Newton-Raphson Method

Newton-Raphson method is an open method. in this method we start with only one initial approximation x(i), the approximation x(i+1) is the x-intercept of the tangent line to the graph of f at ( x(i), f ( x(i+1))). The approximation x(i+2) is the x-intercept of the tangent line to the graph of f at ( x(i+1), f ( x(i+1))) and so on.  From fig. 1 uses the definition of the slope of a function, at x=x(i) 

                                           

Equation (1) is called the Newton-Raphson formula for solving nonlinear equations of the form f(x)=0.  So starting with an initial guess, x(i), one can find the next guess, x(i+1), by using Equation (1).  One can repeat this process until one finds the root within a desirable tolerance.

Maple Setup

[> NewtonsMethod(x^3 + 4*x - 10, x = 1);

                          1.556773264

[> NewtonsMethod(x^3 + 4*x - 10, x = 2, output = sequence);

2, 1.625000000, 1.558650066, 1.556774723, 1.556773264, 1.556773264

[> NewtonsMethod(x^3 + 4*x - 10, x = 0, view = [0 .. 3, DEFAULT], output = animation);