Question

Let \(g(x)=2 x-c x^{2}\) where \(c\) is a positive constant. Prove that if the fixed-point iteration \(p_{n}=g\left(p_{n-1}\right)\) converges to a non-zero limit, then the limit is \(1 / c\)

Short answer

Answer: If convergence occurs, the limiting value of the fixed-point sequence is \(1/c\).

Step-by-step solution

Determine the fixed points of g(x)

To find the fixed points of \(g(x)\), we need to solve the equation \(g(x) = x\). This means $$x = 2x - cx^2.$$ Now we should solve this for x: $$cx^2 - x = 0.$$ Factor out x to get $$x(cx - 1)=0.$$So, x = 0 or x = 1/c are the fixed points of g(x).

Verify conditions for convergence

Now that we have found the fixed points, we want to determine if the fixed-point iteration \(p_{n} = g\left(p_{n-1}\right)\) converges to one of these fixed points. To do this, we need to compute the first derivative of \(g(x)\): $$g'(x) =\frac{d}{dx}(2x - cx^2) = 2 - 2cx.$$Next, we will evaluate \(g'(x)\) at the two known fixed points.

Evaluate g'(x) at the fixed points

Evaluate \(g'(x)\) at x = 0: $$g'(0) = 2 - 2c(0) = 2.$$Evaluate \(g'(x)\) at x = 1/c: $$g'(1/c) = 2 - 2c(1/c) = 2 - 2 = 0.$$Now we have the values of \(g'(x)\) at the fixed points. To ensure convergence, \(|g'(x)| < 1\) for the fixed point.

Determine whether convergence occurs at a non-zero fixed point

We see that at x = 0, \(|g'(x)| = 2\), which is not less than 1, so the fixed-point iteration does not converge to the fixed point x = 0. But at x = 1/c, \(|g'(x)| = 0\), which is less than 1, indicating that the fixed-point iteration converges to the non-zero fixed point x = 1/c.

Conclude the proof

Based on the investigation of the fixed points of \(g(x)\) and the first derivative \(g'(x)\), we can conclude that if the fixed-point iteration \(p_{n} = g\left(p_{n-1}\right)\) converges to a non-zero limit, then the limit is \(1 / c\).

Key concepts

Numerical Analysis

Numerical analysis is a branch of mathematics that deals with algorithms for solving mathematical problems computationally. It often involves iterative methods where solutions are approximated through successive steps. A classic example is the fixed-point iteration, where a function's point, at which it intersects the identity line, is approximated.

Given an initial guess, the method generates a sequence of values that ideally converge to the true fixed point. In our exercise, we examine the function g(x) = 2x - cx^2 and use numerical techniques to determine whether the iteration p_{n} = g(p_{n-1}) converges, and to what value. Understanding numerical analysis concepts is critical for ensuring the safe use of algorithms and interpreting results accurately in real-world applications.

Fixed-Point Theorem

The fixed-point theorem, specifically the Banach fixed-point theorem, asserts that a contraction mapping on a complete metric space has a single fixed point. This theorem provides the conditions under which fixed-point iteration converges.

For the function g(x) under discussion, fixed points are the solutions to the equation g(x) = x. Fixed-point iteration is used to approximate these solutions. While the theorem guarantees the existence of a fixed point for certain types of functions, it is crucial to verify the convergence conditions in practice. In our function g(x), only one of the fixed points (specifically, x = 1/c) satisfies the convergence criteria outlined by the fixed-point theorem.

Convergence Criteria

Convergence criteria in the context of fixed-point iteration involve conditions that iterated values must satisfy to ensure that they converge to a fixed point. The most common criterion is checking the absolute value of the derivative of the function at the fixed point, denoted as |g'(x)|.

For convergence, it is required that |g'(x)| < 1 for the fixed point. In our exercise, we calculated |g'(0)| = 2 and |g'(1/c)| = 0. Since only |g'(1/c)| meets the criterion, we conclude that the iteration converges only to x = 1/c. This analysis is vital for students to understand why certain numerical methods work and under what circumstances they can reliably expect them to produce accurate results.