How should determine solutions to equations of this form? $$e^{-f(x)} + b f(x) = ax$$ Here $f(x)>0$ is real valued. Also $a>0$, $b>0$.
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$\begingroup$ I guess my question might be equivalent to "How to efficiently compute the inverse of $$y \mapsto e^{-y}+by $$" so maybe it belongs elsewhere? $\endgroup$– univariateOct 24, 2014 at 14:25
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4$\begingroup$ Probably you should explain why the usual methods (such as Newton's iteration, etc) for numerical solution of equations are inadequate for your purposes. Otherwise, I think this question is more suitable for math.stackexchange.com (see Help Center for more information). $\endgroup$– Boris BukhOct 24, 2014 at 14:30
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1$\begingroup$ Appreciate the redirect Boris. $\endgroup$– univariateOct 24, 2014 at 15:50
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$f(x) = W(-\exp(-ax/b)/b) + ax/b$ where W is one of the branches of the Lambert W function.