We explore the convergence of Newton's method for f(x)=x^3-3 x^2+2 x Determine which of the thre

We explore the convergence of Newton's method for f(x)=x^3-3...

Key Concepts

Newton's Method

Newton's Method is an iterative algorithm for finding successively better approximations to the roots (or zeroes) of a real-valued function. It uses the function's derivative to generate the next approximation from the current guess, following the formula x??? = x? - f(x?)/f'(x?), and is known for its rapid quadratic convergence near the root if the initial guess is sufficiently close.

Convergence Criteria

Convergence Criteria refer to the conditions under which an iterative method such as Newton's Method will converge to a solution. This includes requirements on the smoothness of the function, the behavior of its derivative, and the choice of the initial guess. Analyzing these factors is important for ensuring that the method converges to the correct root and does so efficiently.

Basins of Attraction

Basins of Attraction are regions in the domain of a function where all initial guesses will eventually converge to the same root under an iterative method. In the context of Newton's Method, different regions of the function’s domain may lead to convergence to different roots, and understanding these regions helps in predicting which root will be approached from a given starting value.

Transcript

00:01 In program 70, we want to test newton's method conversions for this function, and we have already the exit solution of this function, where the zeros are 0, 1 and 2.

00:21 We will test newton's method using many initial values.

00:27 Newton's method depends on iterations based on these initial values.

00:31 The next root of the iteration x n plus 1 equals the previous root x n minus f of x n divided by f dash of x n.

00:44 We have f of x n which is the main function in the problem.

00:49 We can evaluate f dash of x because we will use it.

00:53 F dash of x the differentiation of x cubed which is 3x squared minus 2 multiplied by 3 equals 6 x plus 2 we can start by the initial guess of part a x node equals 0 .54 for clear calculations we make our calculations in a table the number of iteration the value of x the value of x the value of f dash of x we can evaluate f of x by substituting in the main function here can evaluate f dash of x of any number by substituting here the initial value was given in the problem 0 .54 we can get f of x node as we said from here by substituting x equals 0 .54 we get the value of the function 0 .36 26 the value of the differential we can get it from here by substituting x equals 0 .54 equals minus 0 .3652.

02:20 This means we can get x1 by using this equation here x1 equals x node minus f of x node divided by f dash of x node we have these three values in the table by substitution we get x1 equals 1 .5333 we can can see that x1 is not equal to x node then we must continue iterations f of x1 equals minus 0 .382 the value of f dash of x1 equals minus 0 .14755 by using newton's method equation we can get x2 which equals minus 1 .053.

03:20 This sound doesn't sound good because we have we have increased here then we have decreased in the same direction but we will continue iterations until we see what happens the value of f f f of x2 equals minus 6 .6 004 the value of f dash of x2 equals 11 .645 you using nettance method equation, x3 equals minus 0 .4862.

04:06 It sounds good to now.

04:10 We continue iterations, we get f of x4 minus 1 .79665...

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