Predictor-Corrector MethodsDate: 09/07/2002 at 04:08:38 From: Jen Subject: Predictor-Corrector Methods Could you please explain what a predictor-corrector method is, possibly using Euler's and Huen's methods as examples? Please. Thank you so much, Jen Date: 09/07/2002 at 09:05:35 From: Doctor Fenton Subject: Re: Predictor-Corrector Methods Hi Jen, Thanks for sending your question to Dr. Math. The idea of predictor-corrector methods is the following: To solve y' = f(x,y) with y(x0) = y0, and perhaps some additional values of y known: x1 / y(x1) - y(x0) = | f(x,y(x)) dx / x0 If we use a numerical method to compute the integral on the right, then we will have a value for y(x1). Numerical integration formulas are of two types: open formulas, which compute the integral b / | f(x) dx / a by evaluating f at points in the interval (a,b), i.e. not using the value of f at the endpoints; and closed formulas, which use the value of f at the endpoints to compute the integral. An example of an open formula is the Midpoint Rule: b / | f(x)dx ~ f((a+b)/2) (b-a) / a where you approximate the integral by the product of the value of f at the midpoint times the length of the interval. When a and b are very close, this isn't a bad approximation: you are approximating the area between the x-axis and the graph of f, which is a "quadrilateral" with three straight sides, and the fourth side the graph of f, by a rectangle whose height is the height of the curve at the midpoint. Over a very short interval, the graph of f will be almost straight, so the "quadrilateral" will be nearly a trapezoid, and the approximation is very good (it is exact for a true trapezoid). An example of a closed formula is the Trapezoid Rule. Over one interval, you use b / (b-a) | f(x)dx ~ ----- (f(a) + f(b)) / 2 a Back to the DE: suppose we know y(x0) and have computed y(x1) by some method (e.g. Euler's Method). Then we can first approximate the integral on the right in x2 / y(x2) - y(x0) = | f(x,y(x)) dx / x0 with the Midpoint Rule (where x1 is the midpoint of x0 and x2) so that y(x2) = y(x0) + (x2-x0)*f(x1,y(x1)) . The right side is an explicit formula which can easily be computed. However, since the Midpoint Rule isn't as accurate as the Trapezoid Rule, we will regard the value of y(x2) on the left as a preliminary value called a "predictor," which I will denote by y_p(x2). If we had tried to use the Trapezoid Rule initially, we would have (x2-x0) y(x2) = y(x0) + ------- (y(x0)+y(x2)) 2 which is an implicit equation, since y(x2), the unknown, occurs on both sides of the equation. An implicit equation requires us to solve for the unknown, rather than just evaluate an expression, which is inconvenient. However, since we have a value of y(x2) from the "predictor," we can make this equation explicit by using the predicted value on the right side: (x2-x0) y(x2) = y(x0) + ------- (y(x0)+y_p(x2)) . 2 That is, we use the predicted value y_p(x2) on the right to make it easier to solve for a more accurate value of y(x2), which is now called a "corrector," y_c(x2). One can iterate, using the "corrected" value again on the right side to compute a second new "corrected" value, etc. However, it is generally best to use the corrector formula only one or two times. If the results aren't accurate enough, use a smaller step size. Predictor-Corrector methods require a "starting" method to compute the additional initial values for the open integral formula. If you decide to decrease the step size in the middle of the computation, you have to "re-start" the method. Accurate predictor-corrector methods often use a single step method such as Runge-Kutta to get started. They also use very accurate numerical integration formulas. See a numerical analysis text for more details; Conte and de Boor's _Elementary Numerical Analysis_ is fairly easy to read. If you have any questions, please write back. - Doctor Fenton, The Math Forum http://mathforum.org/dr.math/ |
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