compute euclidean norm for large vector

Input:

x = [x_1, x_2, ..., x_n]

when n is very large

compute ||x||_2

Solution:

||x||_2 = \sqrt(x[1]^2 + x[2]^2 + ... + x[n]^2)

One way to compute it is as following:

double m_sum = 0.0;

for (int i=1; i<=n;++i)

{

    m_sum  += x[i]^2;

}

return sqrt(m_sum);

However, it's possible that m_sum  overflow  overflow the MAX_FLOAT or MAX_DOUBLE on the computer (for large n). 

A safer algorithm would be:

 y=max(abs(x)) 

for (int i = 1; i<=n;i++){

      m_sum+=(x(i)/y)^2;

}

 return y*sqrt(m_sum);