11/11/2023 0 Comments Numerical newton raphson method![]() If point A has an x-value of x 0, then its y-value will be f(x 0) because the point is on the curve. We need to calculate the position of point B, where the tangent crosses the x-axis. This diagram shows a curve y = f(x) with its tangent at point A. Notice that at this stage the curve itself is very close to being a straight line, so the tangent is very close to the curve when it hits the x-axis. On the graph below, we have zoomed in on the range 0.5 to 2.5:Īgain, we draw a tangent. The second iteration starts with the new value of 1.5. The formula for calculating the next value of x is shown later in this article. This tangent line hits the x-axis at x = 1.5. We now draw a line that forms a tangent to the curve at x = 2: Here is the graph where we have zoomed in on the region of interest. The first iteration starts with a guess of 2. Once we understand the method it can be calculated without requiring a graph. This is for illustration only, you don't need to draw an accurate graph to use this method. To gain an intuitive understanding of the process, we will go through a couple of iterations and show the results on a graph. A graphical explanation of the Newton Raphson method When the result is accurate enough, the process ends. On each pass, the new guess is usually closer to the required result, so the approximation becomes more accurate. Steps 2 to 4 are repeated until a sufficiently accurate result is obtained, as described in the solving equations article. Repeat from step two with the new value.Find the value of x where the tangent crosses the x-axis.Draw a tangent to the curve at the point x.Start with an initial guess x (2 in this case).The Newton-Raphson method proceeds as follows: The guess doesn't need to be particularly accurate, we can just use the value 2. The Newton-Raphson method starts with an initial guess at the solution. It is useful to sketch a graph of the function: To use the technique we need to have a rough idea of where the solution is. Although we already know the answer to the problem, it is still useful to work through the numerical solution to see how it works (and to gain an approximate value for the square root of 2). We will only look for the first solution. This equation has a solution x = √2 (and an second solution x = -√2). Here is a video on the topic: Example - square root of two using Newton-Raphson methodĪs a simple example, we will solve the equation: ![]() But the Newton-Raphson method has very good convergence, so it can often provide an accurate result very quickly. This method requires us to also know the first differential of the function. The Newton-Raphson method is a numerical method to solve equations of the form f(x) = 0.
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