Week4–SecantMethod

A place to put your answers to assignment 2a. That assignment is as follows:

• Write a program to implement the Secant method.
• Test it, using x0=1 and x1=2 on:
• cos(x)-x
• log(x)-exp(-x)
• Compare it with the performance of Newton-Raphson on the same functions.
• Write a function to show a plot of the iterations of the algorithm.

Push it to your version of this repository as a knitted .Rmd or .md of .pdf file This assignment is due at midnight on Friday Feb 25th.

                      ## Secant Method Function 2

              ## Purpose: Finding roots of nonlinear equations

set.seed(2020)

cos.func <- function(x) { cos(x)-(x) } curve(expr = cos.func, col=‘green’, lwd=2, lty=2, xlim=c(0,3)) abline(h=0) abline(v=0)

                ## Root of function within interval (x=1.5)

x0 <- 1 x1 <- 2 a <- cos(x0)-x0 b <- cos(x1)-x1

                ## Draw Secant Line using determined points

curve(expr = cos.func, col=‘green’, lwd=2, lty=2, xlim=c(0,3)) abline(h=0) abline(v=0) points(x0, a, cex=1.25, pch=21, bg=‘green’, col=‘green’) points(x1, b, cex=1.25, pch=21, bg=‘green’, col=‘green’) segments(x0, a, b, col=‘red’, lwd = 2)

                        ## New Secant Line

x1 <- 1 x2 <- 2 a <- cos(x1)-x1 b <- cos(x2)-x2

curve(expr = cos.func, col=‘blue’, lwd=2, lty=2, xlim=c(0,3)) abline(h=0) abline(v=0) points(x1, a, cex=1.25, pch=21, bg=‘blue’, col=‘blue’) points(x2, b, cex=1.25, pch=21, bg=‘blue’, col=‘blue’) segments(x1, a, x2, b, col=‘red’, lwd = 2)

SMfzero(cos.func, 1, 2) secant.method(cos.func, 1, 2)

                    ## Test Using Newton-Raphson Method

secant.method <- function(f, x0, x1, to1 = 1e-9, n = 500) { for (i in 1:n) { x2 <- x1 - f(x1) / ((f(x1) - f(x0)) / (x1 - x0)) if (abs(x2-x1) < to1) { return(x2) } x0 <- x1 x1 <- x2 } } secant.method(cos.func, 1, 2)

                       ## Secant Method Function 2

func2 <- function(x) { log(x)-exp(-x) }

x0 <- 1 x1 <- 2 a <- log(x0)-exp(-x0) b <- log(x1)-exp(-x1)

               ## Draw Secant Line using determined points
            

curve(expr = func2, col=‘green’, lwd=2, lty=2, xlim=c(0,3)) abline(h=0) abline(v=0) points(x0, a, cex=1.25, pch=21, bg=‘green’, col=‘green’) points(x1, b, cex=1.25, pch=21, bg=‘green’, col=‘green’) segments(x0, a, b, col=‘red’, lwd = 2)

                ## New Secant Line

x1 <- 1 x2 <- 2 a <- log(x1)-exp(-x1) b <- log(x2)-exp(-x2)

curve(expr = func2, col=‘blue’, lwd=2, lty=2, xlim=c(0,3)) abline(h=0) abline(v=0) points(x1, a, cex=1.25, pch=21, bg=‘blue’, col=‘blue’) points(x2, b, cex=1.25, pch=21, bg=‘blue’, col=‘blue’) segments(x1, a, x2, b, col=‘red’, lwd = 2)

SMfzero(func2, 1, 3) secant.method(func2, 1, 3)