Use Newton's method with (a) x0=0.2 and (b) x0=3.0 to find a zero of f(x)=x sinx . Discuss the d

Use Newton's method with (a) x0=0.2 and (b) x0=3.0 to find a...

Key Concepts

Newton's Method

Newton's method is an iterative technique for finding successive approximations to the roots (or zeroes) of a real-valued function. It utilizes the recurrence relation x??? = x? - f(x?)/f'(x?), meaning that at each iteration, the function is approximated by its tangent line at the current estimate and the x-intercept of this tangent is used as the next estimate.

Rate of Convergence (Quadratic Convergence)

When the conditions are ideal — namely, when the function is continuously differentiable near the root and the initial guess is sufficiently close to the actual root — Newton's method exhibits quadratic convergence. This means that once the iterates are in a neighborhood of the root, the number of correct digits approximately doubles at every step, leading to a very rapid approach to the zero.

Importance of Initial Guess

The initial guess in Newton's method critically affects the convergence behavior. A starting value close to the root typically ensures rapid convergence due to the method's quadratic nature, whereas a starting value far from any root can lead to slow convergence, convergence to the wrong root, or even divergence. The behavior emphasizes the sensitivity of the method to the choice of the initial estimate.

Transcript

00:01 In problem 48 we want to use newton's method to get an approximate route for the function f of x using two initial two different initial values for part a the first initial value x node equals 0 .2 let's first recall noughton's method newton's method is used to find a root using iterations and using an estimate for the initial value.

00:39 After the initial value, the next root xn plus 1 equals the previous root xn minus f of x n divided by f dash of xn.

00:54 This means we need to find f dash of x.

00:59 We can get f dash of x by differentiating f of x.

01:03 We have two multiplied functions.

01:06 We set the first function and we multiply by the differential of the second function which is cosine x plus we set the second function sine x and we multiply it by the differential of the first function which is 1 then it equals x cosine x plus sine x now we can apply newton's method equation to get x1 x1 equals x node minus f of x node divided by f -dash of x node and for better calculations and clear calculations we can make a table for our iterations number of iteration the value of x the value of x the value of x the value of f -dash of x the initial value x equals 0 .2 as given in the question f of x node we substitute here by equals x node or x equals 0 .2 to get f of x node which equals 0 .0397 and to get f dash of x node we substitute here by x equals 0 .2 which gives 0 .395 using newton's method here we can get x1 which equals 0 .2 minus 0 .0397 divided by 0 .395 which gives x1 equals 0 .099333.

03:10 We can see that x1 is not equal to x0 or x0.

03:16 This means we continue iterations.

03:20 The second iteration is to get f of x1 as previous.

03:27 F of x1 equals 0 .0098 f dash of x1 equals 0 .198 using newton's method equation we get x2 x2 equals 0 .0 4995 again x2 is not equal to x1 we continue iterations we get f of x2 f of x2 equals 0 .0025 f dash of x2 equals 0 .0991 then we continue iterations we get x3 using the newton's method equation equals 0 .02478 again x3 is not equal to x2 we continue iterations we get f of x3 0 .0 .0661 f dash of x equals if dash of x3 equals 0 .04955 we continue to calculate x4 equals 0 .01 to 4 f of x we continue iterations as long as x4 is not equals to x3 or x n plus 1 is not equal to x n we get f of x4 equals 0 .0 .0015 f dash of x node equals 0 .0...

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