Let fi,.. , fn be n given values satisfying fi f2 fn and let E(C)-ΣΙii - Cl. i=1 Find the minimum of E(C) by graphing E...

Question

Question

Let fi,.. , fn be n given values satisfying fi f2 fn and let E(C)-ΣΙii - Cl. i=1 Find the minimum of E(C) by graphing E as a

Answer 1

Answer #1

$πυ E (C)-Σισ-i i-1$

The function is a piecewise linear sum of functions of the form $C- fi 2$ which is minimum when $C = f$

Of course, each $C$ can't be equal to each $f_i$ but we can make the $C$ equidistant from the $f_i$

So that $C = median(fi)$ is the point where the function attains its minimum (also called the 'average')

In case $n=2k 1$ is odd, we have $C = fk+1$ is the required point

An example is below:

20 = [1,3,5,6,7] = 5 element list a N f(x)abs(x-a»]) 15 n1 mean a 10 = 4.4 (5,9) 4 X k length(a) k = 5 median(a) -5 0 5 10 15

Here, our function is $x 1 -3 x- 5 -7 -9$ whose minimum value occurs at $median(1, 3, 5, 7,9)5$

Answer 2

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