Gram-Schmidt orthogonalization procedure.

We have a set of independent vectors $\{ a_1, a_2, \ldots, a_n \}$. We want to convert it into a set of orthonormal vectors $\{ b_1, b_2 , \ldots, b_n \}$ by the Gram-Schmidt process.
The scalar product between vectors $x$ and $y$ will be noted $<x,y>$

Algorithm 1.

1. Initialization $b_1=a_1$, $k=2$
2. Orthogonalisation

   $$\tilde{b_k}= a_k - \sum_{j=1}^{k} <a_k, b_j> b_j$$ (13.15)

   
   
   We will take $a_k$ and transform it into $\tilde{b_k}$ by removing from $a_k$ the component of $a_k$ parallel to all the previously determined $b_j$.
3. Normalisation

   $$b_k= \frac{ \tilde{b_k} }{ \Vert \tilde{b_k} \Vert }$$ (13.16)

   
   

4. Loop increment $k$. If $k<n$ go to step 2.

Algorithm 2.

1. Initialization k=1;
2. Normalisation

   $$b_k=\frac{ a_k}{ \Vert a_k \Vert}$$ (13.17)

   
   

3. Orthogonalisation for $j=k+1$ to $n$ do:

   $$a_j= a_j - <a_j, b_k> b_k \quad j=k+1, \ldots, n$$ (13.18)

   
   
   We will take the $a_j$ which are left and remove from all of them the component parallel to the current vector $b_k$.
4. Loop increment $k$. If $k<n$ go to step 2.