# Initial guess in Newton's method

• Maged Saeed
In summary, finding an initial guess with high accuracy for Newton's method can be achieved through various methods such as using the mean value theorem or using an educated guess based on the graph of the function. However, there are no specific rules for choosing an initial guess and it may require multiple attempts to find a good estimate.
Maged Saeed
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] ?

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.

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.

## 1. What is an initial guess in Newton's method?

The initial guess in Newton's method is the starting point or estimate for the root of a function. It is used to begin the iterative process of finding the root through successive approximations.

## 2. How is the initial guess chosen in Newton's method?

The initial guess can be chosen arbitrarily or based on prior knowledge about the root. It is usually chosen to be close to the actual root of the function to ensure convergence of the algorithm.

## 3. What happens if the initial guess is not close to the root in Newton's method?

If the initial guess is not close to the root, the algorithm may fail to converge or may converge to a different root of the function. It is important to choose a reasonable initial guess to ensure accurate results.

## 4. Can the initial guess be updated in Newton's method?

Yes, the initial guess can be updated in each iteration of the algorithm. This can help improve the accuracy of the result and ensure convergence to the correct root.

## 5. Is there a way to improve the initial guess in Newton's method?

One way to improve the initial guess is to use a different method, such as the bisection method, to obtain a better estimate for the root. This can then be used as the initial guess in Newton's method for faster convergence.

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