As we'll see, Newton's method can be a very efficient method to approximate a solution to an equation — when it works. We can now describe Newton's method algebraically. Newton's method in the above example is much faster than the bisection algorithm! In only 4 iterations we have 11 decimal places of accuracy!
The number of decimal places of accuracy approximately doubles with each iteration! To how many decimal places is the approximate solution accurate? The number of decimal places accuracy roughly triples with each iteration! Compare the convergence to what you obtained with the bisection method in exercise 5.
While Newton's method can give fantastically good approximations to a solution, several things can go wrong. We now examine some of this less fortunate behaviour. We write them to 2 decimal places. But similar problems will happen applying Newton's method to any curve with a similar shape. Sometimes the choice of initial conditions can be ever so slight, yet lead to a radically different outcome in Newton's method. The computations to 3 decimal places from these initial values are shown in the table below.
As it turns out,. In the Links Forward section we examine this behaviour further, showing some pictures of this sensitive dependence on initial conditions, which is indicative of mathematical chaos. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.
It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. I find many sites explaining how to use Newton's method, but none explaining why it works. Could someone give me the intuition behind it?
The method is easiest to justify in one dimension. Now, it is well-known or at least, ought to be that the tangent line of a function is the "best" linear approximation of a function in the vicinity of its point of tangency:. The first idea of the Newton-Raphson method is that, since it is easy to find the root of a linear function, we pretend that our complicated function is a line, and then find the root of a line, with the hope that the line's crossing is an excellent approximation to the root we actually need.
As you can see, the blue point corresponding to the approximation is a bit far off, which brings us to the second idea of Newton-Raphson: if at first you don't succeed, try again :. As you can see, the new blue point is much nearer to the red point. We then say that we have converged to an approximation of the root. That is the essence of Newton-Raphson. As an aside, the previous discussion should tip you on what might happen if the tangent line is nearly horizontal, which is one of the disastrous things that can happen while applying the method.
This animation from the Wikipedia page for Newton's Method might be useful:. Most root finding methods work by replacing the function, which is only known at a few points, with a plausible model, and finding the root of the model.
For instance, the chord and regula falsi methods work from two known points and hypothetise a linear behavior in between. Newton uses a single known point and the direction of the tangent, and also hypothesizes a linear behavior.
Brent uses three points and a parabolic interpolation. In all cases, the reason why it works is simple: because the new estimate is closer to the root. For this property to hold, the functions must satisfy certain criteria, which are established in the frame of calculus, and essentially mean that the function can be well fitted by the model.
In particular, when the function has Taylor develomments, it locally behaves like a polynomial of some degree. Just to add to J. This shows that the second idea of Newton's method is only a heuristic which could fail in some cases. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
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