0.\ I,4.35846 +0.\ I,4.21276 +0.\ I,4.12436 +0.\ I,4.12268 +0.\ I}\n is not a list of real numbers with dimensions {16} at {a,b} = {1.,1.}. "

FindFit[{{-1.9999999999999993`, -8.71758682396597`}, \
    {-1.7999999999999994`, -8.317969980679658`}, {-1.5999999999999996`, \
    -7.9178913932701125`}, {-1.3999999999999995`, -7.517895681098703`}, \
    {-1.1999999999999997`, -7.117935726160363`}, {-0.9999999999999997`, \
    -6.717970679612214`}, {-0.7999999999999997`, -6.318075235251766`}, \
    {-0.5999999999999999`, -5.918333526484718`}, {-0.3999999999999999`, \
    -5.518978976322539`}, {-0.1999999999999999`, -5.120597037968602`}, \
    {0.`, -4.724630562634576`}, {0.1999999999999997`, \
    -4.334574650968856`}, {0.3999999999999999`, -3.9584626158001357`}, \
    {0.6000000000000001`, -3.6127563661613884`}, {0.7999999999999998`, \
    -3.324357131929245`}, {0.9999999999999994`, -3.1226771789912595`}}, 
     a x^b, {a, b}, x]

I tried the method here, but all didn't work.

Answer 1

data = {{-1.9999999999999993`, -8.71758682396597`}, \
{-1.7999999999999994`, -8.317969980679658`}, {-1.5999999999999996`, \
-7.9178913932701125`}, {-1.3999999999999995`, -7.517895681098703`}, \
{-1.1999999999999997`, -7.117935726160363`}, {-0.9999999999999997`, \
-6.717970679612214`}, {-0.7999999999999997`, -6.318075235251766`}, \
{-0.5999999999999999`, -5.918333526484718`}, {0.`, \
-4.724630562634576`}, {-0.3999999999999999`, -5.518978976322539`}, \
{-0.1999999999999999`, -5.120597037968602`}, {0.1999999999999997`, \
-4.334574650968856`}, {0.3999999999999999`, -3.9584626158001357`}, \
{0.6000000000000001`, -3.6127563661613884`}, {0.7999999999999998`, \
-3.324357131929245`}, {0.9999999999999994`, -3.1226771789912595`}};
model = a x^b;
f = FindFit[data, model, {a, b}, x,NormFunction -> (Norm[#, Infinity] &)]

(* {a -> 1.90367964288013, b -> 1.10590831050973} *)

Use different Norm function. Gradient -> "FiniteDifference" is not really needed, I just left it there, since help says to use it when getting singularity. But it will work without it here if you use infinity norm.

But the fit does not look good. There is a constant offset, and does not generate go to the negative x as the data does:

 Show[ListPlot[data], Plot[model /. f, {x, -1, 2}], PlotRange -> All]

Answer 2

The negative values for the model chosen pose the problem. There are a number of approaches noting the data: 1. a linear model (this give quite a reasonable fit) 2. Using non linear model fit with your model but transforming your data 3. Perhaps, the simplest approach and using your desired FindFit is to transform your data and back transform.

ft = FindFit[# + {2, 9} & /@ data, a x^b, {a, b}, x]

where data is your dataset.

This yields:

{a -> 2.33187, b -> 0.862647}

You can visually assess the fit (the other properties allow diagnostics, R^2 etc):

Show[ListPlot[data], 
 Plot[Evaluate[(a (x + 2)^b - 9) /. ft], {x, -2, 1}]]