Mar 25, 2003 · In order to use Newton's method, we need to be able to calculate the derivative of a function at some point: f'(x1). Sometimes you don't know the derivative; what can you do then? What you can do is estimate the derivative by looking at the change in the function near x1: pick some other point x2 close to x1, and estimate the derivative as

(8 points) This problem is focused on optimization using open methods with convergence criteria based on relative error. For this problem you are required to build separate MATLAB functions for the secant method and Newton’s method to approximate a root r satisfying f (r) = 0 for a given function f .

Error Estimation • To estimate the relative error, we can base it on the true value of root. If our guess is in doubt the error estimate may not be appropriate. • Therefore, we require an error estimate that is not contingent on prior knowledge of the root. One way to do this is by estimating an approximate percent relative error as in

Newton-Raphson Method New estimate is the root of a line tangent to at Ü å,𝑖 Slope of at Ü å,𝑖 is the derivative at Ü å,𝑖: ′ Ü å,𝑖= Δ Δ = Ü å,𝑖 Ü å,𝑖− Ü å,𝑖+1 Solving for the new root estimate: Ü å,𝑖+1= Ü å,𝑖− Ü å,𝑖 ′ Ü å,𝑖

Define approximate percentage relative error as †a = fl fl fl fl fl xnew r ¡xold r xnew r fl fl fl fl fl £100% where xnew r is the estimated root for the present iteration, and xold r is the estimated root from the previous iteration. Define true percentage relative error as †t = fl fl fl fl xt ¡xr xt fl fl fl fl£100% 5

Newton-Raphson Method This is the most commonly used root finding method. If the initial guess at the root is xi , a tangent can be extended from the point [x i, f (x i)]. The point where this tangent crosses the x axis usually represents an improved estimate of the root.

a) Use the bisection method of finding roots of equations to find the depth xto which the ball is submerged under water. Conduct three iterations to estimate the root of the above equation. b) Find the absolute relative approximate error at the end of each iteration, and the number of significant digits at least correct at the end of each ...

Why Root Finding? •Engineering applications: Predict dependent variable (e.g., temperature, force, voltage) given independent variables (e.g., time, position) •Focus on finding real roots

Open methods employ different formulas to predict the root. If the initial guess at the root is xi, a tangent can be extended from the point [xi, f (xi)]. The point where this tangent represents an root.

Apr 16, 2013 · Main Points are: Newton-Raphson Method, Nonlinear Equation, Bracketing Methods, Category of Open Methods, Slope of Function, Improved Estimate of Root, Initial Guess of Root, Absolute Relative Approximate Error