Could $\tanh(x)$ be considered a "static nonlinearity"?

Question

I'm writing about nonlinear functions and I just have a quick question about the hyperbolic tangent.

Could $\tanh$ be considered a nonlinear equation?

IE: Could you use Newton's Method for solving nonlinearity in the same way you can solve the square root of "number" by saying $$f(x) = x^2 - \text {number}$$

How would this be written?

Thanks

Answer 1

You want a sequence that converges to $\tanh k$ by Newton's method?

$\tanh k = x\\ f(x) = \tanh^{-1} x - k = 0$

$x_{n+1} = $$x_n - \frac {f(x_n)}{f'(x_n)}$

$x_{n+1} = $$x_n - \frac {f(x_n)}{1-x^2}$

Here we see where we are going to have a problem. We are going to need to be able to estimate $f(x_n)$ in order to find $x_{n+1}.$

So we can use a Taylor series series: $\tanh^{-1} x = \sum n x^{2n}$

But now you have to use the series to find the next step in your sequence... you could be pretty busy calculating.

Seems like it would be easier to find the Taylor series for $\tanh x$

$\tanh x = x - \frac 13 x^3 + \frac {16}{5!} x^5 - \frac{217}{7!} x^7 \cdots$