Discussion 14

Page:173 - Chapter:4 - Exercise 6

consider \(f(x) = x^2 -2\) and \(x_{0} = 1.5\)

To approximate the value of the root of \(f(x)\) accurately to d decimal places using Newtons Method, we follow this plan:

• Choose a value x0 as an initial approximation of the root.

• Create successive approximations iteractevely. i.e. given an approximation \(x_{n}\) , compute the next approximation \(x_{n+1}\) as:

\[x_{n+1} = x_{n} - \frac{f(x_{n})}{f'(x_{n})}\] - Stop the Iteration when the successive approximations becomes almost the same at d places after the decimal point.

Let’s see what the graph shows:

curve(x^2-2, -4, 4, col = "red", ylab = 'f(x) = x^2 − 2')
abline(v = 0, col = 'black')
abline(v = sqrt(2), col = 'blue')
abline(v = -sqrt(2), col = 'blue')
abline(h = 0, col = 'black')
abline(h = -2, col = 'green')

consider the first derivative of f(x):

\(f'(x) = 2x\)

let’s create our function that will perform the approximation:

library(kableExtra)
library(dplyr)
f <- function(x){return(x^2 -2)}
f1 <- function(x){return(2*x)}
approximTable <- list('iteration' = 0, 'result' = 1.5)
for (i in 2:6){
  approximTable$iteration[i] <- i-1
  approximTable$result[i] <- approximTable$result[i-1] - f(approximTable$result[i-1])/f1(approximTable$result[i-1])
}
approximTable <- data.frame(approximTable)
# kable(approximTable)

approximTable %>%
  kable() %>%
  kable_styling(bootstrap_options = "striped", full_width = F, position = "left") %>%
  row_spec(1:6, bold = T, color = "white", background = "#D7261E") 

iteration	result
0	1.500000
1	1.416667
2	1.414216
3	1.414214
4	1.414214
5	1.414214

our real solution is \(x = \sqrt{2}\)

so let’s see how the difference is calculated between the real solution and the approximation and display everything at a glance.

#real_sol is our real solution
approximTable$real_sol <- sqrt(2)
#difference between the real solution and the calculated approximation
approximTable$diff <- round(approximTable$result - approximTable$real_sol,6)

approximTable %>%
  kable() %>%
  kable_styling(bootstrap_options = "striped", full_width = F, position = "left") %>%
  row_spec(1:6, bold = T, color = "white", background = "#D7261E") 

iteration	result	real_sol	diff
0	1.500000	1.414214	0.085786
1	1.416667	1.414214	0.002453
2	1.414216	1.414214	0.000002
3	1.414214	1.414214	0.000000
4	1.414214	1.414214	0.000000
5	1.414214	1.414214	0.000000