Newton Vs Euler Method of Finding Square Roots

Newton's method, also known as Newton-Raphson method, is an iterative numerical technique used to approximate the roots of a real-valued function, including square roots. When applied to finding the square root of a positive number, it allows us to obtain a progressively more accurate estimate of the square root through a series of iterative steps. Let's say we want to find the square root of a positive number, 'x.' We'll call the initial estimate of the square root 'a.' The goal is to improve this initial guess until we get an approximation of the square root that is close enough to the actual value. The Newton's method can be explained in the following steps: 1. Choose an initial estimate: Start by choosing an initial guess 'a' for the square root of 'x'. This initial guess can be any positive number. 2. Update the estimate: Improve the estimate by using the following formula: a_new = (a + x / a) / 2 Here, 'a_new' is the new estimate, 'a' is the previous estimate, and 'x' is the number for which we want to find the square root. 3. Repeat the process: Continue to apply the formula in Step 2, using the latest estimate as the input for the next iteration. Keep repeating this process until the estimate converges to the desired accuracy. As you perform more iterations, the value of 'a_new' will get closer and closer to the actual square root of 'x'. The more iterations you perform, the more accurate your estimate becomes. It's important to note that Newton's method usually converges very quickly for most positive numbers, meaning it reaches a good approximation with just a few iterations. However, it may not converge for some specific inputs or can converge to a value that's not the square root but another root of the function. By following this iterative process, Newton's method provides an efficient and effective way to approximate the square root of a given number. Example: Step 1: Choose an initial estimate (a) = 3 Step 2: Update the estimate using the formula: a_new = (a + x / a) / 2 where x is the number for which we want to find the square root (in this case, x = 9). First iteration: a_new = (3 + 9 / 3) / 2 a_new = (3 + 3) / 2 a_new = 6 / 2 a_new = 3 Second iteration: a_new = (3 + 9 / 3) / 2 a_new = (3 + 3) / 2 a_new = 6 / 2 a_new = 3 Since the new estimate (3) is the same as the previous estimate, we can conclude that the square root of 9 is approximately 3. Let's verify this: 3 * 3 = 9 So, the square root of 9 is indeed 3, and Newton's method provided us with an accurate approximation in just two iterations.