Question

Fixed Point In Exercises 23-26, approximate the fixed point of the function to two decimal places. [A fixed point \(x_{0}\) of a function \(f\) is a value of \(x\) such that \(\left.f\left(x_{0}\right)=x_{0} .\right]\) $$ f(x)=\cot x, \quad 0

Short answer

The exact fixed point \(x_0\) will depend on the iteration process and the initial guess. Since this is a numerical method, the answer is an approximation. Multiple solutions may exist depending on where the process converges.

Step-by-step solution

Initialize the problem

The problem is to find the fixed point \(x_{0}\) of the function \(f(x) = \cot x\), where \(0 < x < \pi\).

Initial guess

Choose an initial guess, \(x_{1}\), somewhere within the range (0, \(\pi\)). The midpoint, which is \(\frac{\pi}{2}\), can be a reasonable choice.

Fixed-point iteration

Apply the fixed-point iteration using the formula \(x_{n+1} = f(x_n)\). Substituting \(f(x) = \cot x\) into this formula gives \(x_{n+1} = \cot(x_n)\). Use this to generate a sequence of iterates.

Convergence

Stop the iteration when the difference between successive iterates is less than a small pre-specified tolerance (say 0.01), or after a maximum number of iterations has been reached. The value of \(x_{n+1}\) is the desired fixed point.

Verification

Compute \(f(x_{n+1})\) and verify that this value is approximately equal to \(x_{n+1}\) within the specified decimal places to confirm that the found value is indeed a fixed point of the function.

Key concepts

Cotangent Function

Understanding the cotangent function, often written as \text{cot}(x), is critical for solving fixed point iteration problems where this function is involved. The cotangent function is the reciprocal of the tangent function and is defined as the ratio of the adjacent side to the opposite side in a right-angled triangle, or more formally, as \text{cot}(x) = \frac{1}{\text{tan}(x)} = \frac{\text{cos}(x)}{\text{sin}(x)}.

Function \text{cot}(x) is periodic with a period of \( \pi \) and is undefined whenever the sine function equals zero, which occurs at integer multiples of \( \pi \). Within the interval \( 0 < x < \pi \) mentioned in the exercise, \text{cot}(x) is positive, decreasing, and most importantly, continuous, which provides a favorable condition for fixed point iteration to find a solution. An important aspect to consider when initializing with an initial guess \( x_{1} \) is that the function should not be near a discontinuity to ensure effective convergence.

Convergence

The concept of convergence is essential in iterative methods, including fixed point iteration. Convergence refers to the property that an iterative process moves closer to a specific value, known as the limit, with each iteration. In the fixed point iteration context, the sequence produced by iteratively applying the function \( f(x) \) must approach a fixed point \( x_{0} \) as \( n \) approaches infinity.

Convergence can be monotonic, where the sequence solely moves in one direction towards the limit, or it can oscillate before settling down to the fixed point. The speed of convergence is another important aspect and relates to how quickly the iterates reach the desired accuracy. Convergence is assessed by the difference between successive iterates. If this difference falls below a chosen tolerance level, then the process is stopped, and the last iterate is considered a good approximation of the fixed point. However, if convergence cannot be achieved within the specified number of iterations, it suggests a problem with the initial guess, the function, or the iterative method itself.

Iterative Methods

In the realm of mathematics, iterative methods are approaches to find approximations to solutions of various problems when exact solutions are difficult or impossible to obtain. These methods generate sequences that can converge to the exact solution or a very close approximation. Fixed point iteration is one such method that falls under this category.

For the implementation of fixed point iteration, one continuously applies the function \( f \) to an initial guess \( x_{1} \) to produce a sequence \( x_{2}, x_{3}, \) and so on, where \( x_{n+1} = f(x_{n}) \). This method hinges on the mathematical foundation that under certain conditions, specifically if function \( f \) is continuous and meets other criteria related to the derivative, \( x_{n} \) will converge to a fixed point if such a point exists within the interval being considered.

For the fixed point iteration to provide a good solution, several things need to be ensured:

• Good choice of the initial value within the domain of function \( f \).
• The function must be well-behaved in the neighborhood of the fixed point, implying that it shouldn't have any abrupt changes or discontinuities.
• An appropriate level of tolerance for stopping the iteration must be set to balance between accurate results and computational efficiency.

The effectiveness of iterative methods, including the fixed point iteration, greatly depends on these considerations, which are partly addressed in the steps for solving the exercise.