How can you find approximations to the zeros of a function?

Question

Emma Aldridge · Accepted Answer

Use Newton's method to recursively define sequences whose limits are zeros...

If #f(x)# is a continuous, differentiable function then we can usually find its zeros using Newton's method: Given an approximation #a_i# to a zero of #f(x)#, a better one is given by the formula: #a_(i+1) = a_i - f(a_i)/(f'(a_i))# We can use this formula to recursively define a sequence: #a_0, a_1, a_2,...# Then the limit of the sequence is a zero of #f(x)#. By choosing different values for the initial term #a_0#, the resulting sequence will tend to other zeros of #f(x)#. This method is both easy to apply and generally quite effective with polynomial functions. It also works with both real and complex zeros.

Christopher Agresta · Answer

undefined One common method to find approximations to the zeros of a function is the numerical method known as the bisection method. Here's how it works:

1. Choose two initial points, $ a $ and $ b $, such that the function has different signs at these points.
2. Calculate the midpoint $ c $ between $ a $ and $ b $, $ c = \frac{a + b}{2} $.
3. Evaluate the function at the midpoint $ c $.
4. Determine the subinterval where the sign of the function changes (i.e., where the function changes from positive to negative or vice versa).
5. Replace either $ a $ or $ b $ with $ c $, depending on which subinterval contains a zero.
6. Repeat steps 2-5 until the desired level of accuracy is achieved or until a predetermined number of iterations is reached.

The bisection method guarantees convergence to a zero of the function if the initial interval contains exactly one zero and the function is continuous on that interval.

Another commonly used method is the Newton-Raphson method (or Newton's method), which involves iteratively improving an initial guess for the zero of a function using the function's derivative. This method tends to converge faster than the bisection method but requires the function to be differentiable and the derivative to be computable.

Other numerical methods for finding zeros include the secant method, the method of false position, and root-finding algorithms like Brent's method. Each method has its advantages and limitations depending on the specific characteristics of the function and the desired level of accuracy.