Latihan Soal 5.9

Latihan 5.9 (Manual)

library(numDeriv)
quadraticinterpolation <- function(phi.f, l = 0, u = 1, eps = 1e-6)
{
#polynomials evaluated at 3 points(l, m, u) for interpolation: phi_fl, phi_fm, phi_fu
  
#l = lower
#u = upper
  if(l == u)
  {
    l <- 0
  }
  m <- (l + u)/2
  phi.fl <- phi.f(l)
  phi.fm <- phi.f(m)
  phi.fu <- phi.f(u)
  
  
  #a, b and c are coefficients of the polynomial (hx)
  y <- ((l - m)*(m - u)*(u - l))
  a <- (phi.fl*m*u*(u - m) + phi.fm*u*l*(l - u) + phi.fu*l*m*(m - l))/y
  b <- (phi.fl*(m^2 - u^2) + phi.fm*(u^2 - l^2) + phi.fu*(l^2 - m^2))/y
  c <- -(phi.fl*(m - u) + phi.fm*(u - l) + phi.fu*(l - m))/y
  
  alpha <- -b/(2*c)
  falpha <- phi.f(alpha)
  
  ncf <- 4
  
  hx <- function(x)  c*x^2 + b*x + a #environment global
  
  while((abs((hx(alpha) - falpha)) > eps))
  {
    #choice of l, m and C new values
    if(alpha <= m)
    {
      if(phi.fm >= falpha){
        u <- m; phi.fu <- phi.fm
        m <- alpha; phi.fm <- falpha
      }else{
        l <- alpha; phi.fl <- falpha
      }
    }else{
      if(phi.fm >= falpha){
        l <- m; phi.fl <- phi.fm
        m <- alpha; phi.fm <- falpha
      }else{
        u <- alpha; phi.fu <- falpha
      }
      
    }
    
    
    #refitting the function
    y <- ((l - m)*(m - u)*(u - l))
    a <- (phi.fl*m*u*(u - m) + phi.fm*u*l*(l - u) + phi.fu*l*m*(m - l))/y
    b <- (phi.fl*(m^2 - u^2) + phi.fm*(u^2 - l^2) + phi.fu*(l^2 - m^2))/y
    c <- -(phi.fl*(m - u) + phi.fm*(u - l) + phi.fu*(l - m))/y
    
    #quadratic interpolation
    
    alpha <- -b/(2*c)
    falpha <- phi.f(alpha)
    ncf <- ncf + 1
    
  }
  message("Method: Quadratic Interpolation. Number of calls of the objective function: ", ncf)
  return(alpha)
}

f <- function(x){
  0.5/sqrt(1+x^2)-sqrt(1+x^2)*(1-0.5/(1+x^2))+x
  }

quadraticinterpolation(f)

## Method: Quadratic Interpolation. Number of calls of the objective function: 12

## [1] 0.6831249

#Penentuan lower, upper, dan eps subjektif
quadraticinterpolation(f, l = 0, u=1, eps=1e-6)

## Method: Quadratic Interpolation. Number of calls of the objective function: 12

## [1] 0.6831249

#Nilai y
0.5/sqrt(1+0.6831249^2)-sqrt(1+0.6831249^2)*(1-0.5/(1+0.6831249^2))+0.6831249

## [1] 0.2977925

curve(f,xlim=c(0,3), ylim = c(0,0.4), col='purple',lwd=2)
abline(h=0)
abline(v=0)
points(x = 0.6831249, y = 0.2977925, pch = 19)
text(0.7, 0.32, "(0.683, 0.297)")

Latihan 5.9 (Library “Remotes”)

f2 <- function(x)
  {
  0.5/sqrt(1+x^2)-sqrt(1+x^2)*(1-0.5/(1+x^2))+x
  }

quadraticinterpolation(f2, l = 0, u = 1)

## Method: Quadratic Interpolation. Number of calls of the objective function: 12

## [1] 0.6831249

curve(f2,xlim=c(0,3), ylim = c(0,0.4), col='purple',lwd=2)

0.5/sqrt(1+0.6831249^2)-sqrt(1+0.6831249^2)*(1-0.5/(1+0.6831249^2))+0.6831249

## [1] 0.2977925

points(x = 0.6831249, y = 0.2977925, pch = 19)
text(0.7, 0.32, "(0.683, 0.297)")

Kesimpulan

Berdasarkan fungsi manual dan fungsi dari library remotes menghasilkan nilai optimum yang sama yaitu (0.683, 0.297).