Finding the Iterative Function to Solve x3+x-1000=0

[bb.minhtri]

Homework Statement

The iterative method is used to find the approximate root of the equation x³ + x - 1000 = 0 in [9, 10]. What is the suitable iterative function?

Homework Equations

The Attempt at a Solution

How to find the iterative function and is there any conditions for one?
Thanks for helps.

 

[ehild]

The iterative procedure is to calculate the next approximation of x from a function of the previous approximation: x_i+1=f(x_i). 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. ehild

 

[bb.minhtri]

The iterative procedure is to calculate the next approximation of x from a function of the previous approximation: x_i+1=f(x_i). 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.

The second one works ^^ Thank you very much. Your explanation is very clear:D

 

[ehild]

Using cubic root in an iteration process is not too nice. You can find an other method without that. Hint: write x^3-1000 in the form (x-10)(x^2+10x+100), and isolate x from the x-10 factor.

ehild