数值分析2..docVIP

Euler法和预估校正法求解初值问题 摘要 在数学与计算科学中，Euler法是一种一阶数值方法，通常用于对给定初值的常微分方程（初值问题）的求解。Euler法基本思想是迭代就是逐次替代，后求出所要求的解，并达到一定的精度。简单地取切线的端点作为下一步的起点进行计算，当步数增多时，误差会因积累而越来越大因此Euler法一般不用于实际计算。为提高精度，需要在的基础上进行改进的精度为二阶采用区间两端的的作为的斜率。先用Euler法求出预报值，再利用梯形公式求出校正值，局部截断误差比Euler法低了一阶，较大程度地提高了计算精度。 关键字：Euler法，预估校正法，MATLAB软件 EULER METHOD AND FORECAST CORRECTION METHOD FOR SOLVING INITIAL VALUE PROBLEMS ABSTRACT In mathematics and computer science, the Euler method is a numerical method. It is usually used to solve the equations of the given initial value(initial value problems),Euler’s basic method is iterative, that is to say, the ideal is successive substitution, then, find out the required solution and achieved a certain accuracy. Euler method simply means take as the starting point of the next step to calculate the tangent of the end point, when numbers increase, errors due to the accumulation of more and more big. So, the Euler method is generally not used for practical calculation. In order to improve the accuracy, we need to be on the basis of Euler method was improved, the forecast correction method. Forecasts for the second order correction method of the precision, using the average value of a function as a linear equation at each end of the range of the slope. Forecast correction method with Euler method first predicted value, using trapezoid formula to find the correction, the local truncation error lower than the Euler method, greatly improve the calculation accuracy. By write MATLAB program to realize two methods, and through comparing the drawing accuracy, deepen understanding of the two methods. Key words: Euler method, forecast correction method, MATLAB 目录 1 欧拉法........................................................1 1.1 Euler方法简介...........................................1 1.1.1 Euler格式........................................2 1.1.2欧拉方法的误差估计................................2 2 预估校正法.....................................................6 2.