DAPOS said:Homework Statement
Derive cube root formula using Newton-Rhapson's method. x - y^3 = 0.
Homework Equations
xn + 1 = xn - f(xn)/f'(xn)
The Attempt at a Solution
I know that the solution is (2y + (x/y^2))/3
I tried using implicit differentiation and stuff but I can't get this out. Any tips?
LCKurtz said:I think your use of x and y is confusing you. If you want the cube root of ##n## you might solve ##f(x) = x^3 - n## for its roots using your above formula. No implicit differentiation needed and you can switch the names of the variables at the end if you want to.
Newton-Rhapson's Cube Root method is an algorithm used to find the cube root of a number. It is an iterative process that uses calculus to approximate the root of a function.
The method begins with an initial guess for the cube root of the given number. It then calculates the slope of the function at that point and uses it to find a better approximation for the root. This process is repeated until the desired level of accuracy is achieved.
One advantage is that it is a very efficient and fast method for finding cube roots. It also has a high degree of accuracy and can be used for complex numbers as well.
Yes, the method can fail to converge if the initial guess is too far from the actual root or if there are multiple roots for the given function. In addition, it requires knowledge of calculus to understand and implement.
This method is commonly used in fields such as engineering, physics, and finance to solve equations and find the roots of complex functions. It is also used in computer programming to solve equations and optimize algorithms.