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Understanding convergence and stability of the Newton-Raphson method

By Zoltán Kovács


Approximated solution of one and multivariable equations is an important part of numerical mathematics. The easiest case of the Newton-Raphson method leads to the xn+1 = xn − f(xn) f ′ formula which is both easy to prove and (xn) memorize, and it is also very effective in real life problems. However, choosing of the startingx0 point is very important, because convergence may no longer stand for even the easiest equations. Computer aided visualization can give a good picture of “good ” and “bad ” x0 points, and we are also able to study the end point of the convergence. The relationship between the cubic polynomial equations and the Newton fractal is very obvious, and the latter is a marvellous case of self similarity in fractal geometry. To show such behavior we use the XaoS software which is able to demonstrate the common basins of convergence with different colors in real-time visualization, including zooming in or out. The multivariate Newton-Raphson method also raises the above questions. Visual analysis of these problems are done by the Sage computer algebra system. Sage has a large set of modern tools, including groupware and web availability. While Sage is a free software, it is affordable to many people, including the teacher and the student as well. 1

Year: 2013
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