How to Solve Equations with No Analytic Solution Method
Date: 10/26/2005 at 12:02:40 From: Joanne Subject: x(e^x) -> Simple, short, but how to solve? Hi, One of my friends gave me this question several days ago: If x(e^x) = 3, find the value of x. I realized that even if I changed the base to "ln" form, it would not help as the two x's are in different levels: x(e^x)= 3 ln x + x = ln 3 x = ln 3 - ln x x = ln (3/x) Using "ln", "e", or any sort of combination seems fruitless for this case. Please help.
Date: 10/28/2005 at 16:57:52 From: Doctor Fenton Subject: Re: x(e^x) -> Simple, short, but how to solve? Hi Joanne, Thanks for writing to Dr. Math. Simple-looking problems can be difficult to solve. This equation can't be solved analytically, that is, by finding a formula and substituting 3 into it. Instead, you have to use an iterative method, which makes an initial estimate ("guess"), and then finds improved estimates by some algorithm. There are many such algorithms, some better (faster) than others. The simplest, which always works when the problem has a solution, but which is usually the slowest, is bisection. You first write the problem in the form f(x) = 0, so that you are looking for a root of the function f. Then you need to find two numbers a and b such that f(a) and f(b) have different signs. (I am assuming that f is a continuous function, so that it must cross the axis between two values where it has different signs.) Next, you evaluate f at the midpoint m=(a+b)/2 of the interval. If this should happen to be 0, you are done. Otherwise, f(m) will have the same sign as one of the old endpoints. Replace that old endpoint with m, and you have a new interval on which f has different signs at the endpoints, and the new interval is half as large as the original interval. Then you repeat the above process with the new interval. After 10 steps, the interval will be 1/1024 as large as the original interval; after 20, less than 1/1000000 of the original length. "False Position" is a method which takes the straight line through the two points (a,f(a)) and (b,f(b)) and finds where it crosses the axis, say, at x = c. Evaluate f(c). Again, unless f(c) = 0 (possible but extremely unlikely), you get a new, smaller interval on which which f has opposite signs at the endpoints. It generally finds an approximate root faster than bisection. The best method is usually Newton's method (although it sometimes doesn't work), and it works extremely well here. It takes calculus to derive the formulas, but for this problem, you make an initial estimate x_0. Given an estimate x_n, you make an improved estimate x_(n+1) by using xe^x - 3 x_(n+1) = x_n - -------- . (x+1)e^x The function f(x) = xe^x - 3, and f(1) < 0, while f(2) > 0 . Using a spreadsheet, with x_0 = 1, I got 10 decimal place accuracy in 5 steps. With x_0 = 2, I got 8 decimal place accuracy in 5 steps. Newton's method generally doubles the number of correct decimals in each step, once you are "close" to the root. If you have any questions or need more help, please write back and show me what you have been able to do, and I will try to offer further suggestions. - Doctor Fenton, The Math Forum http://mathforum.org/dr.math/
Date: 10/30/2005 at 03:36:39 From: Joanne Subject: Thank you (x(e^x) -> Simple, short, but how to solve?) Dear Dr. Fenton, I would like to express appreciation for the trouble you took to explain the methods of solving this question to me. I have not learned such methods yet, but I'm in the process of understanding them. I'll write again should there be anything that I fail to understand. Thank you again.
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