Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the Fixed-Point Algorithm with \(x_{1}\) as indicated to solve the equations to five decimal places. $$ x=\sqrt{3.2+x} ; x_{1}=47 $$

Step-by-Step Solution

Answer

The fixed-point iteration starting at \( x_1 = 47 \) converges to a solution satisfying \( x = \sqrt{3.2+x} \) to five decimal places.

Step 1: Rearrange the Equation

We need to rearrange the equation to a form suitable for fixed-point iteration. The given equation is \( x = \sqrt{3.2 + x} \). To set it up for iteration, we will express \( x \) in terms of itself: \( g(x) = \sqrt{3.2 + x} \).

Step 2: Initial Setup

Set the initial guess. In this problem, the initial guess is \( x_1 = 47 \). We will use this value in our iteration to find the next approximation of \( x \).

Step 3: Iterative Calculation

Using the function from Step 1, calculate the next estimate: \[x_{n+1} = g(x_n) = \sqrt{3.2 + x_n}\] Start with \( x_1 = 47 \):\[x_2 = \sqrt{3.2 + 47} = \sqrt{50.2}\] Calculate \( x_2 \) and use the result as the next \( x_n \) for further iterations.

Step 4: Continue Until Convergence

Repeat the iterative calculation until the result converges to five decimal places. This means the difference between successive values becomes negligible, specifically smaller than \( 0.00001 \). Calculate further estimates \( x_3, x_4, \ldots \) till required precision is met.

Step 5: Verify the Result

Once the result converges within the required precision, verify that it indeed satisfies the original equation. Substitute the converged value back into \( \sqrt{3.2 + x} \) and check it approximately equals \( x \).

Key Concepts

Convergence Criteria

In the context of Fixed-Point Iteration, convergence criteria are crucial to determine when the iterations can stop.These criteria ensure that the approximation obtained is precise enough within acceptable limits.
The primary convergence criteria is that the difference between successive iterations should be smaller than a certain predefined threshold. In this exercise, the convergence threshold is set to 0.00001, which means computations continue until the absolute difference between two consecutive values, \(|x_{n+1} - x_n|\), is less than this value.

• This criteria ensures that the solution is accurate to the required number of decimal places, here five. The iteration stops when the result maintains a steady state after successive calculations.
• Another consideration is the behavior of the function itself; if the function derivative magnitude is less than 1, the iterations tend to converge.

Understanding when to stop iterations not only aids in computational efficiency but also enhances the reliability of numerical solutions applied in numerous fields.

Numerical Methods

Numerical methods are techniques used to approximate solutions for mathematical problems that cannot be easily solved analytically. Among the wide array of numerical techniques, Fixed-Point Iteration is particularly useful for solving specific types of equations.

• Fixed-Point Iteration is a method used to find an approximate solution by iteratively refining an initial guess.
• The core idea is to reformulate the equation into a function that can be iterated upon, hoping its repeated application converges to a root of the equation.

In mathematical terms, this involves rewriting the equation in the form \(x = g(x)\) and then iterating through \(x_{n+1} = g(x_n)\). Understandably, this method doesn't always guarantee convergence, and sometimes the choice of the initial value or the function's nature might affect the outcome. Many real-world problems in engineering and science heavily rely on numerical methods, making a solid grasp of these techniques indispensable for effective problem-solving.

Root-Finding Algorithms

Root-finding algorithms are designed to find zeros, or 'roots', of real-valued functions. These roots are the solutions of the function set to zero.
The Fixed-Point Iteration is one such algorithm among many others.

• These algorithms typically start with an initial guess and improve it iteratively until achieving a satisfactory level of precision.
• Different root-finding methods may suit different types of problems, each having unique strengths and suitability depending on the function characteristics.

Fixed-Point Iteration is best suited for functions that can be rearranged as \(x = g(x)\) such that the iterative process converges.For other functions or when faster convergence is needed, methods like Newton-Raphson or the Bisection method might be better alternatives.Grasping various root-finding algorithms equips students with the ability to tackle diverse mathematical problems, ensuring they can select the most efficient method for any situation.