Discussion10

11.6

In Example 11.9, assume that we start with a hybrid bred to a hybrid. Find \(u^{(1)}\) , \(u^{(2)}\) and \(u^{(3)}\). What would \(u^{(n)}\) be?

g_matrix = matrix(c(.5,.25,0,.5,.5,.5,0,.25,.5), nrow=3,ncol=3)
g_matrix

##      [,1] [,2] [,3]
## [1,] 0.50  0.5 0.00
## [2,] 0.25  0.5 0.25
## [3,] 0.00  0.5 0.50

As in problem it is stated to start from hybrid, so u vector is:

u = c(0,1,0)
u

## [1] 0 1 0

\[ u^{(n)} = uP^{(n)} \\ u^{(1)} = uP^{1} \]

P1 = g_matrix
u1 = u%*%P1
u1

##      [,1] [,2] [,3]
## [1,] 0.25  0.5 0.25

\[ u^{(n)} = uP^{(n)} \\ u^{(2)} = uP^{2} \]

P2 = g_matrix%*%g_matrix
P2

##       [,1] [,2]  [,3]
## [1,] 0.375  0.5 0.125
## [2,] 0.250  0.5 0.250
## [3,] 0.125  0.5 0.375

u2 = u%*%P2
u2

##      [,1] [,2] [,3]
## [1,] 0.25  0.5 0.25

\[ u^{(n)} = uP^{(n)} \\ u^{(3)} = uP^{3} \]

P3 = g_matrix%*%P2
P3

##        [,1] [,2]   [,3]
## [1,] 0.3125  0.5 0.1875
## [2,] 0.2500  0.5 0.2500
## [3,] 0.1875  0.5 0.3125

u3 = u%*%P3
u3

##      [,1] [,2] [,3]
## [1,] 0.25  0.5 0.25

\[u^{(n)} = (.25, .5 , .25)\]