# To find the new weights of an error function by minimizing it

by Justin   Last Updated January 18, 2018 00:20 AM

My task is to find the closed form solution $\boldsymbol w^*$ to minimize E(W) and hence find $y(x, \boldsymbol w^*)$

Consider the following error function

$E(\boldsymbol w) = \frac{1}{2} \sum\limits_{n=1}^N {(y(x_n,\boldsymbol w)−t_n)^2}$

where w is a vector of weights; $x_n$ and $t_n$ come from two vectors of length N; and y is a polynomial:

y(x,w) = $\sum\limits_{j=0}^M {w_jx^j}$

My task is to find the closed form solution $\boldsymbol w^*$ to minimize E(W) and hence find $y(x, \boldsymbol w^*)$

So, I did $\frac{\partial E(w)}{\partial w}$ = 0 and I obtained :

$\sum\limits_{j=0}^M A_{ij} w_j = T_i$

where, $A_{ij} = \sum\limits_{n=1}^N (x_n)^{i+j}$ and $T_i = \sum\limits_{n=1}^N (x_n)^i t_n$

But I'm not really sure how to solve for $w$ and find $y(x, \boldsymbol w^*)$

Any suggestions on how I can proceed?

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