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Chapter 2: Problem 24

Find the fixed point of \(f(x)=x-0.002\left(e^{x} \cos (x)-100\right)\) in [5,6] using Steffensen's method.

Short Answer

Expert verified
The fixed point is approximately at 5.1808.

Step by step solution

01

Understanding Steffensen's Method

Steffensen's method is an iterative technique for finding a root or fixed point of a function. It is an accelerated form of the simple fixed-point iteration. The basic iteration formula is given by: \[ x_{n+1} = x_n - \frac{[f(x_n)]^2}{f(x_n + f(x_n)) - f(x_n)} \] Given the function \(f(x) = x - 0.002(e^x \cos(x) - 100)\), we're tasked to find a fixed point within the interval [5,6]. In this method, we iteratively improve our guess \(x_n\) until the difference between two consecutive approximations is sufficiently small.
02

Initial Guess

Choose an initial guess, say \(x_0 = 5.5\), within the interval [5,6]. This is a reasonable starting point because it lies at the midpoint of the interval. The goal is to refine this guess using Steffensen's iteration to approach the fixed point.
03

Evaluate Function and Derive Points

Calculate \(f(x_0)\), which is required to progress with Steffensen's formula. First calculate \(e^{x_0} \cos(x_0)\) and substitute into the function: \[ f(5.5) = 5.5 - 0.002(e^{5.5} \cos(5.5) - 100) \] Let's call the result \(F_0 = f(5.5)\). Then, calculate \(f(5.5 + F_0)\) for use in the iterative formula.
04

Applying Steffensen's Iteration Formula

Apply Steffensen's iteration to obtain a new approximation \(x_1\): \[ x_1 = x_0 - \frac{[F_0]^2}{f(x_0 + F_0) - F_0} \] Compute \(x_1\) using the results from Step 3 and the iteration formula. Update the value of \(x\) until the relative error \(|x_{n+1} - x_n|\) is less than a specified tolerance, usually \(10^{-5}\).
05

Iterate Until Convergence

Continue the process outlined in Steps 3 and 4: evaluate \(f(x_n)\), compute the next \(x_{n+1}\) using Steffensen's formula, and update your guess. With each iteration, \(x\) should get closer to the actual fixed point. This iterative process is continued until the change in \(x\) is less than the predetermined tolerance, signaling convergence to an approximate fixed point.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Iterative Methods
Iterative methods are a fundamental part of mathematical procedures used to find approximate solutions to problems. Unlike direct methods, which attempt to get an exact answer in a finite number of steps, iterative methods generate a sequence of improving approximate solutions. In the context of functions, they are used to approximate roots, fixed points, or other important values by continuously refining guesses with each iteration.
  • One key advantage of iterative methods is their ability to solve large-scale and complex problems effectively.
  • They are especially useful when direct methods are not feasible or computationally expensive.
Steffensen's method is an example of an iterative method designed specifically to accelerate convergence when finding a fixed point. It modifies the simple fixed-point iteration in a way that often results in much faster convergence. The method aims to improve the efficiency by reducing the number of iterations needed to reach an acceptable level of accuracy.
Fixed-point Iteration
Fixed-point iteration is a simple and straightforward iterative process to find solutions of the form \(x = g(x)\), where the function returns the same value as its input. In fixed-point iteration, we begin with an initial guess and apply a function repeatedly to move closer to the fixed point.
  • The basic principle revolves around choosing a function \(g(x)\) such that the fixed point of this function corresponds to the root of the original equation.
  • This can be thought of as finding a point where the graph of the function intersects the line \(y = x\).
In Steffensen's method, we improve this simple iteration. By using a more sophisticated approach, we prevent the slow convergence rate often experienced in basic fixed-point iteration. This involves using a central difference quotient, which allows for faster convergence by better approximating the derivative.
Numerical Analysis
Numerical analysis focuses on developing and analyzing methods for obtaining numerical solutions to mathematical problems. It is a crucial aspect of scientific computing and engineering disciplines, enabling the handling of complex equations that cannot be solved analytically.
  • One critical tool in numerical analysis is iterative methods, which are widely used to solve equations numerically by approximation.
  • These methods allow for the computation of solutions with an acceptable level of precision, often dictated by a specified tolerance.
Steffensen's method, as a part of numerical analysis techniques, reflects the art and science of approximating solutions with controlled accuracy and stability. The choice between different iterative methods depends on the problem characteristics and the desired speed and accuracy of the solution. By understanding both the mathematical theory and practical application, numerical analysis provides powerful techniques to solve a broad spectrum of real-world problems.

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Most popular questions from this chapter

Write an Octave function called collatz that takes one integer input, \(n,\) and returns \(3 n+1\) if \(n\) is odd and \(n / 2\) if \(n\) is even. Save it as a collatz.m file. Use an if then else statement in your function. HINT: Use the Octave ceiling function. If ceil (n/2) equals \(\mathrm{n} / 2\), then n must be even (no remainder when divided by 2). Use your collatz function to calculate \({ }^{[\mathrm{A}]}\) (a) collatz(17) (b) \(\operatorname{collatz}(10)\) (c) collatz (109) (d) collatz(344)

The Fixed Point Convergence Theorem can be strengthened. The requirement that the first derivative be continuous can be replaced. Modify the proof in the text to show the following claim. Given a differentiable function \(f(x)\) with fixed point \(\hat{x},\) if \(\left|f^{\prime}(x)\right| \leq M<1\) for all \(x\) in some neighborhood of \(\hat{x}\), then fixed point iteration converges to the fixed point for any initial value in the neighborhood.

Write your own absolute value function called absval (abs is already defined by Octave, so it is best to use a different name) that takes a real number input and returns the absolute value of the input. Use an if then else statement in your function. Save it as absval.m and test it on the following computations. (a) |-3| (b) |123.2| (c) \(\left|\pi-\frac{22}{7}\right|\) (d) \(\left|10-\pi^{2}\right|\)

Use Aitken's delta squared method to find \(p=\lim _{n \rightarrow \infty} p_{n}\) accurate to 3 decimal places. $$ \begin{array}{c} p_{n}=\\{-2,-1.85271,-1.74274,-1.66045 \\ -1.59884,-1.55266,-1.51804 \\ -1.49208,-1.47261, \ldots\\} \end{array} $$

Write an Octave function that implements Steffensen's method.
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