坡地灾害防治技术研究.PPT

Engineering Applications with Computers I(Aspect in Numerical Methods) YUNG-SHAN HONG, Ph.D., PE. Office: E723 Tel:ext. 3260 Thank for your attention ROOTS OF EQUATIONS (Part 2, p.105) Ex. Such as f(x) cannot be solved analytically. In such instance, the only alternative is an approximate solution technique. One method to obtain an approximate solution is to plot the function and determine where it crosses the x axis. This point, which represents the x value for which f(x) = 0, is the root. f(x) x root Copyright ? 2005 by yshong Although graphical method are useful for obtaining rough estimates of roots, they are limited because of their lack of precision. An alternative approach is to use trial and error. This “technique” consists of guessing a value of x and evaluating whether f(x) is zero. Such this methods are obviously inefficient and inadequate for the requirements of engineering practice. Copyright ? 2005 by yshong Ex. Such computations can be performed directly because v is expressed explicitly as a function of time. However, suppose we had to determine the drag coefficient for a parachutist of a given mass to attain a prescribed velocity in a set time period. Ex. There is no way to rearrange the equation so that c is isolated on one side of the equal sign. In such cases, c is said to be implicit. Copyright ? 2005 by yshong Approach of Nonlinear equation solution: Bracketing method (chap. 5) – bisection, false position Open method (chap. 6) – one-point iteration, Newton-Raphson, secant method Roots of polynomials (chap. 7) – Müller’s methos, Bairstow’s method Copyright ? 2005 by yshong Roots within the interval Assumption a nonlinear equation f(x)=0 is a continue function. Two points are “a” and “b” on x-axis, then f(x) is whether solutions between a and b. According to follow as, If f(a)*f(b)=0, then f(x) has a solution. If f(a)*f(b)0, then f(x) has a solution x=r between “a” and “b” to satisfy f(x)=0. If f(a)*f(b)0, then ? Ref. pp.114~115.