ROOT FINDING METHODS

Question 1

is a numerical method, or algorithm, for finding a value x would make f(x) = 0, for a given function f.

Such an x is called a “root” of the function f.

Answer 1

root-finding algorithm

Question 2

it uses iteration, producing a sequence of numbers that
hopefully converge towards the root as a limit.

They require one or more initial guesses of the root as starting values, and then each iteration of the algorithm produces a successively more accurate approximation to the root. Since the iteration must be stopped at some point these methods produce an approximation to the root, not an exact solution.

Answer 2

Numerical root-finding methods

Question 3

it can be a real or complex number

The methods discussed here locate real roots.

Answer 3

root

Question 4

whenever the value of **f(x) **transitions from negative to
positive and vice-versa, as the value of x is incremented at a close interval.

Answer 4

Graphically, the root is found

Question 5

The importance of determining a root?

Answer 5

it is a critical value for a function.

The root can either be the value to want or avoid in some applications.

Question 6

2 sets of methods to get the root(s):

Answer 6

A. Open Methods or “Single Guess” (Fixed Point Iteration Methods)
B. “Two Guess” (Bracket Methods and Secant Methods) Methods

These methods initially start from a “guess” value(s) then using a particular equation (in each method) the next value for the “guess” is determined. This is the iterative or repeated re-substitution process. The process stops when the last “guess” is already the “root”.

Question 7

the iterations may be stopped when:

Answer 7

1. The prescribed number of iterations has been completed.
2. The value of f(x) = 0, when the “guess” value is substituted as “x”.
3. The absolute error criteria have been met. f(x) ≤ Ea
4. The two consecutive iterations have the same “guess” value
   4.1. For two “guess” methods – when both “guess” values become equal in the same iteration.