How to find inverse of a self written integral function?

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Rajendra Gupta 2018-5-11

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编辑： Walter Roberson 2020-11-28

I have defined a function y=iMyF(x) containing an integral written as y=5*log10[4345*\int_0^x du/sqrt(0.3*(1+u)^2+0.7)] so that I can find y for a given value of x. How can I find x for a given value y for this function? I want to use the inverse function to fit a data set using the curve fitting tool.

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Walter Roberson 2020-11-28

编辑：Walter Roberson 2020-11-28

Could you confirm that you want the scaled gamma incomplete and not the unscaled?

It looks plausible to me that the sqrt() could be acting on a negative value, leading to complex, but then converted to real by the abs() ? Does that sound accurate?

Walter Roberson 2020-11-28

I see article https://www.researchgate.net/publication/3388532_Inverse_incomplete_gamma_function_and_its_application . Perhaps some of its ideas could be used as steps in developing the expression you need. It would not be easy though.

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Jeff Miller 2018-5-12

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No doubt there are more efficient methods for your particular function, but a quick and dirty general approach is to make a function using fzero. I don't do this often, but I think it would look something like this:

function x = myinverse(y, guess)
  fun = @(z) (iMyF(z)-y);  % parameterized function
  x = fzero(fun,guess)     % guess is your initial pretty good guess for x
end

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Rajendra Gupta 2018-5-13

Thanks Jeff. It works pretty well provided the 'guess' is reasonable. Problem is when y and x are arrays. One then needs to provide 'guess' also as an array, or have a formula included in the function that can provide a pretty good value of guess for each value of y. I would also like to know more efficient methods if at all possible. Nevertheless, for now I am happy as it solved my immediate problem by embedding a formula for guess in the function.

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