Determining Roots of a FunctionDate: 04/03/2003 at 10:16:44 From: Javier Subject: Determining Roots of a Function Hello, I know that under specific conditions we can calculate the roots of a function using the Newton-Raphson Method. The Newton-Raphson method uses the tangent to approximate the root. I want to know if there is a method to approximate the root with a parabola instead of the tangent. I tried to approximate the root using a parabola like g(x)=(x-x_0)^2 + f(x_0), where f is the original function and x_0 is the initial point (as in Newton-Raphson). The method would be: 1) calculate the roots g, r_1 and r_2. 2) if |f(r_1)|<|f(r_2)| then the new x_0 = r_1 else x_0 = r_2 and repeat again I don't know if I can find a root using this method, and if I find a root, how bad is this method compared with Newton-Raphson? Date: 04/05/2003 at 07:37:20 From: Doctor Jacques Subject: Re: Determining Roots of a Function Hi Javier, You can indeed devise a method based on that idea. First, let us see how Newton-Raphson works. There is a geometric interpretation (the tangent line), but we can also picture it in another way. Assume that a is an approximation of a root of f(x) (a would be x_n in the final formula). We can expand f(x) around a using the Taylor series: [1] f(a+h) = f(a) + hf'(a) + (h^2/2)f"(a)... In this case, h is a small increment. It is the correction to be added to a to get the exact root. The first step is to approximate the series by retaining only the first two terms: [2] f(a+h) = f(a) + hf'(a) (approximately) As h is assumed to be small (since a should be "near the root"), this approximation can be justified, provided that f"(a) and subsequent derivatives are not too large. As we look for a root, we set f(a+h) = 0: [3] 0 = f(a) +hf'(a) from which we get h = -f(a)/f'(a), and the new approximation is: a - f(a)/f'(a) which is Newton's formula. If you want a quadratic variant of the method, all you have to do is keep one more term in [1], i.e. you solve: [4] 0 = f(a) + hf'(a) + (h^2/2)f"(a) for h, and take the next approximation as a + h. This method will certainly converge much more rapidly than Newton's method, since we take a better approximation of the Taylor expansion. However, there are a few practical problems to take into account: * It requires more work to solve a quadratic equation than a linear one. * You must select which of the two roots of the quadratic is the "correct one." If you implement this in a computer program, this is also some additional work, and you can expect some nasty surprises if your initial approximation is not close enough to the root. * You must also compute the second derivative f"(a) - still more work. The conclusion is that, in practice, it is probably more efficient to perform a few more iterations of the normal Newton's method. Does this help? Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Jacques, The Math Forum http://mathforum.org/dr.math/ Date: 04/05/2003 at 11:47:40 From: Doctor Fenton Subject: Re: Determining Roots of a Function Hi Javier, Thanks for writing to Dr. Math. Yes, there are methods that do not use derivatives as Newton's method does, but which employ quadratic interpolation instead. I'm not sure that you can find a root with the method you mention, using g(x)=(x-x_0)^2 + f(x_0). That seems very arbitrary. The secant method finds two values x_1 and x_2 that bracket the root r, so that f(x_1) and f(x_2) have opposite signs, passes a line through (x_1,f(x_1)) and (x_2,f(x_2)), and uses the intercept of the line to replace one of x_1 and x_2. You could use three values x_1,x_2, and x_3, with f(x_1) and f(x_3) having opposite signs, interpolate a quadratic through (x(i),f(x(i)), for i=1, 2, and 3, and use the intercept of the quadratic to replace one of the three points. However, this procedure can apparently give poor results in some cases. The most popular non-derivative method seems to be the one developed by Richard P. Brent, based on Dekker's method, which is given in his book "Algorithms for Minimization Without Derivatives," which has recently been reprinted by Dover books. It is also described in Press, et al., _Numerical Recipes_ (in various computer languages, along with source code). It actually combines bisection and inverse quadratic interpolation to produce rapid convergence. Quadratic interpolation alone can sometimes produce poor results if the function is not sufficiently smooth, according to _Numerical Recipes_. I would recommend you check one of those sources for detailed information. - Doctor Fenton, The Math Forum http://mathforum.org/dr.math/ Date: 04/05/2003 at 23:32:44 From: Javier Subject: Thank you (Determining Roots of a Function) Thank you for the answers. I have also found using Google that there is a method called the Muller Method that approximates the root with a parabola. |
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