Euler方法以及Euler改进.doc

Euler方法以及Euler改进方法的主程序：format longfun=x+y;a=-0.2;b=0.2;h=0.02;n=(b-a)/h;X=a:h:b;Y=zeros(1,n+1);X(1)=a;Y1(1)=1;Y2(1)=1;Y3(1)=1;%Euler方法for i=1:nx=X(i);y=Y1(i);Y1(i+1)=Y1(i)+h*eval(fun);end%Euler 预测-校正方法X(1)=a;Y2(1)=1;for i=1:nx=X(i);y=Y2(i);YP=Y2(i)+h*eval(fun);YC=Y2(i)+h*eval(fun);Y2(i+1)=(YP+YC)/2end%计算精确值fun1=2*exp(x)-x-1;for i=1:nx=X(i+1);Y3(i+1)=eval(fun1)endX,Y1,Y2,Y3%显示结果%绘制结果图形fplot(fun1,a,b)hold onplot(X,Y1,*r,X,Y2,+m)hold onplot(X,Y3,ok)title(a,b取值为-0.2，0.2)return当a，b取不同的值的时候：0,0.1ans =0 1.000000000000000 1.000000000000000 1.000000000000000 0.020000000000000 1.020000000000000 1.020400000000000 1.020402680053512 0.040000000000000 1.040800000000000 1.041616080000000 1.041621548384776 0.060000000000000 1.062416000000000 1.063664724816000 1.063673093090719 0.080000000000000 1.084864320000000 1.086562752257283 1.086574135349917 0.100000000000000 1.108161606400000 1.110327319852880 1.110341836151295-0.1,0.1ans =-0.100000000000000 1.000000000000000 1.000000000000000 1.000000000000000 -0.080000000000000 1.018000000000000 1.018380000000000 0.926232692773272 -0.060000000000000 1.036760000000000 1.037535276000000 0.943529067168497 -0.040000000000000 1.056295200000000 1.057481488575200 0.961578878304646 -0.020000000000000 1.076621104000000 1.078234614644420 0.980397346613511 0 1.097753526080000 1.099810953860237 1.000000000000000 0.020000000000000 1.119708596601600 1.122227135128214 1.020402680053512 0.040000000000000 1.142502768533632 1.145500123257804 1.041621548384776 0.060000000000000 1.166152823904305 1.169647225747611 1.063673093090719 0.080000000000000 1.190675880382391 1.194686099707713 1.086574135349917 0.100000000000000 1.216089397990039 1.220634758921809 1.110341836151295 0,0.2ans =0 1.000000000000000 1.000000000000000 1.000000000000000 0.020000000000000 1.020000000000000 1.020400000000000 1.020402680053512 0.040000000000000 1.040800000000000 1.041616080000000 1.041621548384776 0.060000000000000 1.062416000000000 1.063664724816000 1.063673093090719 0.080000000000000 1.084864320000000 1.086562752257283 1.086574135349917 0.100000000000000 1.108161606400000 1.110327319852880 1.110341836151295 0.120000000000000 1.132324838528000 1.134975931713908 1.134993703158751 0.140000000000000 1.157371335298560 1.160526445534529 1.160547597714455 0.160000000000000 1.183318762004531 1.186997079734327 1.187021741983620 0.180000000000000 1.210185137244622 1.214406420744960 1.214434726243620 0.200000000000000 1.237988839989515 1.242773430444008 1.242805516320340-0.2,0.2ans =-0.200000000000000 1.000000000000000 1.000000000000000 1.000000000000000 -0.180000000000000 1.016000000000000 1.016360000000000 0.850540422822544 -0.160000000000000 1.032720000000000 1.033454472000000 0.864287577932423 -0.140000000000000 1.050174400000000 1.051298252334400 0.878716470797612 -0.120000000000000 1.068377888000000 1.069906477031555 0.893840873434315 -0.100000000000000 1.087345445760000 1.089294587867593 0.909674836071919 -0.080000000000000 1.107092354675200 1.109478338542518 0.926232692773272 -0.060000000000000 1.127634201768704 1.130473800981077 0.943529067168497 -0.040000000000000 1.148986885804078 1.152297371760895 0.961578878304646 -0.020000000000000 1.171166623520160 1.174965778670465 0.980397346613511 0 1.194189955990563 1.198496087399608 1.000000000000000 0.020000000000000 1.218073755110374 1.222905708365080 1.020402680053512 0.040000000000000 1.242835230212582 1.248212403674055 1.041621548384776 0.060000000000000 1.268491934816833 1.274434294228271 1.063673093090719 0.080000000000000 1.295061773513170 1.301589866971682 1.086574135349917 0.100000000000000 1.322563008983434 1.329697982284510 1.110341836151295 0.120000000000000 1.351014269163102 1.358777881526657 1.134993703158751 0.140000000000000 1.380434554546364 1.388849194733496 1.160547597714455 0.160000000000000 1.410843245637292 1.419931948467112 1.187021741983620 0.180000000000000 1.442260110550037 1.452046573826148 1.214434726243620 0.200000000000000 1.474705312761038 1.485213914617436 1.242805516320340 1.从以上四张图来看，若x的取值区间为-a,a，则在起点处euler方法以及euler预测方法都是与真值相差较大的。而x的取值区间为0,a时，拟合程度比较高，同时，从图中可以看出用+标注的紫色线拟合程度更好，而且+与真值线的o重合。2.紫色线是Euler预测-校正方法的数值解，Euler改进方法比Euler方法的精确度要高，这与这两种方法的基本原理紧密相关。下图为0,0.2的部分放大图： 3.对区间0,0.1有euler方法：1