Question

Consider the function \(g(x)=x^{2}+\frac{3}{16}\). (a) This function has two fixed points. What are they? (b) Consider the fixed point iteration \(x_{k+1}=g\left(x_{k}\right)\) for this \(g .\) For which of the points you have found in (a) can you be sure that the iterations will converge to that fixed point? Briefly justify your answer. You may assume that the initial guess is sufficiently close to the fixed point. (c) For the point or points you found in (b), roughly how many iterations will be required to reduce the convergence error by a factor of \(10 ?\)

Short answer

Answer: It takes approximately 5 iterations.

Step-by-step solution

Finding the Fixed Points

To find the fixed points, we need to set \(g(x) = x\) and solve for x. Let's solve the equation: $$ x^2 + \frac{3}{16} = x $$ Subtract x from both sides to obtain a quadratic equation: $$ x^2 -x + \frac{3}{16} = 0 $$ Now, we can use the quadratic formula to find the fixed points: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where \(a = 1\), \(b = -1\), and \(c = \frac{3}{16}\). Plugging the values of a, b, and c in the formula, we get: $$ x = \frac{1 \pm \sqrt{(-1)^2 - 4(1)(\frac{3}{16})}}{2(1)} $$ Simplifying, $$ x = \frac{1 \pm \sqrt{\frac{13}{16}}}{2} $$ So, the two fixed points are: $$ x_1 = \frac{1 + \sqrt{\frac{13}{16}}}{2} \quad \text{and} \quad x_2 = \frac{1 - \sqrt{\frac{13}{16}}}{2} $$

Checking Convergence of Fixed Points

In this step, we will determine which of the fixed points found in step 1, if any, guarantee convergence using the fixed point iteration method. Enforcing the condition that \(|g'(x^*)| < 1\) for convergence, we need to find the first derivative of \(g(x)\) and evaluate it at both fixed points: $$ g'(x) = \frac{\mathrm{d}}{\mathrm{d}x} \Big(g(x)\Big) = \frac{\mathrm{d}}{\mathrm{d}x} \Big( x^2 + \frac{3}{16} \Big) = 2x $$ Now let's evaluate \(g'(x)\) at both fixed points, \(x_1\) and \(x_2\): $$ g'(x_1) = 2\left(\frac{1 + \sqrt{\frac{13}{16}}}{2}\right) = 1 + \sqrt{\frac{13}{16}} > 1 $$ $$ g'(x_2) = 2\left(\frac{1 - \sqrt{\frac{13}{16}}}{2}\right) = 1 - \sqrt{\frac{13}{16}} < 1 $$ From the inequalities above, we can conclude that the fixed point iteration will converge to \(x_2\).

Estimating the Number of Iterations for Convergence

Finally, we will roughly estimate the number of iterations needed to reduce the convergence error by a factor of 10 for the converging fixed point \(x_2\). Since \(|g'(x_2)| < 1\), we know that the error decreases by a factor of \(|g'(x_2)|\) in each iteration. Therefore, we can denote the decrease factor as: $$ \text{Decrease factor} = |g'(x_2)| = \Big|1 - \sqrt{\frac{13}{16}}\Big| $$ Because we want the error to decrease by a factor of 10, we can use the following equation: $$ (\text{Decrease factor})^n \le \frac{1}{10} $$ Now we will solve for n: $$ \Big(1 - \sqrt{\frac{13}{16}}\Big)^n \le \frac{1}{10} $$ Taking the logarithm of both sides, we obtain: $$ n \cdot \log{\Big(1 - \sqrt{\frac{13}{16}}\Big)} \le \log{\frac{1}{10}} $$ Divide both sides by \(\log{\Big(1 - \sqrt{\frac{13}{16}}\Big)}\): $$ n \ge \frac{\log{(\frac{1}{10})}}{\log{\Big(1 - \sqrt{\frac{13}{16}}\Big)}} $$ By calculating the right-hand side, we get \(n \ge 4.49\). Since n has to be an integer, we must round up to get the smallest integer greater than or equal to 4.49. Therefore, it takes approximately 5 iterations to reduce the convergence error by a factor of 10.

Key concepts

Convergence of Iterations

The idea of convergence in iterations is fundamental in numerical methods, particularly when looking for solutions to equations. Convergence refers to the process of getting closer to a fixed point—an exact solution—after each iteration or repetition of an algorithm.

In the context of the exercise, we focus on the fixed point iteration, which is a method used to find an approximative solution to equations. It's a sequential process, where we start with an initial guess and apply a function repeatedly until the process converges to a point. However, the key is that not all points will lead to convergence. The convergence depends on the behavior of the function at that point.

A common criterion for convergence in fixed point iterations is to check whether the absolute value of the derivative of the function at the fixed point, |g'(x*)|, is less than one, as it suggests that applying the function repeatedly will pull values towards the fixed point, reducing the error each time. This was used in the provided solution to select which fixed point from part (a) the iterations will converge to in part (b). Understanding when and why iterations converge is crucial in the study of numerical analysis because it saves computational resources and helps in understanding the stability of the methods used.

Numerical Methods

Numerical methods are algorithms or techniques designed to solve mathematical problems that may be challenging or impossible to address analytically. They are often the only feasible approach when dealing with complex equations, irregular shapes in geometry, or systems with many degrees of freedom.

Fixed point iteration, the method used in the given problem, is just one example of a numerical method. Numerical methods also include bisection methods, Newton’s method, and the Runge-Kutta methods for differential equations, among others.

The application of numerical methods frequently requires a balance between accuracy and computational efficiency. The step-by-step solution not only demonstrates the practical application of a numerical method, the fixed point iteration, but also underscores the importance of understanding the underlying theory to ensure convergence and hence the accuracy of the results.

Educators often emphasize the importance of checking and estimating the error in numerical methods. This is seen in the final part (c) of the solution, which involves estimating the number of iterations required to achieve a desired level of accuracy. Being able to determine the rate at which errors decrease with each iteration is a pivotal skill in applying numerical methods efficiently.

Quadratic Equation

A quadratic equation is a second-order polynomial equation of the form ax² + bx + c = 0. Finding the roots, or solutions, of a quadratic equation is a fundamental task in algebra. The solutions to the quadratic equation can be found analytically using the quadratic formula, as shown in step 1 of the exercise.

The quadratic equation in the exercise appears in a context that might be unusual for some students—the determination of fixed points—but understanding its roots is just as crucial as in more conventional applications. It illustrates how algebraic knowledge is applied in numerical analysis and other advanced levels of mathematics.

The quadratic formula used in the solution provides the fixed points that inform the rest of the exercise. It's worth noting that every quadratic equation has two solutions – in this case, representing two potential fixed points. However, as the solution to part (b) shows, not every fixed point is valid for the convergence of iterations in numerical methods. This integration between different areas of mathematics highlights the beauty and complexity of the subject.