Newton's Method & Error Analysis

In summary, the conversation discusses using Newton's Method to solve an equation and how to estimate the absolute error in the solution. It suggests directly approximating the error using a formula and using the difference between the solutions accurate to 3 and 4 significant figures. It is found that these approximations are similar.
  • #1
Physics197
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Homework Statement



(i) Use Newton’s Method and apparent convergence
to solve x ln(x) = 5 accurate to 3 and 4 significant figures. Start out with x0 = 2. (ii) Directly
approximate the absolute error on f, i.e. _f = f(x) − f(˜x). (iii) Use the difference between
the 4 significant figures and 3 significant figures results for x and the error formula to estimate
_f. You should find that they approximate each other.

Homework Equations



_f = ef'(x)

The Attempt at a Solution



I understand (i), I got 3.77 & 3.769.

For (ii), I don't understand what f(x) and f(~x) are. I know x is the measured value and ~x is the true value, but what are they in this case?

For (iii) do we take the difference to be the absolute error and the multiple it by the derivative at the true value?
 
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  • #2
x is the solution you get using Newton's method and ~x is the "true" value. You can't know ~x, that's why they say "approximate".
 

FAQ: Newton's Method & Error Analysis

1. What is Newton's Method?

Newton's Method is a mathematical algorithm for finding the roots of a function. It is a type of iterative method, meaning it uses a sequence of repeated calculations to approach a more accurate solution.

2. How does Newton's Method work?

First, an initial guess for the root of the function is made. Then, using the derivative of the function, the algorithm calculates a new approximation for the root by drawing a tangent line to the function at the initial guess and finding where it intersects the x-axis. This process is repeated until the desired level of accuracy is achieved.

3. What is the benefit of using Newton's Method?

Newton's Method is a fast and efficient way to find the roots of a function, especially compared to other methods such as bisection or regula falsi. It also allows for relatively easy implementation and can handle a wide range of functions.

4. What is error analysis in the context of Newton's Method?

Error analysis in Newton's Method involves examining the accuracy of the calculated solution by comparing it to the actual root of the function. This can help determine the level of precision and number of iterations needed for the algorithm to produce an acceptable result.

5. What are the limitations of Newton's Method?

Newton's Method may not converge or may converge to the wrong root if the initial guess is too far from the actual root, if the function has multiple roots, or if the function has a steep slope or sharp turns. Additionally, it can be computationally expensive for certain types of functions.

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