source('utils.R')

# 3.1-1:
#
xs = seq( -5, +5, length.out=100 )
fs = xs^2 - 2 * xs - 2
plot( xs, fs, type='l', xlab='x', ylab='x^2 - 2 x - 2' )
abline(h=0, col='black')
grid()

truth = polyroot( c( -2, -2, 1 ) )
truth = Re(truth)

# For the left-most root (negative one):
#
bm_res = bisection_method(-2, 0, function(x){x^2-2*x-2}, xtol=1.e-2)
rf_res = regula_falsi(-2, 0, function(x){x^2-2*x-2}, xtol=1.e-2)
mrf_res = modified_regula_falsi(-2, 0, function(x){x^2-2*x-2}, xtol=1.e-2)
print(sprintf('Truth= %f; Bisection= %f; Regula Falsi= %f; Modified Regula Falsi= %f', truth[1], bm_res[1], rf_res[1], mrf_res[1]))

# For the right-most root (positive one):
#
bm_res = bisection_method(1, 4, function(x){x^2-2*x-2}, xtol=1.e-2)
rf_res = regula_falsi(1, 4, function(x){x^2-2*x-2}, xtol=1.e-2)
mrf_res = modified_regula_falsi(1, 4, function(x){x^2-2*x-2}, xtol=1.e-2)
print(sprintf('Truth= %f; Bisection= %f; Regula Falsi= %f; Modified Regula Falsi= %f', truth[2], bm_res[1], rf_res[1], mrf_res[1]))


# 3.1-3:
#
xs = seq( 0, +3, length.out=100 )
fs = xs^3 - 2 * xs - 1
plot( xs, fs, type='l', xlab='x', ylab='x^3 - 2 x - 1' )
abline(h=0, col='black')
grid()

truth = polyroot( c( -1, -2, 0, 1 ) )
truth = Re(truth)

secant_res = secant_method(1, 2, function(x){x^3-2*x-1}, xtol=1.e-2)
print(sprintf('Truth= %f; Secant= %f', truth[3], secant_res[1]))


# 3.1-6:
#
fn_1 = function(x){ 1/(3*x-1) }

xs = seq( 0, +1, length.out=100 )
fs = fn_1(xs)
plot( xs, fs, type='l', xlab='x', ylab='1/(3x-1)' )
abline(h=0, col='black')
grid()

fn_2 = function(x){ cos(10*x) }

xs = seq( 0, +1, length.out=100 )
fs = fn_2(xs)
plot( xs, fs, type='l', xlab='x', ylab='cos(10*x)' )
abline(h=0, col='black')
grid()

fn_3 = function(x){ if(x>=0.3){ 1.0 }else{ -1.0 } }

xs = seq( 0, +1, length.out=100 )
fs = rep(1.0, length(xs))
fs[xs<0.3] = -1.0
plot( xs, fs, type='l', xlab='x', ylab='step function' )
abline(h=0, col='black')
grid()



# 3.1-7:
#
root_fn = function(x){ exp(2*x) - exp(x) - 2 } 
root_fn_prime = function(x){ 2*exp(2*x) - exp(x) } 

xs = seq( 0, +1, length.out=100 )
fs = root_fn(xs)
plot( xs, fs, type='l', xlab='x', ylab='exp(2x) - exp(x) - 2' )
abline(h=0, col='black')
grid()

newton_res = newtons_method(0.6, root_fn, root_fn_prime, xtol=1.e-4)
print(sprintf('Newton= %f', newton_res[1]))


# 3.1-8:
#
root_fn = function(x){ 4*sin(x) - exp(x) }

xs = seq( 0, 0.5, length.out=100 )
fs = root_fn(xs)
plot( xs, fs, type='l', xlab='x', ylab='4sin(x) - exp(x)')
abline(h=0, col='black')
grid()

secant_res = secant_method(0, 0.5, root_fn, xtol=1.e-4)
print(sprintf('Secant= %f', secant_res[1]))


# 3.1-9:
#
root_fn = function(x){ 2*x^3 - 3*x - 4 }

xs = seq( 0, 3, length.out=100 )
fs = root_fn(xs)
plot( xs, fs, type='l', xlab='x', ylab='2x^3 - 2 x - 1')
abline(h=0, col='black')
grid()

truth = polyroot( c(-4, -3, 0, 2 ) ) 

bm_res = bisection_method(0, 3, root_fn, xtol=1.e-3)
print(sprintf('Truth= %f; Bisection= %f', truth[3], bm_res[1]))