source('utils.R')

#
# For verification we duplicate the numbers found in the table on EPage 113:
#
# Generate the iterates x_n given in the text:
#
iter_fn = function(x){ (0.1 + tan(x))/1.5 }
x0 = 0.0
n = 30
xn = rep(NA, n+1)
xn[1] = x0 # =x_0
for( k in 2:(n+1) ){
    xn[k] = iter_fn(xn[k-1])
}
a_res = aitkens_delta2(xn)

T = cbind( seq(0, n), xn, a_res$xhatn, a_res$rn )
colnames(T) = c('n', 'x_n', 'x_hat_n', 'r_n')
head(T)
tail(T)


# Exercise 3.4-2:
#
# Generate the iterates given in the text:
#
root_fn = function(x){ 0.5 - x + 0.2 * sin(x) }
iter_fn = function(x){ 0.5 + 0.2 * sin(x) }
x0 = 0.5
n = 10
xn = rep(NA, n+1)
xn[1] = x0
for( k in 2:(n+1) ){
    xn[k] = iter_fn(xn[k-1])
}
a_res = aitkens_delta2(xn)

T = cbind( seq(0, n), xn, a_res$xhatn, a_res$rn )
colnames(T) = c('n', 'x_n', 'x_hat_n', 'r_n')
print(T)

g_prime_xi = 1/a_res$rn[n]
print(g_prime_xi)


# Exercise 3.4-3:
#
x0 = 0.5
res = steffensen_iteration(x0, iter_fn)
print(sprintf('Steffensens on g(x) = 0.5 + 0.2 sin(x) gives: x= %10f n=%3d', res[1], res[2]))


# Exercise 3.4-4:
#
# Generate the iterates given in the text:
#
iter_fn = function(x){ sqrt(2+x) }
x0 = 0.0
n = 5
xn = rep(NA, n+1)
xn[1] = x0
for( k in 2:(n+1) ){
    xn[k] = iter_fn(xn[k-1])
}

a_res = aitkens_delta2(xn)

T = cbind( seq(0, n), xn, a_res$xhatn, a_res$rn )
colnames(T) = c('n', 'x_n', 'x_hat_n', 'r_n')
print(T)