source('utils.R')

## 2.3-1:
##
xs = c(1.0, 1.5, 2.0, 3.0, 3.5, 4.0)
ds = log10(xs)

## Get the Newton form coefficients:
##
as = coefficients_of_newton_form(xs, ds)

xs = rev(xs) ## convert into the orderning needed by newton_form_nested_multiplication
as = rev(as)

## 2.3-2:
##
## Evaluate our interpolating polynomial at some points:
##
x_test = c(2.5, 1.25, 3.25)
f_hat_test = c()
for( x in x_test ){
    res = newton_form_nested_multiplication(as, xs[-length(xs)], x)
    f_hat_test = c( f_hat_test, res$as_prime[1] )
}

## Package results:
##
error = log10(x_test) - f_hat_test
res = data.frame(x=x_test, log10_x=log10(x_test), log10_x_hat=f_hat_test, error=error)
print(res)

##
## Duplicate the results above using the "short cut" function:
##
xs = c(1.0, 1.5, 2.0, 3.0, 3.5, 4.0) ## x values
ds = log10(xs) ## y values
x_test = c(2.5, 1.25, 3.25) ## where to evaluate our interpreting polynomial

print(newton_form_interpolating_polynomial(xs, ds, x_test)) ## duplicate

## 2.3-3:
##
y = log10(2) + log10(1.25)
y_hat = res[1, 3]
print(sprintf('y= %f; y_hat= %f; abs_err= %f; rel_err= %f', y, y_hat, y-y_hat, (y-y_hat)/y))