Question

Explain how the iteration formula for Newton's method works.

Short answer

Answer: Newton's method finds an approximation to the root of a real-valued function by iteratively improving the guess using the iteration formula derived from the tangent line to the function at the current approximation. The iteration formula is: x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}. The method starts with an initial guess and repeats this process until converging to the actual root or reaching a specified tolerance level or maximum number of iterations. The convergence rate depends on the initial guess and the behavior of the function and its derivative around the root.

Step-by-step solution

Understand the problem

Newton's method is used to find an approximation to a root of a given function, f(x). The goal is to iteratively improve the approximation of the root until it converges to the actual root.

Write the tangent line equation

To derive the iteration formula for Newton's method, we will use the tangent line to the function f(x) at a given point. The tangent line equation is given by: y = f'(x_n)(x-x_n) + f(x_n) where x_n is the current approximation of the root, f(x_n) is the function value at x_n, and f'(x_n) is the derivative of f(x) at x_n.

Find the x-intercept of the tangent line

The x-intercept of the tangent line is the point at which the tangent line crosses the x-axis. At this point, y = 0, so we can find the x-intercept by setting y = 0 and solving for x: 0 = f'(x_n)(x-x_n) + f(x_n) We want to find the value of x for which this equation holds.

Derive the iteration formula

To derive the iteration formula, solve for x in the tangent line equation: x = x_n - \frac{f(x_n)}{f'(x_n)} This is the formula for the next approximation of the root, denoted as x_{n+1}. We can use this formula iteratively, starting from an initial approximation, to find successively better approximations of the root of the function.

Implement Newton's method

To implement Newton's method, follow these steps: 1. Choose an initial guess for the root, denoted as x_0. 2. Calculate the next approximation of the root using the iteration formula: x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}. 3. Repeat step 2 until the difference between successive approximations falls below a specified tolerance level or the maximum number of iterations has been reached.

Understand convergence properties

Newton's method converges to the root quickly when given a good initial guess, and it may fail to converge or converge to a different root when given a poor guess. In general, the convergence rate depends on the choice of the initial guess and the behavior of the function f(x) and its derivative f'(x) around the root.