Initial guess in Newton's method

Click For Summary

Discussion Overview

The discussion revolves around finding an appropriate initial guess for Newton's method, focusing on the accuracy of the guess and alternative strategies to improve it. Participants explore theoretical and practical aspects of selecting initial values for this numerical method.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the need for Newton's method if a high accuracy initial guess is already known, suggesting that this would yield the solution directly.
  • Another participant emphasizes that while they do not know the exact solution, they are aware of the interval containing it, implying that any initial guess within this interval could potentially work.
  • A different viewpoint suggests that the choice of initial guess may not significantly impact the outcome, as a random guess could still lead to a solution in many cases.
  • Some participants propose using graphical methods to visualize the function and identify potential roots, although this does not guarantee fewer iterations in Newton's method.
  • It is noted that there are no strict rules for selecting an initial guess, but certain methods exist for specific cases, such as polynomials, to help bound the roots.
  • Concerns are raised about the possibility of divergence in Newton's method if the function has a problematic shape, recommending limits on iterations and trying different initial guesses if necessary.

Areas of Agreement / Disagreement

Participants express differing opinions on the necessity and effectiveness of various strategies for selecting an initial guess, indicating that there is no consensus on a singular approach or rule for this process.

Contextual Notes

Some limitations are acknowledged, such as the dependence on the specific characteristics of the function and the potential for divergence in certain cases, which may affect the choice of initial guess.

Maged Saeed
Messages
123
Reaction score
3
How to find the Initial guess for Newton method with high accuracy ??

Is there a way rather than using mean value theorem [which is used to test whether there is a solution on a closed interval to the equation] ?
 
Physics news on Phys.org
That seems odd to me. A high accuracy initial guess seems to me an initial guess which is very near to the solution. But if you find such a thing, then you have the solution with a high accuracy. So why should you use Newton's method then?
 
I don't don't the solution yet , But I know the interval in which the solution lies .

Sorry for my English.

:)
 
Maged Saeed said:
I don't don't the solution yet , But I know the interval in which the solution lies .

Sorry for my English.

:)
My point is, it doesn't matter what point you choose. If you are lucky enough(which you are with a high probability because troubles are not very probable), that initial guess will be good enough to give you the solution with desirable accuracy in a good number of steps.
But if you want to have an educated guess, you can try to draw(or imagine!) the graph of the function and see where are the zeroes. This way you can have a very good guess but it doesn't garuantee that the steps will be fewer because sometimes you get far from the solution and reach it again. So just make a guess.
 
  • Like
Likes   Reactions: Maged Saeed
Okay , Thanks
 
In general, I think there are no rules. For special cases, there may be methods to bound the roots to an interval. For polynomials, there are several rules to help locate the roots. See http://en.wikipedia.org/wiki/Properties_of_polynomial_roots This may also help if you have a polynomial approximation to the function you are working with.
 
There really isn't a ``rule" for which ##x##-value to pick for the intial ``guess" in Newton's Method. Just look at the function and use some analytic techniques (like FactChecker mentioned) to try to reason where the roots might be, then pick the nearest integer around there.
 
Be aware that Newton's method can diverge if the function has a bad shape. For those functions, it can take several tries to get an answer. So put a limit on the number of iterations and start again with a different initial guess when the limit is exceeded.
 

Similar threads

Graduate How to use the Newton-Raphson method while keeping the sum constant?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Understanding Euler Method: Finding Initial Condition of y(0)=1
  • · Replies 7 ·
Replies
7
Views
5K
Undergrad Make Intelligent Initial Guesses for Newton-Raphson Programmatically
  • · Replies 16 ·
Replies
16
Views
4K
Graduate Can Newton's method work with an approximated integral
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Quasilinear Equation but with non-zero initial condition?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Finding the time for the first shock for a quasilinear first order PDE
  • · Replies 4 ·
Replies
4
Views
4K
Undergrad Why Lagrange’s method of solving Pp + Qq=R works?
  • · Replies 3 ·
Replies
3
Views
1K
MHB Determine the position using an iteration method
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad The algorithms involved in finding a solution with numerical methods
  • · Replies 14 ·
Replies
14
Views
3K
Undergrad Error bounds for Newton's Method
  • · Replies 13 ·
Replies
13
Views
3K