# Newton raphson method example problems. 6a 2022-12-07

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The Newton-Raphson method is a powerful and widely used technique for finding approximate solutions to equations. It is particularly useful for finding the roots of nonlinear equations, which cannot be solved using traditional algebraic methods. In this essay, we will explore the Newton-Raphson method through a series of example problems.

The basic idea behind the Newton-Raphson method is to iteratively improve an initial guess for the solution to an equation by using information about the slope of the equation at that point. This process is repeated until the solution is found to within a specified tolerance.

To illustrate the Newton-Raphson method, consider the equation x^2 - 3x + 2 = 0. This equation has two roots, x = 1 and x = 2. To find these roots using the Newton-Raphson method, we first need to rewrite the equation in the form f(x) = 0, where f(x) is a differentiable function. In this case, we can simply let f(x) = x^2 - 3x + 2.

Next, we need to choose an initial guess for the solution, which we will call x0. This initial guess should be reasonably close to the actual solution, as the Newton-Raphson method relies on the assumption that the function is well-behaved in the vicinity of the solution. In this example, we will choose x0 = 1.5.

Using the initial guess x0, we can then calculate the next approximation for the solution, x1, using the following formula:

x1 = x0 - f(x0) / f'(x0)

Where f'(x0) is the derivative of f(x) evaluated at x0.

Substituting the values from our example, we get:

x1 = 1.5 - (1.5^2 - 31.5 + 2) / (21.5 - 3)

= 1.5 - (-0.25) / (-0.5)

= 1.5 - 0.5

= 1

We can repeat this process to find the second root of the equation. Choosing x0 = 2, we get:

x1 = 2 - (2^2 - 32 + 2) / (22 - 3)

= 2 - (2) / (-1)

= 2 + 2

= 4

Since the solution x = 4 is not within the specified tolerance of the actual solution x = 2, we can repeat the process using x1 as the new initial guess. Continuing this process, we will eventually converge on the solution x = 2 to within the desired tolerance.

This simple example illustrates the basic principles of the Newton-Raphson method. In practice, the method can be applied to a wide range of equations, including those with multiple roots and those with more complicated functional forms. It is an essential tool for solving nonlinear equations, and is widely used in fields such as engineering, physics, and finance.

## 11 Highly Instructive Examples for the Newton Raphson Method Again, we can ask about the dependence of the number of necessary steps on the initial guess. The figures show the iteration converges to a unique solution in 7 steps. Just imagine that you are computing a derivative of a function with two real variables instead of one. In particular, we see that around 30 iterations are needed to arrive at a converged result for the double root, while 5 or 6 suffice to get to a simple root with the same relative precision. That seems easy enough, right? You are encouraged to use different initial x o to get a feel of how the method works.

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## Newton . So far, we have seen only simple roots in our examples. But how does this computational algorithm help to compute square roots? All of them are very close and it is hard to see anything: The function is plotted in blue and hard to see underneath the red-dashed tangents. Now that might be a problem, because sometimes — finding the exact solution roots of an equation algebraically is easier said than done. Or maybe the inverse square rule for gravitational pull; but then again I'm not sure where we use it NASA? The first which root is translated into the color of the starting point on the grid, and the second how many iterations can be used to additionally shade the color of that point. In general, the Newton-Raphson method requires making several attempts before "all" the solutions can be found.

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## Newton Raphson Method Practice Problems Online Homework Statement Hi, an undergrad engineering presentation question: As a presentation, I am plus a group mate tasked to present a real world application of the Newthon-Raphson method of finding a root. There is a region in the middle, around the origin, where the method fails. I show you the equations and pictures you need to understand what happens, and I give you a piece of python code so that you can try all that yourself. This means you need to set your equation equal to zero and solve for x. Here is an example. First, we try to find that root value by plotting the curve by plugging different values of x for example starting from 0 to 3. Assume we want to find the root i.

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## Newton Raphson Method Find the break-even point of the firm, that is, how much it should produce per day in order to have neither a profit nor a loss. I used Newtons method to compute shock polars for materials, does that help? So our root should exist between 0 and 1. Check it out here: My domain computingskillset. While for this particular choice of the initial guess, the convergence is fast, it is different for other initial guesses. Right now I'm reading on shock polars since i'll have to explain what they are i have no idea what they are myself. Therefore, we always need to check for this issue in the code. Testing this and plotting this number across the entire range plotted in the overview, we get: The low regions in the red-dot-curve provide a good solution after a low number of iteration steps, while the high regions take longer.

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## Real Work Application of Newthon We can clearly see that a plateau at 20 appears at the maximum value of our function and is there for the entire asymptotic region of the exponential tail. So when faced with solving an equation that seems impossible, we need this method! Next, we will calculate the first derivative and substitute both the function and derivative into our formula. Convergence plot Converged solution Codes A snapshot of the Mathematica code for both solved problems are shown below, along with the download links to the Mathematica notebook. Well, you know that a root is synonymous with x-intercept, so you are being asked to find where the function crosses the x-axis. The result is the following figure: These red dots indicate areas of fast convergence, where they are low, and slow convergence, where they are high. For this purpose, a polynomial is kind of ideal: it is easily constructed, can have several roots at specific locations, and derivatives are simple to get, too.

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## Calculus I Well it seems like a perfect real world engineering example but it would be fantastic if you could provide me with a simple example of doing that. At least, I learn more easily from examples. This worked really well and fairly quickly, because we were close to the root and the function is well-behaved. The answer lies in the previous example. So far, we have only looked at the number of iterations it takes to arrive at a converged solution — any solution.

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## Newton's Method (How To w/ Step Think about the following situation: you tell me about some interesting phenomenon. However, meaningful values are produced on the way to this failure, so we can still feel content about having avoided the asymptotic trap of this function. Excel Moment Template excelNR. Hey very sorry for the late reply, this tread just slipped my mind. For the second iteration 4- Get the value of f 2. For this, our usual plot with the number of necessary iterations as a function of the starting point comes in handy. So, perhaps you do, too.

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## 6a As a result, running this as a numerical exercise can only help to expose missing maximum numbers of iteration, or producing numerical overflows. I guess that main thing I would like to know is where do we actually use a root of a number in engineering? Compute f solTrial 2. One more step, ok? For this case, I show the third kind of plot again like in the first example , where the number of iteration steps needed for a solution is added to the overview. This will not be the case with complex or multi-variable functions, however. Ok, first we pick an initial guess. Add your own as needed.

Next Together we will walk through several examples, in great detail, so we can become human calculators and wield this new superpower. One difficulty with the method is that convergence is not always guaranteed unless the function fis well behaved. The figures show that the iteration converges to a unique solution in 9 steps. The result is getting closer to the actual root of our equation really quickly now. In general, a small slope of the function in question at a certain initial guess leads to slower convergence. And then it could well happen that there is no root for this function.

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