Hayldiburasomas' question via email about Secant Method

In summary, using the Secant Method, we can solve the equation $\displaystyle \sin{\left( 1.8\,x \right) } =\frac{1}{2}\,x^2 - 10 $ with initial estimates $\displaystyle x_0 = 4.43 $ and $\displaystyle x_1 = 4.63 $. After three iterations, the approximate solution is $\displaystyle x_4 = 4.66053 $. The calculator's answer also matches this solution.
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Use three iterations of the Secant Method to find an approximate solution of the equation

$\displaystyle \sin{\left( 1.8\,x \right) } =\frac{1}{2}\,x^2 - 10 $

if your initial estimates are $\displaystyle x_0 = 4.43 $ and $\displaystyle x_1 = 4.63 $.

The Secant Method is a numerical scheme to solve equations of the form $\displaystyle f\left( x \right) = 0 $, so we must rewrite the equation as $\displaystyle 0 = \frac{1}{2}\,x^2 - 10 - \sin{ \left( 1.8\,x \right) } $.

Thus $\displaystyle f\left( x \right) = \frac{1}{2}\,x^2 - 10 - \sin{ \left( 1.8\,x \right) } $.

The Secant Method is $\displaystyle x_{n+1} = x_n - f\left( x_n \right) \left[ \frac{x_n - x_{n-1}}{f\left( x_n \right) - f\left( x_{n-1}\right) } \right] $.

I have used my CAS to solve this problem.

View attachment 9651

View attachment 9652

So after three iterations your solution is approximately $\displaystyle x_4 = 4.66053 $.

I also included the calculator's answer, which matches.
 

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Thanks for the help and support as usual Hayden!
 

FAQ: Hayldiburasomas' question via email about Secant Method

1. What is the Secant Method and how does it work?

The Secant Method is a numerical algorithm used to find the roots of a function. It is an iterative process that uses two initial points on either side of the root to approximate the root. The algorithm then uses the slope of the line connecting these two points to find a better approximation of the root. This process is repeated until the desired level of accuracy is achieved.

2. What are the advantages of using the Secant Method over other root-finding methods?

The Secant Method has several advantages over other root-finding methods, including its simplicity and efficiency. Unlike other methods, it does not require the derivative of the function, making it more versatile and applicable to a wider range of functions. It also converges faster than the Bisection Method and does not require an initial guess like the Newton-Raphson Method.

3. What are the limitations of the Secant Method?

While the Secant Method has many advantages, it also has some limitations. It may not always converge to the root if the initial points are not chosen carefully. It also requires two initial points, which can be time-consuming to find for more complex functions. Additionally, the Secant Method may not work for functions with multiple roots or roots that are close together.

4. How do you choose the initial points for the Secant Method?

The initial points for the Secant Method should be chosen carefully to ensure convergence. One way to choose these points is by plotting the function and visually identifying points on either side of the root. Another method is to use an interval-halving method, where the interval is divided into smaller intervals until the root is bracketed. The points at the ends of the final interval can then be used as the initial points.

5. What are some applications of the Secant Method in real-world problems?

The Secant Method has many practical applications, including finding the roots of equations in physics, engineering, and economics. It can also be used to solve optimization problems, such as finding the maximum or minimum of a function. Additionally, the Secant Method is commonly used in computer programming and numerical analysis to solve complex equations and systems of equations.

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