Numerical Approximation in Java

Question

I am trying to solve an equation which I believe cannot be done analytically, but can use a numerical approximation to get a result. The equation is: $$\frac{2*\sqrt{\pi}*h*s*e^{m^{2}/(2*s^2)}}{\sqrt{2}} = a$$ where $h, m,$ and $a$ are given constants. This is for a project that I'm writing in Java, and it would be infeasible to grab the solution output from another program such as Matlab, Mathematica, or some other scientific computing language for a couple reasons, one of which is that I will have to solve this equation hundreds of times with different values. Basically I am wondering if anybody knows of a library I can easily import and use for this kind of problem, and to possibly spell out how I would go about doing that. So far I have been looking through math libraries such as Apache Commons Math but cannot find out if they are appropriate. Thanks!

Answer 1

With $p:=\dfrac ms$, squaring and collecting all the constants to avoid the mess, the equation reads

$$e^{p^2}=\lambda p^2$$ or $$-p^2e^{-p^2}=-\lambda^{-1}.$$

Hence using the Lambert function,

$$s=\frac m{\sqrt{-W(-\lambda^{-1})}}.$$

The positive square root must be taken, and the two branches of $W$ yield two real solutions (provided $\lambda>e$).

You can probably find ready-made implementations of that function in Java, or adapt from a C/C++ version.

Answer 2

You are asking how to solve $e^{ax^2}=b$ for $x$, given constants $a,b$. The solution is $x=\sqrt{\frac{1}{a} \log b}$, which can be computed (approximately) using standard algorithms for computing the log and the square root.

In particular, your specific equation has the solution

$$m = \sqrt{2s^2 \log (a/(\sqrt{2\pi} hs))}.$$

You can verify the correctness of this solution by plugging into the equation and verifying it works.

Answer 3

Quote: "I don't think it's that simple. I am fairly certain that you cannot find a symbolic solution for x with the equation I provided above. "

$$\frac{2*\sqrt{\pi}*h*s*e^{m^{2}/(2*s^2)}}{\sqrt{2}} = a$$

Multiply both sides by sqrt (2).

Divide both sides by 2 sqrt(pi) h s.

Take the natural logarithm.

Multiply by 2 s^2.

Take the square root.

Now m is alone on the left side, and the right side is the solution:

$$m = \sqrt {\ln (\frac {a \cdot \sqrt{2}}{2*\sqrt{\pi}*h*s}) \cdot (2\cdot s^2)}$$

(Of course it would help if you posted the problem you have, and not a different problem).