Question

The process described in part ( \(c\) ) of Exercise 22 can of course also be applied to functions that map \((0, \infty)\) to \((0, \infty)\). Fix some \(\alpha>1\), and put $$ f(x)=\frac{1}{2}\left(x+\frac{\alpha}{x}\right), \quad g(x)=\frac{\alpha+x}{1+x} $$ Both \(f\) and \(g\) have \(\sqrt{\alpha}\) as their only fixed point in \((0, \infty)\). Try to explain, on the basis of properties of \(f\) and \(g\), why the convergence in Exercise 16, Chap. 3 , is so much more rapid than it is in Exercise 17. (Compare \(f^{\prime}\) and \(g^{\prime}\), draw the zig-zags suggested in Exercise 22.) Do the same when \(0<\alpha<1\).

Short answer

In summary, the convergence of Exercise 16, Chap. 3, is more rapid than in Exercise 17 because the function \(f(x)\) is a contraction mapping. It has a derivative \(f'(x)\) with values between 0 and 1/2 on the entire interval \((0, \infty)\). On the other hand, the function \(g(x)\) has a derivative \(g'(x)\) that depends on how far \(x\) is from the fixed point \(\sqrt{\alpha}\), which results in varying converging speeds depending on the starting point \(x_0\).

Step-by-step solution

Calculate the derivatives of \(f(x)\) and \(g(x)\)

To determine the derivatives of given functions, we will apply basic differentiation rules. For function \(f(x)\): $$ f(x) = \frac{1}{2}(x+\frac{\alpha}{x}) $$ Differentiate \(f(x)\) with respect to \(x\): $$ f'(x) = \frac{1}{2}(1 - \frac{\alpha}{x^2}) $$ For function \(g(x)\): $$ g(x)=\frac{\alpha+x}{1+x} $$ Differentiate \(g(x)\) with respect to \(x\): $$ g'(x) = \frac{(1)(1+x) - (\alpha+x)(1)}{(1+x)^2} = \frac{1 - \alpha}{(1+x)^2} $$

Analyzing and comparing the behavior of \(f'(x)\) and \(g'(x)\)

We found: $$ f'(x) = \frac{1}{2}(1-\frac{\alpha}{x^2}) $$ and $$ g'(x) = \frac{1-\alpha}{(1+x)^2} $$ For \(\alpha > 1\), we notice that: - Both \(f'(x)\) and \(g'(x)\) are continuous functions on the interval \((0, \infty)\). - For \(f'(x)\), \(\frac{\alpha}{x^2} > 0\) implies that \(1-\frac{\alpha}{x^2} < 1\). Thus, \(0 \leq f'(x) < \frac{1}{2}\) on \((0, \infty)\). This means that \(f(x)\) is a contraction mapping. - For \(g'(x)\), we can see that the derivative function depends on how far \(x\) is from the fixed point \(\sqrt{\alpha}\). This results in a differing converging speed depending on the starting point \(x_0\). Therefore, the convergence in Exercise 16 is more rapid due to \(f(x)\) being a contraction mapping. In contrast, \(g(x)\) has different convergence behavior depending on the starting value of \(x_0\).

Analyzing and comparing when \(0 < \alpha < 1\)

When \(0 < \alpha < 1\), we have: For \(f'(x)\): $$ f'(x) = \frac{1}{2}(1-\frac{\alpha}{x^2}) $$ Since \(\alpha < 1\), we get \(1-\frac{\alpha}{x^2} > 0\). Therefore, \(0 < f'(x) < \frac{1}{2}\) on \((0, \infty)\), and \(f(x)\) remains a contraction mapping. For \(g'(x)\): $$ g'(x) = \frac{1-\alpha}{(1+x)^2} $$ Since \(\alpha < 1\), \(1-\alpha > 0\), and thus the derivative \(g'(x)\) becomes positive on \((0, \infty)\). However, the convergence behavior also depends on how far \(x\) is from the fixed point \(\sqrt{\alpha}\), similar to the previous scenario. In conclusion, depending on the starting point, the convergence of \(g(x)\) can be rapid, but overall, the convergence of \(f(x)\) is more rapid because it remains a contraction mapping.

Key concepts

Fixed Point

A fixed point in mathematical analysis is a value that remains unchanged when a function is applied to it. It's a fundamental concept in various branches of mathematics, including iterative methods and dynamical systems. Specifically, if you have a function \( f \), then a number \( x \) is called a fixed point of \( f \) if \( f(x) = x \).

The fixed point theorem states that under certain conditions, every continuous function has at least one fixed point within a given domain. A typical example is the intersection point in the graphical representation of a function where the curve \( y = f(x) \) and the line \( y = x \) meet. Fixed points have practical applications in areas like economics, computer science, and physics, where they are used to find equilibrium states, solve recursive relations, and model stable systems.

Contraction Mapping

A contraction mapping is a function that brings points closer together, making it a valuable tool for solving equations and understanding dynamical systems. In more technical terms, a function \( f \) is a contraction mapping on an interval \( I \) if there exists a constant \( 0 \leq k < 1 \) such that for any two points \( x \) and \( y \) in \( I \) the following inequality holds:

\[ \|f(x) - f(y)\| \leq k\|x - y\| \]

This means that the distance between \( f(x) \) and \( f(y) \) is strictly less than the distance between \( x \) and \( y \). The essence of contraction mappings is that they progressively move closer to a single fixed point with each iteration, making them highly useful for computing approximations and proving existence and uniqueness of solutions.

The contraction mapping principle, also known as the Banach fixed-point theorem, guarantees that a contraction mapping on a complete metric space has exactly one fixed point. This principle is especially useful in numerical analysis, where it can ensure the convergence of an iterative process towards a solution.

Derivatives

Derivatives represent one of the core pillars of calculus and are essential for understanding rates of change. The derivative of a function at a given point quantifies how sensitive the function's output is to small changes in its input, which is often phrased as the 'slope' of the function at that point.

Mathematically, if \( f \) is a function that depends on \( x \) then the derivative of \( f \) with respect to \( x \) is denoted by \( f'(x) \) or \( \frac{df}{dx} \) and defined as the limit:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \]

Derivatives have many applications, from calculating velocity in physics to finding the maximum profit in economics. When it comes to functions with a fixed point, the value of the derivative at the fixed point can give us insights into the behavior of iterative sequences and the stability of the system. For instance, a derivative with an absolute value less than 1 at the fixed point suggests that the function is a contraction mapping, as seen in the given exercise's function \( f(x) \).