第二章 线性方程组的直接解法 .......................................................................................................... 2 第三章 解线性方程组的迭代法 .......................................................................................................... 7 第五章 非线性方程和方程组的数值解法 ........................................................................................ 10 第六章 插值法与数值微分 ................................................................................................................ 14 第七章 数据拟合与函数逼近 ............................................................................................................ 19 第八章 数值积分 ................................................................................................................................ 23 第九章 常微分方程的数值解法 ........................................................................................................ 28
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第二章 线性方程组的直接解法
1、用LU分解法求如下方程组的解
?335??1??323??5??????X??3? 220（1）?359????0?，（2）??????5917??1????????3012???7??解：（1）
（2）
?????5?A??1??33?11????24??L?U ???52??321??????3??L?Y?(101)T?Y?(1,?1,4)T3UX?Y?X?(392,?2,2)T??3??323???1??32??220????2?31??2?2???3012??????3??1?31????3????1??2??5??5??1???3?y???3?1???y????3?? ?1?31????7????1????323?????1???2???5?1????2?X??1?3??3???X????
?3????1???2??1??3??
2
?5?b???3????7??
??2426?2、对4阶矩阵A??49615???26918??进行LU分解 ?6151840????2426???1??2426?解：A??49615??26918?????21??121????123??36?? ?6151840????3321????1??3、用高斯列主元素消去法解线性方程组
?2x1?x2?3x3?1①??4x1?2x2?5x3?4 ??x1?2x2?7?11x1?3x2?2x3?3②???23x1?11x2?x3?0 ??x1?2x2?2x3??1解：对增广矩阵进行初等行变换
????2?131?r??1r3?(?5)r2?2?131?①??4254?2+(-2)r?2?131??8?2?? ????04?12???04?1?1207??r13?(?2)r1??5313??721??02?22????00?84????2x1?x2?3x3同解方程组为??1?4x2?x3?2 ????78x213?4回代求解得X?(9,?1,?6)T
此种方法叫高斯消去法，下面用高斯列主元素消去法
??2?13154??r?r?42?25r?(?1?42)r1??4254?21?131???2?21?120????27????1207????0?r23?(?14)r1??3?02?54 3
4????1??6??? ??54??r3?3?42??4r2??0?21?2?1??
??00?721?84??得同解方程组
???4x1?2x2?5x3?4???2x1?2?2x3??1 ????7218x3?4回代求解得
X?(9,?1,?6)T
??11?3?23?11??2311r?r??231110?r2?r1?②?52??231110???21?11?3?23??23?122?1??????122?1??0??r?1r?233231??57?023???231110?????231110??r3?r2????5747?r3?(?5257)r2??0?1??2323?1????5747?0?
?5235??2323223??023?233????00?1935757??得同解方程组
????23x1?11x2?x3?0??0?57x?47x?232233??1 ???0?0?(?19322357)x3?57回代得
X?(0.212435,0.549222,?1.15544)T4
10???35?233??47?23?1??
4、用Jordan消去法解矩阵方程，AX?B其中：
?11?1??10??，B??01? A??12?2????????211???10??解：容易验证A?0，故A可逆，有X?A?1?B .因此，写出方程组的增广矩阵，对其进行初等变换得
??11?1?10??11?110??11?1?12?2?01???01?1?11???01??211??30????1??10????03?1???002?????10?1002?1???02?1??01?1?11??????01????0013?3??102?
2??2????0013?3?2??????2?1???X?A?1?B??1?2??2??
??3?3?2???25?6??x1??10?5、用LU分解法求解如下方程组??413?19??x???19? ??6?3?6???2???????x3?????30???100?解：A?LU???210???25?6?03?7? ??341????????004??(1)解Ly?b??1??y1??10??21??y???19??41???2?????3???y 3?????30??得y1?10,y2?19?20??1,y3?34?30?4即y?(10,?1,4)T 5
10??11? 6?3???