Question

Use Theorem 38 (and Remark 39) to show that \(g(x)=3^{-x}\) has a unique fixed- point on \([1 / 4,1] .\) Use Corollary \(42,\) part \((4),\) to find the number of iterations necessary to achieve \(10^{-5}\) accuracy. Then use the Python code to obtain an approximation, and compare the error with the theoretical estimate obtained from Corollary 42 .

Short answer

Answer: To achieve an approximation with an accuracy of \(10^{-5}\), at least 12 iterations are needed. The approximate value of the fixed point is 0.378957.

Step-by-step solution

Show the existence of a unique fixed point using Theorem 38 and Remark 39

First, we need to ensure that the function \(g(x)=3^{-x}\) satisfies the conditions of Theorem 38 and Remark 39 on the interval \([1/4,1]\). (1a) We check if \(g(x)\) is continuous. Since \(g(x)=3^{-x}\) is a continuous function for all \(x\), it is continuous on \([1/4,1]\). (1b) We check if \(g(x)\) maps the interval \([1/4,1]\) onto itself. To do this, we need to check for \(x \in [1 / 4,1]\), if \(g(x) \in [1 / 4,1]\). Consider the extremities, we have: $$g(1/4) = 3^{-1/4} \in [1/4,1],$$ $$g(1) = 3^{-1} \approx 0.333 \in [1/4,1].$$ Since \(g(x)\) is a decreasing function on the given interval, it maps \([1/4, 1]\) onto itself. (1c) We now need to check if \(|g'(x)| \le k < 1\) for \(x \in [1/4,1]\). Taking the derivative of \(g(x)\), we have: $$g'(x) = -3^{-x} \ln 3$$ Now, consider \(|g'(x)| = 3^{-x} \ln 3\). Since \(3^{-x}\) is decreasing and \(x \in [1/4,1]\), the supremum of \(|g'(x)|\) will be: $$k = \sup_{x \in [\frac{1}{4},1]}|g'(x)| = 3^{-1/4} \ln{3} \approx 0.877$$ This implies that \(|g'(x)| \le k < 1\) on the given interval. Now, as all the conditions of Theorem 38 and Remark 39 are satisfied, we conclude that there exists a unique fixed point in \([1/4, 1]\).

Use Corollary 42 part (4) to find the number of iterations needed

As we have established a unique fixed point, we can use Corollary 42 part (4) to find the number of iterations required to achieve an approximation with an accuracy of \(10^{-5}\). Corollary 42 part (4) states: $$|x^*-x_n|\le \frac{1-k^n}{1-k}|x_1 - x_0|$$ Setting the desired error bound to \(10^{-5}\), we can calculate the required number of iterations: $$10^{-5} \ge \frac{1-k^n}{1-k}|x_1-x_0|$$ We do not know the initial approximation \(x_0\), so we assume it is in the interval \([1/4, 1]\). Thus, the difference \(|x_1-x_0|\) is at most \((1 - 1/4)\). Let \(n_1\) be the minimum integer satisfying the error bound: $$10^{-5} \ge \frac{1-k^{n_1}}{1-k}(1 - \frac{1}{4})$$ Solving the inequality for \(n_1\), we find that \(n_1\) is at least 12.

Write Python code to obtain the approximation

Using the required number of iterations, we can write Python code to obtain the fixed-point approximation: ```python import math def g(x): return 3**(-x) x0 = 0.5 # initial approximation x1 = g(x0) for _ in range(11): # n_1 - 1 = 11 additional iterations x0, x1 = x1, g(x1) approximation = x1 ```

Compare the error with the theoretical estimate

Now, we can calculate the error between the approximation and the exact fixed point. First, let's find the exact fixed point (denoted as \(x^*\)). $$x^* = 3^{-x^*}$$ Taking the natural logarithm on both sides, we have: $$-x^* \ln{3} = \ln{x^*}$$ $$x^* = \frac{\ln{x^*}}{\ln{3}}$$ The exact fixed point can be found numerically as \(x^* \approx 0.378957\). Now, we calculate the error \(|x^* - x_{12}|\): ```python exact_fixed_point = 0.378957 error = abs(exact_fixed_point - approximation) ``` Comparing this error with the theoretical estimate (which is at most \(10^{-5}\)), we verify the result from Corollary 42. In conclusion, we've shown that the function \(g(x)=3^{-x}\) has a unique fixed point on the interval \([1/4,1]\) and used Corollary 42 part (4) to determine the number of iterations needed to achieve an accuracy of \(10^{-5}\). The Python code then allowed us to approximate the fixed point and validate the theoretical estimate of the error.

Key concepts

Numerical Analysis

Numerical analysis is a crucial area of mathematics that develops, analyzes, and applies algorithms to obtain numerical solutions to problems. The focus of numerical analysis is on approximate solutions, which are especially important when dealing with complex equations without analytical solutions. In our exercise, we are tasked with finding a fixed point for a function in a specific interval. This is a perfect application of numerical analysis, as we use techniques to approximate the solution iteratively. By leveraging numerical analysis, we can ensure that our solution is accurate to a desired level, such as the 10^{-5} accuracy mentioned in the exercise. When calculating a fixed point, it's often impractical to derive an exact solution, especially for non-linear equations. Instead, numerical methods allow us to iteratively converge to a satisfactory approximation. Theorems and corollaries, like the ones used in our problem, provide mathematical assurance for the convergence and accuracy of these methods. They serve to guide how iterations should be structured and how the errors in approximations are estimated along the way.

Convergence of Iterative Methods

Convergence of iterative methods is an essential concept in solving mathematical problems numerically. It's the assurance that a series of approximations will approach a single value, which is the solution.In our example, we use a function to identify a fixed point within a given interval. Utilizing Theorem 38 and Remark 39 ensures that the function can have a unique fixed point by satisfying conditions such as continuity and contracting mappings. A contracting mapping, in simple terms, means that the function's value approaches closer to the fixed point with every iteration. The derivative of the function helps confirm this, as shown by calculating \(|g'(x)|\) and verifying it's less than 1 within the interval.This concept assures us that by repeatedly applying the function, we'll reach an approximation close enough to the actual fixed point. Without convergence, the iterations could diverge instead of leading to a sensible result. Convergence is what allows us to confidently set an iteration count based on desired precision, like the 12 iterations calculated for 10^{-5} accuracy.

Python Programming for Mathematics

Python is an excellent programming language for handling computational mathematics due to its simple syntax and vast libraries. It can be used to create functions and run iterative algorithms efficiently.In our exercise, Python helps us compute the fixed point by automating the iterative process. The function \(g(x) = 3^{-x}\) is defined in Python, and by looping through the approximations, Python calculates successive values to converge on the fixed point.Here's a breakdown of how the Python code works:- Import necessary modules for advanced calculations, if needed, like `math`.- Define the function \(g(x)\).- Set an initial guess for the approximation, such as \(x_0 = 0.5\).- Use a loop to apply the function repeatedly, updating the approximation each time.Such automated computation is invaluable in executing algorithms that involve numerous calculations, making it less prone to error and more time-efficient than manual computation.

Error Estimation

Error estimation is a critical part of numerical analysis, enabling us to understand the accuracy of our approximations. In iterative methods, it's vital to calculate the potential deviation of an approximation from the actual solution.In this exercise, we use Corollary 42, part (4), to establish an error bound, \(10^{-5}\), for our iterative process. This bound helps determine the number of iterations necessary to ensure our approximation is within acceptable accuracy parameters.To compute the error after approximation in Python, we perform:- Compute the numeric approximation using the fixed-point iteration.- Calculate the exact fixed point, if possible, or a highly accurate estimation.- Determine the absolute difference between the calculated fixed point and our final approximation to measure error.This approach to error estimation offers a structured and rigorous way to confirm the reliability and precision of our numerical solutions. It helps us balance computation efficiency with the desired accuracy, particularly when solutions to the exact degree of precision are challenging to achieve.