Editing Neighbor joining (section)

=== First step ===
====First joining====
We calculate the <math>Q_1</math> values by equation ({{EquationNote|1}}). For example:
:<math>Q_1(a,b) = (n-2)d(a,b) - \sum_{k=1}^5 d(a,k) - \sum_{k=1}^5 d(b,k)</math>
:<math>= (5-2) \times 5 - (5+9+9+8) - (5+10+10+9) = 15-31-34 = -50</math>

We obtain the following values for the <math>Q_1</math> matrix (the diagonal elements of the matrix are not used and are
omitted here):

{| class="wikitable" style="text-align: center"
|-
! style="width: 20px;" | 
! style="width: 20px;" | a
! style="width: 20px;" | b
! style="width: 20px;" | c
! style="width: 20px;" | d
! style="width: 20px;" | e
|-
! a
| 
| style=background:#ffffcc; | &minus;50
| &minus;38
| &minus;34
| &minus;34
|-
! b
| style=background:#ffffcc; | &minus;50
| 
| &minus;38
| &minus;34
| &minus;34
|-
! c
| &minus;38
| &minus;38
| 
| &minus;40
| &minus;40
|-
! d
| &minus;34
| &minus;34
| &minus;40
| 
| &minus;48
|-
! e
| &minus;34
| &minus;34
| &minus;40
| &minus;48
| 
|-
|}

In the example above, <math>Q_1(a,b)=-50</math>. This is the smallest value of <math>Q_1</math>, so we join elements <math>a</math> and <math>b</math>.

====First branch length estimation====
Let <math>u</math> denote the new node. By equation ({{EquationNote|2}}), above, the branches joining  <math>a</math> and <math>b</math> to <math>u</math> then have lengths:

:<math>\delta(a,u)=\frac{1}{2}d(a,b)+\frac{1}{2(5-2)} \left [ \sum_{k=1}^5 d(a,k) - \sum_{k=1}^5 d(b,k) \right ] \quad =\frac{5}{2} + \frac{31-34}{6} = 2</math>
:<math>\delta(b,u)=d(a,b)-\delta(a,u) \quad = 5-2 = 3</math>

====First distance matrix update====
We then proceed to update the initial distance matrix <math>D</math> into a new distance matrix <math>D_1</math> (see below), reduced in size by one row and one column because of the joining of <math>a</math> with <math>b</math> into their neighbor <math>u</math>. Using equation ({{EquationNote|3}}) above, we compute the distance from <math>u</math> to each of the other nodes besides <math>a</math> and <math>b</math>. In this case, we obtain:

:<math>d(u,c)=\frac{1}{2} [d(a,c)+d(b,c)-d(a,b)] = \frac{9+10-5}{2} = 7</math>
:<math>d(u,d)=\frac{1}{2} [d(a,d)+d(b,d)-d(a,b)] = \frac{9+10-5}{2} = 7</math>
:<math>d(u,e)=\frac{1}{2} [d(a,e)+d(b,e)-d(a,b)] = \frac{8+9-5}{2} = 6</math>

The resulting distance matrix <math>D_1</math> is:

{| class="wikitable" style="text-align: center"
|-
! style="width: 20px;" | 
! style="width: 20px;" | u
! style="width: 20px;" | c
! style="width: 20px;" | d
! style="width: 20px;" | e
|-
! u
| 0
| '''7'''
| '''7'''
| '''6'''
|-
! c
| '''7'''
| 0
| ''8''
| ''7''
|-
! d
| '''7'''
| ''8''
| 0
| ''3''
|-
! e
| '''6'''
| ''7''
| ''3''
| 0
|-
|}

Bold values in <math>D_1</math> correspond to the newly calculated distances, whereas italicized values are not affected by the matrix update as they correspond to distances between elements not involved in the first joining of taxa.