Numerical Root Finding: Bisection Method in R

Question

I need to do numerical root finding using bisection method, and print the values of variables involved at every iteration until it reaches a certain value.

 bisection <- function(x1, x2){
  l <- vector(mode="integer")
    l[1] <- x1
  r <- vector(mode="integer")
    r[1] <- x2
  m <- vector(mode="integer")
  gl <- vector(mode="integer")
  gr <- vector(mode="integer")
  gm <- vector(mode="integer")
  
  root <- 5e-8  
  i <- 1
  repeat{
    m[i] <- (l[i]+r[i])/2
    gl[i] <- gx(l[i])
    gr[i] <- gx(r[i])
    gm[i] <- gx(m[i])
        
    if (isTRUE(abs(gm[i]) > root) && isTRUE(gl[i]*gm[i] < 0)){
      l[i+1] <- l[i]
      r[i+1] <- m[i]
    }
    if (isTRUE(gm[i] > root) && isTRUE(gr[i]*gm[i] < 0)){
      l[i+1] <- m[i]
      r[i+1] <- r[i]
    }
    else if (isTRUE(abs(gm[i]) <= root)){
      j <- c(0:(length(gm)-1))
      df <- data.frame(j, l,r,m,gl,gr,gm)
      names(df) <- c("i", "xl","xr","xm", "gxl","gxr", "gxm")
      print(df)
      break
    }
  }
}

When I try running this function with bisection(1,1.5), its output is only one row of iteration even tho solving for it manually would result in at least 12 iterations. It also hangs(?).

I don't know where I'm going wrong. Please help.

Edited to say the gx function is this: gx <- function(x){x^3-x-1}

Answer 1

There are two issues with your code:

1. In your second if statement you forgot abs, it has to be if (isTRUE(abs(gm[i]) > root)
2. You forgot to increase the value of your counter, i.e. i remains 1.

Additionally I made a small adjustment to your code. Instead of using if-if-else-if first check wether the root is found and if not update l and r using one if-else. Makes it easier to read and understand.

bisection <- function(x1, x2){
  l <- vector(mode="integer")
  l[1] <- x1
  r <- vector(mode="integer")
  r[1] <- x2
  m <- vector(mode="integer")
  gl <- vector(mode="integer")
  gr <- vector(mode="integer")
  gm <- vector(mode="integer")
  
  root <- 5e-8  
  i <- 1
  repeat{
    m[i] <- (l[i]+r[i])/2
    gl[i] <- gx(l[i])
    gr[i] <- gx(r[i])
    gm[i] <- gx(m[i])
    
    if (isTRUE(abs(gm[i]) <= root)){
      j <- c(0:(length(gm)-1))
      df <- data.frame(j, l,r,m,gl,gr,gm)
      names(df) <- c("i", "xl","xr","xm", "gxl","gxr", "gxm")
      print(df)
      break
    }
    
    if (isTRUE(abs(gm[i]) > root) && isTRUE(gl[i]*gm[i] < 0)){
      l[i+1] <- l[i]
      r[i+1] <- m[i]
    } else {
      l[i+1] <- m[i]
      r[i+1] <- r[i]
    }
    
    i <- i +1
  }
}

gx <- function(x) x^2 - 2

bisection(1, 1.5)
#>     i       xl       xr       xm           gxl          gxr           gxm
#> 1   0 1.000000 1.500000 1.250000 -1.000000e+00 2.500000e-01 -4.375000e-01
#> 2   1 1.250000 1.500000 1.375000 -4.375000e-01 2.500000e-01 -1.093750e-01
#> 3   2 1.375000 1.500000 1.437500 -1.093750e-01 2.500000e-01  6.640625e-02
#> 4   3 1.375000 1.437500 1.406250 -1.093750e-01 6.640625e-02 -2.246094e-02
#> 5   4 1.406250 1.437500 1.421875 -2.246094e-02 6.640625e-02  2.172852e-02
#> 6   5 1.406250 1.421875 1.414062 -2.246094e-02 2.172852e-02 -4.272461e-04
#> 7   6 1.414062 1.421875 1.417969 -4.272461e-04 2.172852e-02  1.063538e-02
#> 8   7 1.414062 1.417969 1.416016 -4.272461e-04 1.063538e-02  5.100250e-03
#> 9   8 1.414062 1.416016 1.415039 -4.272461e-04 5.100250e-03  2.335548e-03
#> 10  9 1.414062 1.415039 1.414551 -4.272461e-04 2.335548e-03  9.539127e-04
#> 11 10 1.414062 1.414551 1.414307 -4.272461e-04 9.539127e-04  2.632737e-04
#> 12 11 1.414062 1.414307 1.414185 -4.272461e-04 2.632737e-04 -8.200109e-05
#> 13 12 1.414185 1.414307 1.414246 -8.200109e-05 2.632737e-04  9.063259e-05
#> 14 13 1.414185 1.414246 1.414215 -8.200109e-05 9.063259e-05  4.314817e-06
#> 15 14 1.414185 1.414215 1.414200 -8.200109e-05 4.314817e-06 -3.884337e-05
#> 16 15 1.414200 1.414215 1.414207 -3.884337e-05 4.314817e-06 -1.726433e-05
#> 17 16 1.414207 1.414215 1.414211 -1.726433e-05 4.314817e-06 -6.474773e-06
#> 18 17 1.414211 1.414215 1.414213 -6.474773e-06 4.314817e-06 -1.079981e-06
#> 19 18 1.414213 1.414215 1.414214 -1.079981e-06 4.314817e-06  1.617417e-06
#> 20 19 1.414213 1.414214 1.414214 -1.079981e-06 1.617417e-06  2.687177e-07
#> 21 20 1.414213 1.414214 1.414213 -1.079981e-06 2.687177e-07 -4.056319e-07
#> 22 21 1.414213 1.414214 1.414214 -4.056319e-07 2.687177e-07 -6.845708e-08
#> 23 22 1.414214 1.414214 1.414214 -6.845708e-08 2.687177e-07  1.001303e-07
#> 24 23 1.414214 1.414214 1.414214 -6.845708e-08 1.001303e-07  1.583661e-08