Is the Newton Raphson Method accurate for finding roots of equations?

In summary, the Newton Raphson Method is an efficient iterative numerical method used to find the roots of a nonlinear equation. It works by using tangent lines to approximate the roots of a function and converges quickly towards the root with a close initial guess. It is commonly used in various fields such as engineering, physics, economics, optimization problems, and machine learning. However, it may fail to converge if the initial guess is too far from the actual root and requires the function to be differentiable and continuous.
  • #1
peterianstaker
2
0
Can someone check this is correct?

Using the Newton Raphson method with X0=2 to find the root of the equation:

x^3-x-1=0 (correct to 4.d.p)

My answer is:

f'(x)= 3x^2-1

xn+1= 2-x^3-x-1/3x^2-1

xn+1= 2-2^3-2-1/3(2^2)-1

x1= 17/11

x2= 17/11-(17/11^3)-17/11-1/3x(17/11^2)-1

= 1.3596
 
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  • #2
No, you need to iterate the method until two successive approximations agree to 5 decimal places, and then round if necessary. :D
 

1. What is the Newton Raphson Method?

The Newton Raphson Method is an iterative numerical method used to find the roots of a nonlinear equation. It is also known as the Newton's method, and it is based on the idea of using tangent lines to approximate the roots of a function.

2. How does the Newton Raphson Method work?

The method starts with an initial guess for the root of the function. Then, using the derivative of the function, it calculates the slope of the tangent line at that point and finds the point where the tangent line intersects the x-axis. This new point becomes the new guess for the root, and the process is repeated until the desired level of accuracy is achieved.

3. What are the advantages of using the Newton Raphson Method?

The main advantage of this method is its efficiency in finding accurate roots of a function. It converges quickly towards the root, especially when the initial guess is close to the actual root. It also works well for functions with multiple roots, as it can be used to find all the roots by starting with different initial guesses.

4. What are the limitations of the Newton Raphson Method?

One limitation of the method is that it may fail to converge if the initial guess is too far from the actual root. It also requires the function to be differentiable and continuous. If the function has multiple roots that are close together, the method may converge to a different root than the intended one.

5. In what fields is the Newton Raphson Method commonly used?

The Newton Raphson Method is widely used in various fields, including engineering, physics, and economics. It is commonly used to solve equations in optimization problems, finding critical points, and solving differential equations. It is also used in machine learning algorithms and in financial modeling.

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