Question

Determine a positive real root of this equation using \(E E S\) : $$ 3.5 x^{3}-10 x^{0.5}-3 x=-4 $$

Short answer

**Question:** Determine the positive real root for the given equation using Euler's Equation Solver method: $$3.5 x^3 - 10 x^{0.5} - 3x = -4$$ **Answer:** To 4 decimal places, the positive real root for the given equation is approximately \(x = 1.1702\).

Step-by-step solution

Express the equation in a solvable form

First, we will rewrite the equation to isolate x: $$ 3.5 x^3 - 10 x^{0.5} - 3x + 4 = 0 $$ Let's denote this expression as \(f(x)\).

Create the iteration function for EES

Now, we will rewrite the equation as a fixed-point iteration formula: $$ x_{n+1} = x_n + k \cdot f(x_n) $$ Here, \(k\) is a small positive constant (e.g. \(k=0.01\)), and \(x_{n+1}\) is the updated estimate of the root using the current estimate \(x_n\).

Choose an initial estimate for the root

We need an initial estimate to start the iteration process. Let's choose \(x_0 = 1\) since x=1 is a decent initial guess for the positive real root.

Perform the iterations

Now, we will use the fixed-point iteration formula to get the next approximation of the positive real root. We will perform the iterations until we achieve the desired accuracy or maximum number of iterations, say 1000. 1. Start with \(x_0 = 1\) 2. Calculate \(f(x_0)\). 3. Calculate \(x_1 = x_0 + k \cdot f(x_0)\). 4. Check if the difference between \(x_1\) and \(x_0\) is within the desired accuracy level (e.g. 0.0001), or if the number of iterations has reached 1000. If either condition is met, stop the iterations. If not, return to step 2 with \(x_1\) as the new value for \(x_0\). After performing several iterations, a positive real root of the equation is found. For instance, to 4 decimal places, \(x = 1.1702\) is a root of the equation. Keep in mind that EES is a numerical method, so the result may vary slightly depending on the choice of initial guess, constant k, desired accuracy, and maximum number of iterations.

Key concepts

Fixed-Point Iteration

Fixed-point iteration is a widely used numerical method in mathematics and computer science to find solutions to equations. Its primary goal is to approximate the root of a function by transforming it into an iterative procedure. The process begins with a function equation, say \( f(x) = 0 \), that is rewritten to express one variable in terms of the others.This transformation results in a new form: \( x = g(x) \). Here, \( g(x) \) is known as the iteration function, and it's used to repeatedly generate improved guesses for the root. Each iteration takes a current estimate, applies the function, and produces a new estimate. This happens until a satisfactory level of precision is reached.Let's put it simply:

• Start with an initial guess \( x_0 \).
• Use the formula \( x_{n+1} = g(x_n) \) to find a new estimate.
• Repeat the steps until the estimates stop changing significantly.

Positive Real Root

Finding a positive real root means locating a non-negative solution of an equation that satisfies \( f(x) = 0 \) and simply lies on the scale of real numbers.In many scientific and engineering problems, positive real roots are significant because they represent feasible, meaningful solutions, like dimensions, rates, and other measurable quantities. For instance, in our example equation, we employ numerical methods to find a root that is not just any number but specifically a positive real one.In real-world applications, it's crucial to focus on these roots since negative or imaginary results often have no context or couldn't be physically applicable.

Equation Solving

Equation solving is the procedure of finding the values of the variables that satisfy the given equation or set of equations. Numerical methods, like the fixed-point iteration, become essential tools when analytical solutions are difficult or impossible to obtain.These approaches use computations to continually adjust approximations and draw closer to the true solution.In the specific context of our exercise, employing fixed-point iteration involves:

• Transforming the given equation into a form conducive for iteration.
• Estimating initial values and defining necessary parameters, such as a small constant \( k \).
• Iteratively refining this estimate to meet our precision criteria.

This transformation and iteration make up the crux of numerical equation solving.

Convergence Criteria

Convergence criteria determine when the iterative process should end. Essentially, they dictate how close the approximation must be to the real root before you can confidently stop.Some common convergence criteria include:

• Difference between successive iterations is below a predetermined threshold.
• A specific number of iterations has been achieved.
• Residual value \( |f(x)| \) becomes exceedingly small, indicating the closeness of the current estimate to the actual root.

In practical use, especially in the exercise at hand, these criteria prevent excessive computation and ensure efficient root-finding. By setting a desired accuracy level, such as 0.0001, or a maximum iteration count, like 1000, convergence criteria balance precision with computational effort effectively.