Use Newton's method with (a) x0=-1.1 and (b) x0=2.1 to find a zero of f(x)=x^3-3 x-2 . Discuss t

Use Newton's method with (a) x0=-1.1 and (b) x0=2.1 to find...

Key Concepts

Convergence Behavior Analysis

Understanding the differences in convergence behavior based on starting values is critical. When the initial guess is near a simple root with a well-behaved derivative, Newton's method converges rapidly due to its quadratic nature. However, if the guess is far from the root or near a region where the derivative is small or has incorrect sign, the convergence can be slow or erratic, illustrating the importance of selecting appropriate initial values.

Quadratic Convergence

Quadratic convergence describes the property of an algorithm where the error decreases proportionally to the square of the previous error near the root. In the context of Newton's method, once the iterates are sufficiently close to the actual root and the derivative is nonzero, the method typically exhibits quadratic convergence, meaning that the number of correct digits roughly doubles with each iteration.

Newton's Method

Newton's Method is an iterative numerical technique used to approximate the roots of a real-valued function. It works by taking an initial guess and repeatedly applying the formula x??? = x? - f(x?)/f'(x?) to generate successively better approximations. The method relies on evaluating both the function and its derivative, making it efficient for functions that are differentiable and where a good initial guess is known.

Initial Guesses and Basins of Attraction

The choice of the initial guess in Newton's method is crucial because it determines which root the iteration will converge to, if convergence occurs at all. The initial guess places the process within a particular basin of attraction, and starting points closer to the actual root generally lead to faster convergence. Conversely, poor initial guesses may lead to slow convergence, divergence, or even converging to a different root.

Transcript

00:01 In problem 49 we want to use newton's method to get an approximate route for this function using two different initial guess part a the initial guess equals minus 1 .1 first let's recall newton's method newton's method is an iterative approach to get an approximate route for any function using an initial guess the next iteration or the next iteration or the next root x n plus 1 equals the brevest root x n minus f of x n divided by f dash of x n this means we need to get f dash of x of x equals 3x squared minus 3 and applying newton's method we can get x1 equals x node minus f of x node divided by f dash of and to make our calculations clear we can make a table for all calculated values number of iteration the initial value the value of x which is given minus 1 .1 the value of f of x and the value of f dash of x the value of f of x can be calculated from this equation we substitute by x equals minus 1 .1.

01:47 Which gives f of x node equals minus 0 .031 and f dash of x we can get its value from here by substituting x equals x node or x equals minus 1 .1 which gives 0 .63.

02:07 Using nathan's method equation we can get x1 we have x node we have now f of x node we have f of x node we have f dash of x node by substituted we can get x1 equals minus 1 .0508 we can see that x1 is not equal to x node to some sort of accuracy which means we continue iterations we get f of x1 as previous equals minus 0 .0787 and we get f dash of x1 equals 0 .3 125 we use the equation of naitans method to get x2 equals minus 1 .0256 we can see that x2 is not equal to x1 for 3 or 4 decimal places we continue iterations we get f of x2 minus 0 .0 0 .0498 f dash of xx equals 0 .156 .6 we use nenten's method equation to get x3 which equals minus 1 .01286 x3 is not equal to x2 for two decimal places then we should continue iterations f of x3 equals minus 0 .0 .025 f f f b...

Please give Ace some feedback