现代数值计算方法(matlab版)第6章(1)new

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2、4íúªg¦!:xnþ)¼êy(xn)Cqyn,¡y0,y1,···,yn,···Ð¯K(6.1)ê).福建师范大学数计学院~©§Ð¯Kê){©üaµ3/17(1)Ú{µùa{3Oyn+1^xn+1,xnÚyn,=cÚ.Ïd3kÐÒ±ÅÚeO,ÙL´9-¥©(Runge-Kutta){.(2)õÚ{µùa{3Oyn+1Ø^xn+1,xnÚyn±,^xn−p,yn−p(p=1,···,k;k>0),=c¡kÚ.ÙL´JJCêd(Adams){.IIJIBackClose§6.1î.{9ÙU?§6.1.1î.ªÚÛªî.ª福建师范大学dê©cûúª±)

3、ûЯK(6.1)¥êy0数计学院êO¯Kµ4/170y(xn+h)−y(xn)y(xn+1)−y(xn)y(xn)≈=,hhdd0y(xn+1)≈y(xn)+hy(xn).(6.1)¢SþÑJJII00y(x)=f(x,y(x))⇒y(xn)=f(xn,y(xn)).JIu´kBacky(xn+1)≈y(xn)+hf(xn,y(xn)).Close2dyn≈y(xn),yn+1≈y(xn+1)yn+1=yn+hf(xn,yn),n=0,1,···(6.2)福建师范大学数计学院4íúª(6.2)¡î.ª.Ó,dûúªÑe¡©ªµ5/17yn+1=yn+hf

4、(xn+1,yn+1),n=0,1,···(6.3)úª(6.3)'uyn+15§,¡Ûªî.ª.Ûªª¦^ØB,§'w«ªäkÐê½5.~6.1ÄЯKJJ0y=−y+x,0≤x≤0.5IIy(0)=0,J−xIÙ°()y(x)=e+x−1.Á©O^î.ªÚÛªî.ªBackOÙê),¿°()?1'.Close)K¥f(x,y)=−y+x,î.ªyn+1=yn+hf(xn,yn)=(1−h)yn+hxn,n=0,1,···,y0=0.福建师范大学数计学院Ûªî.ª6/17yn+1=yn+hf(xn,yn+1)=yn−hyn+1+hxn+1,n=0,1

5、,···,y0=0,n1yn+1=(yn+hxn+1),n=0,1,···,y0=0,1+hh=0.1,O(JXeL:JJxn0.00.10.20.30.40.5IIî.ª00.00000.01000.02900.05610.0905JÛªî.ª00.00910.02640.05130.08300.1209IBack°()00.00480.01870.04080.07030.1065Close~©§ê)Ø©Û'(J,Ï~Ä1n+1Ú¤¢/ÛÜ0äØ.·ÄkÑÛÜäؽÂ.福建师范大学½Â6.1éu¦)ЯK(6.1),©ª,hÚ.b数计学院y1,··

6、·,yn´O(,¡7/17εn+1=y(xn+1)−yn+1(6.4)T©ªÛÜäØ.ε=O(hp+1),¡T©ªn+1äkp°Ý.~6.2?Øî.ª(6.2)ÚÛªî.ª(6.3)°Ý.JJII)òy(x)3xn?VÐmJy00(x)y000(x)Iy(x)=y(x)+y0(x)(x−x)+n(x−x)2+n(x−x)3+···nnnnn2!3!BackClose=kh2h3000000y(xn+1)=y(xn)+hy(xn)+y(xn)+y(xn)+···(6.5)26福建师范大学数计学院(1)éAuî.ª(6.2),yn=y(xn),8/170yn

7、+1=yn+hf(xn,yn)=y(xn)+hf(xn,y(xn))=y(xn)+hy(xn),l'(6.5)2y(xn+1)−yn+1=O(h),=î.úª1°Ý.JJ(2)éAÛªî.úª(6.3),d¼êVЪIIJf(x,y)=f(xn,yn)+fx(xn,yn)(x−xn)I22+fy(xn,yn)(y−yn)+O[(x−xn)+(y−yn)],BackCloseyn=y(xn),f(xn+1,yn+1)福建师范大学=f(x,y)+hf(x,y)+f(x,y)(y−y)+O[h2+(y−y)2]数计学