ehild said:The iterative procedure is to calculate the next approximation of x from a function of the previous approximation: xi+1=f(xi). The iterative process can converge in a range of x where the derivative |df/dx |<1.
You can try the ways: x=1000-x3 or x=(1000-x)^1/3.
Which one works? And you can find other iterative functions for this equation.
To find the iterative function, we can use the Newton-Raphson method. This method involves repeatedly plugging in initial guesses for the root of the function and using the derivative of the function to refine the guess until we get a more accurate approximation.
The Newton-Raphson method is an iterative numerical method used to find the roots of a function. It involves using initial guesses and the derivative of the function to refine the guess until a satisfactory approximation for the root is found.
You can use the error formula to determine the error between the actual root and your approximation. If the error is small enough, then your initial guess is close enough to the root and you can stop iterating.
Yes, you can use any initial guess for the Newton-Raphson method. However, choosing a good initial guess can help speed up the convergence of the method.
The accuracy of the Newton-Raphson method depends on the initial guess and the function itself. In general, the method can provide very accurate approximations for the root of a function.