![]() ![]() construct the tangent to \(f\) at this point.note the value of \(x\) at this point and call it \(x_0\).pick a point on the function close enough to that zero (you’ll see exactly what “close enough” means).figure out, where one of the zeros roughly is (Newton’s method finds zeros one at a time).convince ourselves that the function \(f(x)\) does indeed have a zero (or more).rewrite the equation such that it is of the form \(f(x)=0\).Given any equation in one real variable to solve, we do the following: repeat until we are done) the following idea: Such a repetition in a mathematical procedure or an algorithm is called iteration. In particular, a set of steps is repeated over and over again, until we are satisfied with the result or have decided that the method failed. Newton’s method is a step-by-step procedure at heart. The Newton-Raphson method as an iterative procedure But before we do, we need to understand the outline of what the method does. These two requirements become clear as soon as we start using Newton’s method, and we’ll get to that soon. There are two more things we need to be able to do when using the method: one is calculating derivatives and the other is finding reasonable points for the function to start looking for the solution in the neighborhood of these points. The examples will show perfectly and intuitively, why that is. Newton’s method cannot find complex-valued solutions to such an equation. This goes for parameters and coefficients, but also for the variable \(x\) itself. There is another prerequisite, namely that we are dealing with real numbers in the equation. At the end of such a procedure, we’ll arrive at a form similar to our example, which is\ If need be, we set them to example values. All other variables (or parameters, in this case), have to assume numerical values. We need to make sure that the only variable in the equation is the one we want to know a solution for. This leads us to Prerequisites for the use of Newton’s method So, in our case, we could have chosen \(a =1\) and \(b =1\) and arrived at the upper form of the equation. If we cannot set these parameters to numerical values, we have to choose some to use the method. ![]() Then, we would need to know concrete numerical values for the parameters \(a\) and \(b\) in order to be able to apply Newton’s method. Imagine, though, that the equation above would look like this originally:\ What do I mean by numerical values for parameters in the equation? In the our example there is a single variable, \(x\), for which we’ll try and find a solution later, but there is nothing else that needs a value. The only thing that matters is that you know its form and that you basically have numerical values for all parameters in the equation. Well, in fact it doesn’t matter, how complicated the equation is, either. Let’s agree not to try and solve it exactly. This equation has a rather simple structure, but it’s complicated enough. It doesn’t matter where it comes from, and it doesn’t even matter how complicated the equation looks. Someone gave it to you to solve or you found it yourself in a cool model for some phenomenon that you are investigating (like in your artificial-intelligence research, or something like that). ![]() Let’s say you have an equation in front of you. Fun reading on the Newton-Raphson method A typical situation to use Newton’s method in.Further information on the Newton-Raphson method.Python example code for the Newton-Raphson method.Using Newton’s method without a computer.Summary: the algorithm for Newton’s method.Regions of instability for the Newton-Raphson method.Convergence in the Newton-Raphson method for different initial guesses.Waiting for convergence in the Newton-Raphson method.Repeating the tangent construction for Newton’s method.Intersecting the tangent with zero to find the next step in the Newton-method.Constructing the tangent to the function for the next Newton-method step.Finding the starting point \(x_0\) for the Newton-Raphson method.Before starting Newton’s method: getting an idea what the function looks like.Transforming the equation to be solved into a function, whose zeros Newton’s method should find.The Newton-Raphson method as an iterative procedure.Prerequisites for the use of Newton’s method.A typical situation to use Newton’s method in. ![]()
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