If you have the error formula for the root finding method shown in the image where xn is the approximation of r at step n, en = xn - r is the error and cassi is a value between xn and r, how do you show that the method converges to r. Also what would the convergence rate be?
If you have the error formula for the root finding method shown in the image where xn is the approximation of r at step n, en = xn - r is the error and cassi is a value between xn and r, how do you show that the method converges to r.
Also what would the convergence rate be?
![The equation shown is:
\[ e_{n+1} = \frac{e_n^2}{f(\xi_n) + 1}, \quad n \geq 0 \]
This formula is used in numerical methods, typically for iterative processes where \( e_{n+1} \) represents an error term at iteration \( n+1 \), and \( e_n \) represents the error at the current iteration \( n \). The function \( f(\xi_n) \) is evaluated at some intermediate point \( \xi_n \) within the domain, which affects the convergence of the sequence. The equation highlights how the error at the next iteration is computed based on the current error, a function of an intermediate point, and an adjustment factor of 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2Fc478f835-6cdd-49b9-adc1-61e3bf43e276%2Fn1c1b8_processed.png&w=3840&q=75)
To demonstrate that a root-finding method converges to a root "r," you would typically need to show two things:
Convergence Criteria: You would need to establish a condition that ensures that as you iterate the method, the sequence of approximations {xn} gets closer and closer to the actual root "r." This condition could be something like |xn - r| approaches zero as n (the number of iterations) increases.
Rate of Convergence: To determine the convergence rate, you would examine how quickly the method's error (en = xn - r) decreases with each iteration. Common methods include determining whether the error decreases linearly, quadratically, or at a different rate.
Different root-finding methods have different convergence rates, and the specific method you're using will influence the rate of convergence. For instance, the Newton-Raphson method often exhibits quadratic convergence when certain conditions are met.
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