JPark CUNY 605 Discussion Week 13

Exercise 4.1 Question 8

Use Newton’s Method to approximate all roots of the given functions accurate to 3 places after the decimal. If an interval is given, find only the roots that lie in that interval. Use technology to obtain good initial approximates.

\[f(x) = x^3 + 5x^2 - x - 1\]

Using Newton’s Method as described in Chapter 4 in the Apex Calculus Textbook.

\[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\]

eqn <- function(x){
  return(x**3 + 5 * x**2 - x - 1)
}

# Reference: https://www.math.ucla.edu/~anderson/rw1001/library/base/html/curve.html
plot(eqn, from = -10, to = 10)
abline(0, 0, col = 'blue')

# Reference: https://www.youtube.com/watch?v=C4Zoc0Gh-es
f <- expression(x**3 + 5 * x**2 - x - 1)

newton_method <- function(eqn, x_function, initial_value){
  # Gives the approximate roots after user gives an initial value
  # Reference from page 161 from Apex Calculus textbook
  x <- initial_value
  f_prime <- function(x) {}
  body(f_prime) <- D(x_function,'x')
  while (TRUE) {
    oldx <- x
    x <- x - (eqn(x)/f_prime(x))
    print(x)
    if (abs(x-oldx) < .001){
      break
    }
  }
}

print("Initial Guess: -5 ")

## [1] "Initial Guess: -5 "

newton_method(eqn, f, -5)

## [1] -5.166667
## [1] -5.156366
## [1] -5.156325

print("Initial Guess: 0 ")

## [1] "Initial Guess: 0 "

newton_method(eqn, f, 0)

## [1] -1
## [1] -0.5
## [1] -0.3809524
## [1] -0.3692266
## [1] -0.3691024

print("Initial Guess: 1 ")

## [1] "Initial Guess: 1 "

newton_method(eqn, f, 1)

## [1] 0.6666667
## [1] 0.5449735
## [1] 0.5259008
## [1] 0.5254279

The above function demonstrates the three roots using the Newton Method of Approximation. The roots to the above function are: -5.167, -0.369, and 0.525. Let’s confirm this with R.

if (!require(rootSolve)) install.packages("rootSolve")

## Loading required package: rootSolve

library(rootSolve)

# Function polyroot requires a vector of polynomial coefficients in increasing order
v_coefficient <- c(-1, -1, 5, 1)

ans <- polyroot(v_coefficient)
print(paste0("Roots of this function are: ", round(ans,3)))

## [1] "Roots of this function are: 0.525+0i" 
## [2] "Roots of this function are: -0.369+0i"
## [3] "Roots of this function are: -5.156+0i"