![]() Thus, thanks to this similarity, one might use #x=1# or #x=-2# as guesses to start Newton's Method with f(x). This is similar to another function #g(x) = x^2 + x - 2#, whose roots are #x=1# and #x=-2#. For example, suppose one is presented with the function #f(x) = x^2 +x -2.5#. The method is constructed as follows: given a function #f(x)# defined over the domain of real numbers #x#, and the derivative of said function ( #f'(x)#), one begins with an estimate or "guess" as to where the function's root might lie. Newton's Method is a mathematical tool often used in numerical analysis, which serves to approximate the zeroes or roots of a function (that is, all #x: f(x)=0#). You can apply this same logic to whatever cube root you'd like to find, just use #x^3 - a = 0# as your equation instead, where #a# is the number whose cube root you're looking for. You can see that with only 8 iterations, we've obtained an approximation of #root 3(3)# which is correct to 8 decimal places! Then we substitute each previous number for #x_n# back into the equation to get a closer and closer approximation to a solution of #x^3 - 3 = 0#. Now, we pick an arbitrary number, (the closer it actually is to #root3(3)# the better) for #x_0#. Substituting for #f(x) = x^3 - 3# gives us: Now we will recall the iterative equation for Newton-Raphson. Therefore,įor the Newton-Raphson method to be able to work its magic, we need to set this equation to zero. And let's say that #x# is the cube root of #3#. ![]() ![]() Let's say we're trying to find the cube root of #3#. So, we need a function whose root is the cube root we're trying to calculate. The Newton-Raphson method approximates the roots of a function. ![]()
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