0
$\begingroup$

I am new to math. How to approach the following problem?

$\min_{a,b} \sum_{t=1}^N (-4aX_t\sin(Z_tb) -4aY_t\cos(Z_tb)+a^2Z_t^2 + X_t^2 + Y_t^2)$

where $X_t,Y_t,Z_t$ for $t\in \{1,...N\}$ are given. Say $N$ is around 800.

Do I need software? Which? I am not having luck with simple optim() in R, but maybe I am using wrong parameters.

Would this be a problem that could be solved in Mathematica? (I don't have access to Mathematica, and am also not familiar with it. I just heard that it is powerful.)

1 Answers 1

0

With a little bit of work you can convert your two-dimensional minimization problem into a one-dimensional root-finding problem.

Call the quantity to be minimised $L(a,b)$. Then computing the partial derivatives and setting to zero gets you

$\frac{\partial L}{\partial a} = 0 \;\Rightarrow\; \frac{a}{2} \sum_{t=1}^n Z_t^2 - \sum_{t=1}^N X_t\sin(Z_tb) + Y_t\cos(Z_tb) = 0$

$\frac{\partial L}{\partial b} =0 \;\Rightarrow\; a \sum_{t=1}^N X_t\cos(Z_tb) - Y_t\sin(Z_tb) = 0$

and hence either $a=0$ or the sum in the second equation is zero. If you can find $b$ such that the sum is zero (using some numerical root finder) then $a$ is determined by the first equation. On the other hand, if $a$ is zero then you determine $b$ from the first equation using a numerical root finder.

  • 0
    Thanks, had hoped there was some way to avoid this.2011-05-27