Solving implicit equations in R

Question

I have an equation which looks like this: $$\frac{1}{\sqrt{}}=−2\log⁡\left(+\frac{}{\sqrt{}}\right)$$ where $A$ and $B$ are constants

How do I solve this in $\mathbb{R}$? Can this be solved iteratively?

Answer 1

First, recall that under the substitution $y=\frac{1}{\sqrt{x}}$ ($y>0$), you obtain the equivalent equation $$ y=-2\log(A+By). $$

Up to some simplifications, you can also realize that the solution of the above is equivalent to find the zeros of the function $$g(y)=e^{-\frac{y}{2}}-A-By.$$

The above equation is non-linear so that there is not direct available on the literature that gives you a direct solution. In any case, you can use the so-called "perturbation of the inverse approach" (fixed point theorem in disguise) to prove for which $A \& B$, you have a solution (I will come back to this later, depending on your reply).

A simple strategy is try to apply recursively the Bolzano theorem. For this approach, under the fact that your function is continuous, you only need to start with a first estimate for the interval $[a,b]$ for which $g(a)g(b)<0$.

Maybe it would be nice that you tell me in which course this problem have appeared to provide you a goal-oriented answer to your reply.

Answer 2

This looks like the Colebrook equation for the friction factor in a pipe.

First, let $\frac{1}{\sqrt{}}=t$ to make the equation $$t=-2\log(A+Bt)$$ the solution of which being given in terms of Lambert function $$t=2 W\left(\pm\frac{1}{2} \sqrt{\frac{e^{A/B}}{B^2}}\right)-\frac{A}{B}$$

Take care that, in the real domain, $W(y)$ is real if $y \geq -\frac 1 e$ and that two branches $(W_0(y)\text{ and } W_{-1}(y))$ exist.