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Working on a problem related to effect size, I get this relation where

$$ Q = \log(m) \left(m - \frac{1}{m} \right) $$

The domain of $m$ is $]0, \infty[$. For a given Q, whenever I find a $m = m^*$ satisfying the equality, the equality is also satisfied with $m = 1/m^*$. Therefore, I can limit the domain of $m$ to $]1, \infty[$.

Is there a formula that can isolate $m$ so that given a value $Q$ ($Q \in \mathbb{R}^+$), $m$ follows?

I plotted the relation and it is univocal (for a given $Q$, there is a single $m > 1$ that satify the equality). I can do a numerical search for $m$, but a formal relation would be more satisfying.

It might be that the relation can be obtained using an infinite series? If it converges, then using a limited number of terms might provide an adequate solution?

Denis Cousineau
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1 Answers1

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Looking at this graph you can see that there are only two solutions for a given $Q$. If you want to solve the equation using iterative method would work $$m_{n+1} = \frac{\log(m_n)(m_n^2-1)}{Q}$$

Use an initial guess $m_0$ in $(0,1)$. As this will make sure that square term doesn't explode. You can get arbitrary accuracy from this.

Sonal_sqrt
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  • (i) I do not see the graph you are refering to; (ii) This mathematica code m = 0.9; For[i = 0, i < 10, i++, m = Log[m] (m^2 - 1)/0.4]; m returns a huge number, whereas a numerical search suggest the solutions 0.643973 and 1.55286. Did I made an error in the code? – Denis Cousineau Jan 30 '22 at 12:28