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

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Homework Help Overview

The discussion revolves around finding a suitable iterative function to approximate the root of the equation x³ + x - 1000 = 0 within the interval [9, 10]. Participants explore different forms of iterative functions and the conditions for convergence.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the iterative method and its requirement for convergence, specifically the condition |df/dx| < 1. Various potential iterative functions are proposed, including x = 1000 - x³ and x = (1000 - x)^(1/3). There is also a suggestion to consider alternative methods that avoid using cubic roots.

Discussion Status

Some participants have shared their thoughts on the effectiveness of the proposed iterative functions, with one noting that the second function works well. Others are exploring different approaches and questioning the suitability of using cubic roots in the iteration process.

Contextual Notes

Participants are considering the implications of the iterative method and its convergence criteria, as well as the potential for alternative formulations of the problem. There is a focus on ensuring that the chosen iterative function meets the necessary conditions for convergence.

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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.
 
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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. ehild
 
Last edited:
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.

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

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