METHODS Flashcards
uses one initial guess (x) for the root of the function
FIXED POINT ITERATION METHODS
Open methods employ a formula to predict the root.
Such a formula can be developed for simple fixed-point iteration (or, as it is also called, one-point iteration or successive substitution) by rearranging the function f (x) = 0 so that x is on the left-hand side of the equation: x = g(x)
SIMPLE FIXED-POINT ITERATION or METHOD OF SUCCESSIVE SUBSTITUTION (MOSS)
employ a formula to predict the root
Open methods
Newton-Raphson named after
Isaac Newton and Joseph Raphson
is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
Given f(x), here you would need to determine and use the first derivative f’(x) of the given function.
Newton-Raphson
In this method the number of iterations, are fewer than MOSS and by changing the initial or guess values, multiple real roots can be found.
In selecting the initial value, you may need to graph the function first to see the closest value to the root.
NEWTON-RAPHSON METHOD
a simplified version of the Secant Method
MODIFIED SECANT METHOD (MSM)
In this method the derivative of the function is no longer required.
However a perturbation fraction (δ) is needed. The value of δ usually takes the value of our absolute error requirement.
MODIFIED SECANT METHOD (MSM)
TWO INITIAL GUESSES METHODS
(BRACKET METHODS)
1. Bisection Method
2. Regula-Falsi or False Position Method (RFM)
(Not a Bracket method)
3. Secant Method (SM)
uses two initial guesses (a & b) that “bracket” the root of the function.
BRACKET METHODS
The rules are similar to the bisection method, except
for the iteration formula above and there is no need to
switch the values of a, only b is replaced by the value
of xr in the iterations.
Regula-Falsi or False Position Method (RFM)
is much powerful than bisection. It results in fewer numbers of iterations
Regula-Falsi or False Position Method (RFM)
not a bracketing method due to the fact that the root is not within the two initial guesses; however you still need two guesses.
Secant Method (SM)
is less complicated than the bracketing methods, and gives the root(s) in fewer iterations.
Secant Method (SM)
is most useful when the function involve does
not change signs after zero.
Secant Method (SM)
