Nonlinear method to solve an equation with the error function in it

Question

My question is to find a method to solve the following non-linear equation. I know it should be an iterative method, but I don't know what would be the best method to use. Any help is highly appreciated.

erf(x) + a*x = b ; where both a and b are constants.

Thanks in advance.

Answer 1

I assume you're using a computer and have the tools to evaluate semi-complicated expressions involving the error function. If so, simply use Newton's method and iterate

$$ n(x) = x - \frac{f(x)}{f'(x)} = x-\frac{a x-b+\text{erf}(x)}{a+\frac{2 e^{-x^2}}{\sqrt{\pi }}} $$

For example, if $a=2$ and $b=1$, then starting at $x=1.0$, you'll generate the sequence

$$1.0, 0.237011, 0.322921, 0.323604, 0.323604.$$

Alternatively, if you've just got a few cases to solve, you could enter it into WolframAlpha: