demo(plotmath)
par(mar = c(4, 4, 2, 0.1))
plot(rnorm(100), rnorm(100),
  xlab = expression(hat(mu)[0]), ylab = expression(alpha^beta),
  main = expression(paste("Plot of ", alpha^beta, " versus ", hat(mu)[0])))

par(mar = c(4, 4, 2, 0.1))
x_mean <- 1.5
x_sd <- 1.2
hist(rnorm(100, x_mean, x_sd),
  main = substitute(
    paste(X[i], " ~ N(", mu, "=", m, ", ", sigma^2, "=", s2, ")"),
    list(m = x_mean, s2 = x_sd^2)
  )
)

install.packages("rgl")

library(rgl)
data(volcano)
z <- 2 * volcano # Exaggerate the relief
x <- 10 * (1:nrow(z)) # 10 meter spacing (S to N)
y <- 10 * (1:ncol(z)) # 10 meter spacing (E to W)
zlim <- range(z)
zlen <- zlim[2] - zlim[1] + 1
colorlut <- terrain.colors(zlen,alpha=0) # height color lookup table
col <- colorlut[ z-zlim[1]+1 ] # assign colors to heights for each point
open3d()
rgl.surface(x, y, z, color=col, alpha=0.75, back="lines")

مشتق پذیر

 R

X0

h

R=x0+h

f<- function(x) {
  exp(2 * x) - x - 6
}


curve(f, col = 'blue', lty = 2, lwd = 2, xlim=c(-5,5), ylim=c(-5,5), ylab='f(x)')
abline(h=0)
abline(v=0)

newton <- function(f, tol=1E-12,x0=1,N=20) {

        h <- 0.001

        i <- 1; x1 <- x0

        p <- numeric(N)

        while (i<=N) {

                df.dx <- (f(x0+h)-f(x0))/h

                x1 <- (x0 - (f(x0)/df.dx))

                p[i] <- x1

                i <- i + 1

                if (abs(x1-x0) < tol) break

                x0 <- x1

        }

        return(p[1:(i-1)])

}

f <- function(x) {

  exp(2 * x) - x - 6

}

p <- newton(f, x0=1, N=20)

 p

> p <- newton(f, x0=1, N=10)

>  p

[1] 0.9717930 0.9708719 0.9708700 0.9708700 0.9708700 0.9708700

> 

> p <- newton(f, x0=1, N=20)

>  p

[1] 0.9717930 0.9708719 0.9708700 0.9708700 0.9708700 0.9708700

> 

newton.raphson <- function(f, a, b, tol = 1e-5, n = 1000) {
  require(numDeriv) # Package for computing f'(x)
  
  x0 <- a # Set start value to supplied lower bound
  k <- n # Initialize for iteration results
  
  # Check the upper and lower bounds to see if approximations result in 0
  fa <- f(a)
  if (fa == 0.0) {
    return(a)
  }
  
  fb <- f(b)
  if (fb == 0.0) {
    return(b)
  }

  for (i in 1:n) {
    dx <- genD(func = f, x = x0)$D[1] # First-order derivative f'(x0)
    x1 <- x0 - (f(x0) / dx) # Calculate next value x1
    k[i] <- x1 # Store x1
    # Once the difference between x0 and x1 becomes sufficiently small, output the results.
    if (abs(x1 - x0) < tol) {
      root.approx <- tail(k, n=1)
      res <- list('root approximation' = root.approx, 'iterations' = k)
      return(res)
    }
    # If Newton-Raphson has not yet reached convergence set x1 as x0 and continue
    x0 <- x1
  }
  print('Too many iterations in method')
}

f<- function(x) {
  exp(2 * x) - x - 6
}

uniroot(f, c(.5, 1.5))

> uniroot(f, c(.5, 1.5))
$root
[1] 0.9708565

$f.root
[1] -0.000175188

$iter
[1] 6

$init.it
[1] NA

$estim.prec
[1] 6.103516e-05

> 

density

d <- density(mtcars$mpg)

plot(d, main="Kernel Density of Miles Per Gallon")

polygon(d, col="red", border="blue") 

cor

cov

> x <- mtcars[1:3]
> y <- mtcars[4:6]
> cor(x, y) 
             hp       drat         wt
mpg  -0.7761684  0.6811719 -0.8676594
cyl   0.8324475 -0.6999381  0.7824958
disp  0.7909486 -0.7102139  0.8879799
> 

cor.test 

 stats