Use Newton's Method to find out an approximation to the solution to cos x = x which lies in the interval [0,2]. Determine the approximation to six decimal places.
Solution
Firstly note that we weren't given an initial guess. However, we were given an interval in which to look. We will utilize this to get our initial guess. As noted down above the general rule of thumb in these cases is to take the initial approximation to be the midpoint of the interval. Thus, we'll utilize x0 = 1 as our initial guess.
Next, recall that we ought to have the function in the form f ( x ) = 0 . Thus, first we rewrite the equation as,
cos x - x = 0
Now we can write down the general formula for Newton's Method. Doing this will frequently simplify up the work a little so generally it's not a bad idea to do this.
xn +1 = xn - (cos x - x /(- sin x -1))
Now let's get the first approximation.
x1 = 1 -( cos (1) -1/- sin (1) -1) = 0.7503638679
At this point we have to point out that the phrase "six decimal places" does not mean only get x1 to six decimal places & then stop. Rather than it means that we continue till two successive approximations agree to six decimal places.
Given that stopping condition we obviously have to go at least one step farther.
x 2 = 0.7503638679 - (cos (0.7503638679) - 0.7503638679/- sin (0.7503638679) -1)
= 0.7391128909
We've got the approximation to 1 decimal place. Let's accomplish another one, leaving the details of the computation to you.
x3 = 0.7390851334
We've got it to three decimal places. We'll require another one.
x4 = 0.7390851332
And now we've got two approximations that agree to 9 decimal places and therefore we can stop. We will suppose that the solution is approximately x4 = 0.7390851332 .