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seen Jun 15 at 10:19

May
24
comment Finding inverse of polynomial in a field
Oh, sorry. You're right. I haven't noticed that.
May
22
comment Finding inverse of polynomial in a field
I think 3rd line is wrong: (x+1) = (2x)*2 + 1. Did you mean (x+1) = (x/2)*2 + 1? Of course, the following solution is wrong too. The correct solution is (x^2-x+1)/2.
May
15
awarded  Caucus
Jun
19
awarded  Teacher
Jun
19
revised What is the usefulness of matrices?
added 25 characters in body
Jun
19
answered What is the usefulness of matrices?
May
31
awarded  Scholar
May
31
accepted Refraction equation, quartic equation
May
31
comment Refraction equation, quartic equation
I've tested it and it works great! Only 10-20 iterations of bisection search is enough. I could also improve the solution using Newton's method.
May
30
comment Refraction equation, quartic equation
Right, converting the original equation into a polynomial introduces unwanted ambiguity. The polynomial can have as many as four roots while there's only one correct solution to the original problem. I just hoped that I could find a descent analytic solution but it turns out that a numerical method is the only way to solve the problem efficiently.
May
29
awarded  Commentator
May
29
comment Refraction equation, quartic equation
Yes, but deriving $x$ from Snell's Law gives the same result - quartic equation. In fact, my problem is to find the point of refraction of light according to Snell's Law.
May
29
comment Refraction equation, quartic equation
That's a good idea. Finding a root for $0<x<d$ is guaranteed, since $f(0)=-d^2h^2<0$ and $f(d)=d^2n^2w^2>0$ and $f$ is continous.
May
29
revised Refraction equation, quartic equation
added 5 characters in body; edited tags
May
29
comment Refraction equation, quartic equation
Edit to my previous comment: when starting with $x=0$, it converges to $-5.7$ (which means $n=-1.33$). The correct result can be obtained starting for example with $x=1$ or $x=12.5$. Do you have any idea how to choose the starting point which guarantees the convergence and the correct result?
May
29
comment Refraction equation, quartic equation
Although it generally works well for $x=dh/(h+w)$, there is a problem for this. Starting with $x=25$ the result is getting close to $x=30$ while $x=30$ is not a root. The correct result can be obtained starting with $x=0$ which gives approx. $5.7$. Also I'm not sure if starting with $x=0$ always gives the correct result.
May
29
comment Refraction equation, quartic equation
Thank you! I will check this method if it's accurate enough.
May
29
comment Refraction equation, quartic equation
@joriki: Right, thanks for fixing it ;)
May
28
comment Refraction equation, quartic equation
@Valentin: I need a solution for $n>1$ (or at least for $n=1.33$). I'd prefer to find an exact solution.
May
28
revised Refraction equation, quartic equation
edited tags