source('utils.R')

# 3.3-2 (a):
#
xs = seq( 1, 1.5, length.out=100 ) # our interval I
fs = xs^3 - xs - 1

true_roots = polyroot( c( -1, -1, 0, 1 ) )
truth = Re(true_roots[3])

par(mfrow=c(1,2))
plot( xs, fs, type='l', xlab='x', ylab='f(x)')
abline(h=0, col='black')
grid()

#gs = xs^3 - 1
gs = ( xs + 1 )^(1/3)
plot( xs, gs, type='l', xlab='x', ylab='g(x)')
abline(h=0, col='black')
grid()
par(mfrow=c(1,1))

g_prime_max = 1 / ( 3*(2^(2/3)) )

root_fn = function(x){ x^3 - x - 1 }
iter_fn = function(x){ (x+1)^(1/3) }
res = fixed_point_iteration(1.25, root_fn, iter_fn)
print(sprintf('Part (a): Truth: %f; FP iteration: %f', truth, res[1]))


# 3.3-2 (b):
#
xs = seq( 4.45, 4.55, length.out=100 ) # our interval I
fs = xs - tan(xs)

par(mfrow=c(1,2))
plot( xs, fs, type='l', xlab='x', ylab='f(x)')
abline(h=0, col='black')
grid()

gs = atan(xs) + pi
plot( xs, gs, type='l', xlab='x', ylab='g(x)')
abline(h=0, col='black')
grid()
par(mfrow=c(1,1))

g_prime_max = 1 / ( 1 + min(xs)^2 )

root_fn = function(x){ x - tan(x) }
iter_fn = function(x){ atan(x) + pi }
res = fixed_point_iteration(4.5, root_fn, iter_fn)
print(sprintf('Part (b); FP iteration: %f', res[1]))


# 3.3-2 (c):
#
xs = seq( 0.5, 2, length.out=100 ) # our interval I
fs = exp(-xs) - cos(xs)

par(mfrow=c(1,2))
plot( xs, fs, type='l', xlab='x', ylab='f(x)')
abline(h=0, col='black')
grid()
gs = acos(exp(-xs))
plot( xs, gs, type='l', xlab='x', ylab='g(x)')
abline(h=0, col='black')
grid()
par(mfrow=c(1,1))

g_prime_max = exp(-min(xs)) / sqrt( 1 - exp(-2*min(xs)))

root_fn = function(x){ exp(-x) - cos(x) }
iter_fn = function(x){ acos(exp(-x)) }
res = fixed_point_iteration(4.5, root_fn, iter_fn)
print(sprintf('Part (c); FP iteration: %f', res[1]))


# 3.3-4:
#
a = 0.5
xs = seq( -5, +5, length.out=100 )
gs = 2*xs - 2*a*xs^2
plot(xs, gs, type='l', xlab='x', ylab='g(x)')
grid()