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I've been trying to find a way to get an idea about the root for the implicit function:

$ \frac{1}{\sqrt{x}}=4\log_{10}(\sqrt{x}\,Re) - 0.4 $ where $Re>0$ is some constant (Reynolds number).

I've tried to express this function to a function of $x$ and that makes the $Re$ value the function value. However, I'm interested to know what value $x$ is around when $Re = 10^4$.

Any ideas?

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    Yep, it is. But I don't know anything about it, the equation is just given in an excercise.2012-12-19

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For this example simple fixed-point iteration works. Reformulate the problem, such that $x=F(x)$. One(!) example of doing so would be \begin{align} x = \frac{1}{(4\log_{10}(Re\cdot \sqrt{x})-0.4)^2}=F(x) \end{align} Fixed-point iteration works like this: $x^{k+1}=F(x^k)$. That means you simply plug in the result of $F(x^k)$ once more into $F$. Now you have to prove that this does converge, see i.e. Banach fixed-pint theorem.

Another method to use would be Newton's Method, but it turns out that the fixed-point iteration just works fine. With the Matlab code below, you reach a fixed-point in 5 iterations.

function fixedPointIteration  x = 1; tol=1e-5; while abs(x-f(x))>tol     i=i+1     x=f(x); end sprintf('Result of fixed-point iteration is x=%s',x) end  function y=f(x) Re = 1e4; y =1/(4*log10(Re*sqrt(x)-0.4))^2; end 

Which gives $x=0.0072742$. It should be easy to adapt the code into other languages. Hope this helps.

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    Yes, of course! Thank you!2012-12-19