R programming

Below is a modification of newtonraphson that plots intermediate results, analogous to fixedpoint_show above. Use it to investigate the roots of the following functions:

(a). cos(x) x using x0 = 1, 3, 6 (b). log(x) exp(x) using x0 = 2 (c). x3 x 3 using x0 = 0 (d). x3 7x2 + 14x 8 using x0 = 1.1, 1.2, . . . , 1.9 (e). log(x) exp(x) using x0 = 2.

# program spuRs/resources/scripts/newtonraphson_show.r # loadable spuRs function newtonraphson_show <- function(ftn, x0, xmin = x0-1, xmax = x0+1) { # applies Newton-Raphson to find x such that ftn(x)[1] == 0 # x0 is the starting point # subsequent iterations are plotted in the range [xmin, xmax]

# plot the function x <- seq(xmin, xmax, (xmax - xmin)/200) fx <- c() for (i in 1:length(x)) { fx[i] <- ftn(x[i])[1] } plot(x, fx, type = "l", xlab = "x", ylab = "f(x)", main = "zero f(x) = 0", col = "blue", lwd = 2) lines(c(xmin, xmax), c(0, 0), col = "blue")

# do first iteration xold <- x0 f.xold <- ftn(xold) xnew <- xold - f.xold[1]/f.xold[2] lines(c(xold, xold, xnew), c(0, f.xold[1], 0), col = "red")

# continue iterating while user types "y" cat("last x value", xnew, " ") continue <- readline("continue (y or n)? ") == "y" while (continue) { xold <- xnew; f.xold <- ftn(xold) xnew <- xold - f.xold[1]/f.xold[2] lines(c(xold, xold, xnew), c(0, f.xold[1], 0), col = "red") cat("last x value", xnew, " ") continue <- readline("continue (y or n)? ") == "y" } return(xnew)

}