## 4.5-10
##

## Get the matrix A and A^{-1} from Exercise 4.2-8 (the exact solution is [1, 1, 1, 1]):
##
source('./section_4_4_ex_2.R')
x_exact = matrix(c(1, 1, 1, 1), nrow=4, ncol=1)

## Get the right-hand-side from Exercise 4.2-8:
b = matrix(c(0.96, 0.71, 1.26, 1.53), nrow=4, ncol=1)
print(b)

## Compute the conditions numbers for A: WWX: TODO
##
one_norm = norm(A, 'O') * norm(AInv, 'O')
inf_norm = norm(A, 'I') * norm(AInv, 'I')
print(sprintf('Condition numbers: One-norm= %f; Inf-norm= %f', one_norm, inf_norm))

## What was the solutions found for x in Exercises 4.2-8 and 4.3-4:
##
source('./section_4_2_ex_8.R')
x_ex_4_2_8 = x 

source('./section_4_3_ex_4.R')
x_ex_4_3_4 = x

Xs = cbind(x_ex_4_2_8, x_ex_4_3_4, x_exact)
colnames(Xs) = c('ex_4_2_8_sol', 'ex_4_3_4_sol', 'true_solution')
print(Xs)

## Compute the relative errors:
##
print(sprintf('Relative errors (one-norm): ex_4_2_8= %f; ex_4_3_4= %f', norm(x_ex_4_2_8 - x_exact, 'O') / norm(x_exact, 'O'), norm(x_ex_4_3_4 - x_exact, 'O') / norm(x_exact, 'O')))


## Compute the residuals:
##
r_ex_4_2_8 = b - A %*% x_ex_4_2_8
r_ex_4_3_4 = b - A %*% x_ex_4_3_4

Rs = cbind(r_ex_4_2_8, r_ex_4_3_4)
colnames(Rs) = c('ex_4_2_8_residual', 'ex_4_3_4_residual')
print(Rs)

## Compute A^{-1} r:
##
results = GE_w_scaled_partial_pivoting(A, r_ex_4_2_8)
e_hat_ex_4_2_8 = results$X

results = GE_w_scaled_partial_pivoting(A, r_ex_4_3_4)
e_hat_ex_4_3_4 = results$X

EHats = cbind(e_hat_ex_4_2_8, e_hat_ex_4_3_4)
colnames(EHats) = c('ex_4_2_8', 'ex_4_3_4')
print(EHats)

## Add in our correction: 
## 
print(x_ex_4_2_8 + e_hat_ex_4_2_8)
print(x_ex_4_3_4 + e_hat_ex_4_3_4)