Question

Use the Fixed-Point Algorithm with \(x_{1}\) as indicated to solve the equations to five decimal places. $$ x=\sqrt{2.7+x} ; x_{1}=1 $$

Short answer

The solution is approximately 2.21525.

Step-by-step solution

Rewrite the Equation

The given equation is \( x = \sqrt{2.7 + x} \). For fixed-point iteration, rewrite this equation in the form \( x = g(x) \) such that \( g(x) = \sqrt{2.7 + x} \).

Initial Approximation

We are given the initial approximation \( x_1 = 1 \). We'll start the iteration process from this point.

Iteration Process

Use the iteration formula \( x_{n+1} = g(x_n) = \sqrt{2.7 + x_n} \). Compute successive approximations until the difference between consecutive approximations is less than the desired accuracy (0.00001).

First Iteration

Compute \( x_2 \): \[ x_2 = g(x_1) = \sqrt{2.7 + 1} = \sqrt{3.7} \approx 1.92354 \]

Second Iteration

Compute \( x_3 \): \[ x_3 = g(x_2) = \sqrt{2.7 + 1.92354} = \sqrt{4.62354} \approx 2.15031 \]

Third Iteration

Compute \( x_4 \): \[ x_4 = g(x_3) = \sqrt{2.7 + 2.15031} = \sqrt{4.85031} \approx 2.20234 \]

Fourth Iteration

Compute \( x_5 \): \[ x_5 = g(x_4) = \sqrt{2.7 + 2.20234} = \sqrt{4.90234} \approx 2.21313 \]

Fifth Iteration

Compute \( x_6 \): \[ x_6 = g(x_5) = \sqrt{2.7 + 2.21313} = \sqrt{4.91313} \approx 2.21518 \]

Check for Convergence

Continue in this manner until the difference between two consecutive approximations is less than 0.00001.

Conclusion

After several iterations, the value of \( x \) stabilizes. The approximate solution to the equation to five decimal places is \( x \approx 2.21525 \).

Key concepts

Numerical Methods

Numerical methods are powerful techniques used in mathematics to find approximate solutions to problems. These techniques play a crucial role when an exact solution is impossible or hard to obtain analytically. One common application is in solving equations where you seek to find values or roots accurately.

These methods involve iterative procedures, which means computations are repeated with ever-improving approximations. The Fixed-Point Algorithm is one such method used for solving equations numerically. It requires rewriting the equation in a form suitable for iteration, often such as \( x = g(x) \), and then applying iterative steps to converge to a solution.

Different numerical methods offer various advantages. Fixed-point iteration, for example, is simple and easily implemented but may not always converge or may converge slowly. Understanding which method to use and how to apply it effectively is pivotal in achieving accurate results.

Convergence in Iterations

Convergence in iterations is a fundamental concept in numerical methods. It refers to the process through which a sequence of approximations gets increasingly closer to the true solution. For the Fixed-Point Algorithm, convergence means that repeated application of the iteration formula leads the approaches to stabilize around a specific value.

Convergence depends on various factors, such as the choice of initial approximation, the nature of the function, and the manner in which the function was modified for iteration. For efficient convergence, good initial guesses and smoothly varying functions increase the likelihood of success. The rate of convergence, or how quickly the approximations stabilize, is also significant. Faster convergence is generally more desirable.

• Convergence Criteria: Ensure that the difference between successive approximations is smaller than a predetermined tolerance level.
• Function Behavior: A poorly chosen function form can lead to divergence instead of convergence.

Achieving convergence guarantees that our numerical method is successfully approximating the root of the equation.

Root-finding Algorithms

Root-finding algorithms are essential in mathematics for solving equations where you want to find the values that satisfy the equation \( f(x) = 0 \). At its core, root-finding involves identifying values, known as roots, which make the formula equal zero or satisfy the given equation.

Among many root-finding strategies, the Fixed-Point Algorithm is a method that seeks to achieve convergence by rewriting the original equation in the form \( x = g(x) \) and iterating towards a stable solution. This transformation allows the method to find roots through a simple iterative process.

Key factors that affect root-finding include:

• Choice of Strategy: Other methods include the Newton-Raphson method and the Bisection method, which may offer different advantages depending on the context.
• Form of the Function: The form in which the function is expressed for iteration can influence the reliability and speed of achieving a solution.

Root-finding is practical in diverse scientific and engineering problems where direct solutions are difficult, making these algorithms invaluable tools for computations.