Bisection method
Bisection Method Theorem :An equation f (x) = 0 , where f (x) is a real continuous function, has at least one root between x/ and xu if f(xl) .f(xu) < 0.What is the bisection method and what is ...
Newton raphson method for finding root
Newton Raphson Method
The Newton Raphson method is for solving equations of the form f(x) = 0. We make an initial guess for the root we are trying to find, and we call this initial guess x0.The sequence x0, x1, x2 ,x3, . . . generated in the manner described below should con-verge to the exact root.
To implement it analytically we need a formula for each approximation in terms of the previous one, i.e. we need xn+1 in terms of xn.
The equation of the tangent line to the graph y = f(x) at the point (x0, f(x0)) is
Its application to solving equations of the form f(x) = 0, as we now demonstrate, is called the Newton Raphson method.It is guaranteed to converge if the initial guess x0 is close enough, but it is hard to make a clear statement about what we mean by ‘close enough’ because this is highly problem specific. A sketch of the graph of f(x) can help us decide on an appropriate initial guess x0 for a particular problem.