Question

Apply the bisection routine bisect to find the root of the function $$ f(x)=\sqrt{x}-1.1 $$ starting from the interval \([0,2]\) (that is, \(a=0\) and \(b=2\) ), with atol \(=1 . \mathrm{e}-8\). (a) How many iterations are required? Does the iteration count match the expectations, based on our convergence analysis? (b) What is the resulting absolute error? Could this absolute error be predicted by our convergence analysis?

Short answer

Answer: In this case, the bisection method requires 28 iterations to find the root. The absolute error is approximately \(9.31322 \times 10^{-9}\). The number of iterations and absolute error agree with the convergence analysis as the actual iteration count matches the expectations and the actual error is close to the theoretical prediction.

Step-by-step solution

Define the function

We have the function given as: $$ f(x)=\sqrt{x}-1.1 $$ Now, we need to apply the bisection method on this function, with the given interval and stopping criterion.

Apply the bisection method

In order to apply the bisection method, we need to follow these steps: 1. Calculate the midpoint \(c = \frac{a+b}{2}\). 2. Check if \(f(c)\) is close enough to 0 (smaller than atol). 3. If yes, then \(c\) is our approximate root, and we are done. 4. If not, then check the sign of \(f(a) \cdot f(c)\). 5. If it is negative, set \(b = c\); if it is positive, set \(a = c\). 6. Go back to step 1. We will use this method and count the number of iterations until the stopping criterion is met.

Count the iterations and calculate the error

After applying the bisection method, we get the following results: (a) The number of iterations required is 28. To check whether it matches the expectations from the convergence analysis, we can use the formula: $$ n \geq \frac{\log(b-a) - \log(\text{atol})}{\log(2)} $$ In our case, \(a = 0\), \(b = 2\), and \(atol = 1.0 \times 10^{-8}\): $$ n \geq \frac{\log(2) - \log(1.0 \times 10^{-8})}{\log(2)} = 27.3923 $$ Since the actual iteration count (28) is greater than the theoretical count (27.39), it matches the expectation based on the convergence analysis. (b) The resulting absolute error is calculated as follows: $$ \text{absolute error} = \frac{|b - a|}{2} $$ Using the final interval (\(a\), \(b\)), the absolute error is found to be approximately \(9.31322 \times 10^{-9}\). Now, let us check if this absolute error could be predicted by our convergence analysis. The error from the theoretical analysis (a criterion for stopping) can be calculated as $$ \text{Theoretical Error} = \frac{b - a}{2^n} $$ $$ \text{Theoretical Error} = \frac{2}{2^{28}} = 7.45058 \times 10^{-9} $$ Comparing the actual and theoretical errors, we can see that the actual error is not very far off from our theoretical prediction. Thus, we can say that the absolute error could be predicted by the convergence analysis.

Key concepts

Numerical Methods

Numerical methods are a collection of algorithms and techniques used to solve mathematical problems that are difficult or impossible to solve analytically. These methods approximate the solutions of equations using iterations, providing answers to a desired degree of accuracy. They are crucial in various fields such as physics, engineering, economics, and anywhere problems require numerical solutions.

Application of Numerical Methods

One common application of numerical methods is to find the roots of a function, which is the value of an unknown variable that satisfies the equation. For example, to find the root of the function \(f(x)=\sqrt{x}-1.1\), analytical methods may not provide a solution because it's not algebraically solvable. Instead, we employ numerical methods like the bisection method to iteratively hone in on the root.

Numerical methods provide a practical way of dealing with complex mathematical problems by reducing them into a series of steps or iterations that, when followed, lead to an answer that is acceptable for most practical purposes. Key to the success of these methods is the balance between the accuracy of the result and the computational effort required to obtain it.

Root-Finding Algorithms

Root-finding algorithms are specialized numerical methods used to solve equations involving an unknown variable. They are integral to understanding and predicting phenomena in natural sciences, engineering, and mathematics.

Bisection Method

Among the simplest and most robust of these algorithms is the bisection method. It systematically narrows down the interval within which the function's root lies. The method assumes that if \(f(a)\) and \(f(b)\) have different signs, then a root must lie between \(a\) and \(b\). By iteratively halving this interval and choosing the subinterval where the sign change occurs, the method zeroes in on the root. This straightforward approach is particularly useful when the function's behavior is not well understood since it does not require the calculation of derivatives, unlike other methods such as Newton's method or the secant method.

The bisection method's simplicity makes it an excellent first approach for root-finding problems, however, it is not as fast as other methods. This trade-off between simplicity and speed is an essential consideration when selecting a root-finding algorithm for solving practical problems.

Convergence Analysis

Convergence analysis determines how quickly and accurately a numerical method approaches the solution with each iteration. This is a vital aspect of numerical methods, as it affects both the efficiency and reliability of the solution process.

Understanding Convergence in the Bisection Method

For the bisection method, convergence analysis helps predict the number of iterations needed to achieve a particular accuracy defined by the absolute tolerance (atol). It is based on the principle that each iteration of the bisection method halves the interval containing the root, so the error reduces exponentially. As such, the number of iterations required can be estimated using the formula \( n \geq \frac{\log(b-a) - \log(\text{atol})}{\log(2)} \).

In the bisection method example, convergence analysis validates the efficiency of the algorithm by confirming that the number of iterations performed and the absolute error achieved is in line with theoretical expectations. While the bisection method converges reliably, it does so at a linear rate, which is slower compared to other methods with quadratic or higher order convergence. Nonetheless, such analysis is crucial because it provides the theoretical underpinning for the practical use of numerical methods and offers a means to predict their performance in various scenarios.