計(jì)算方法之解線性方程組

1、 nnnnnnnnnnbxaxaxabxaxaxabxaxaxa22112222212111212111 nnnnnnaaaaaaaaaA212222111211 nxxxx21 nbbbb21nninninnnniiniiiaabaaaaabaaaaabaaaD1,1,2121,221,2222111, 111, 11211 nnnnbbbxaxaxa 21222111nnnnnnbbbxlxlxlxlxlxl 212211222121111iiijjijiilxlbx/ )(11 nnnnnnnnbxubxuxubxuxuxu2222211212111iinijjijiiuxubx/ )(

2、1 nnnnnnnbaaabaaabaaabA21222221111211,754217743322321321321 xxxxxxxxx 754217743322,bA 13122rrrr486051303322 660051303322 232rr 66533322332321xxxxxx 16/623/ )5(2/ )323(332321xxxxxx )1()1()1(2)1(2)1(2)1(22111211)1()1(00,nnnnnnbaabaabaaabAniblbbanjalaaiiiijiijij,.3 , 20,.,3 , 2,11)1()1(111)1( 0)1(22 a)

3、1(2)1(32naa)1(22)1(22/aalii )1(22)1(3232/aal 0)1(22)1(22)1(32)1(32 aaaa )2()2()2(3)2(3)2(3)2(33)1(2)1(2)1(23)1(2211131211)2()2(00000,nnnnnnnbaabaabaaabaaaabAniblbbanjalaaiiiijiijij,.,4 , 30,.,4 , 3,)1(22)1()2()2(2)1(22)1()2( 0)1( kkka)1(,)1(, 1 kknkkkaa )()()(1,)(1)(, 1)(1, 1)1()1(,)1(1,)1()1(2)1(2)

4、1(1,2)1(2)1(22111, 111211)()(000000,knknnkknkkknkkkkkkknkkkkkkknkknkkkkbaabaabaaabaaaabaaaaabAnkiaalblbbankjalaakkkkikikkkikkikikikkkjikkijkij,.,1/0,.,1,)1()1()1()1()()()1()1()( njibbaaiiijij,.,2 , 1,0)0( )1()1()1()1()1(1,)1()1(2)1(2)1(1,2)1(2)1(22111, 111211)1()1(,nnnnnkkkknkkkkkknkknkknnbabaaabaaa

5、abaaaaabA1 , 2,.,1,/ )(1)1()1()1( nnkaxabxnkjkkkjkkjkkk 1,1,1,1,221,22221, 111, 111211nnnnnkknkkkknnkknnkkaugaugaugaugaugaugaugaugaugaugaugaugaugaugaugaugaugAug )1 , 2 , 1(/ )(/1,1,1,niaugxaugaugxaugaugxiinijjjiniinnnnn)1( kkka)1( kkka)1()1(, 1. knkkkkaa)1()1(/ kkkkikikaal 1434873.128127. 1238916321

6、321321xxxxxxxxx 14134873.128127. 12381916,bA 13131212)1()1(,rlrrlrbAbA 5 . 475. 025. 50123. 8125. 8002. 003819163,/)1(22)1(22 iaalii2625002. 0/25. 5/)1(22)1(3232 aal 2323)1()1()2()2(,rlrbAbA 213262132700123. 8125. 8002. 00381916123. 8125. 8002. 005 . 475. 025. 50381916)1 (),1 (32rrbA 1247. 81247. 80

7、5 . 475. 025. 50381916,)2()2(bA)1( kkka)1()1(, 1)1(,., knkkkkkkkaaa)1()1(max kiknikkrkaa nnnnnnnbbbxxxaaaaaaaaa21321212222111211 )1 , 2 , 1(/ )(/1,1,1,niaugxaugaugxaugaugxiinijjjiniinnnnn )3(4)2(3)1(214321)3(44)2(34)2(33)1(24)1(23)1(22141312114321432144434241343332312423222114131211bbbbxxxxaaaaaaaaa

8、abbbbxxxxaaaaaaaaaaaaaaaa 4)3(44)0(1414)3(44)1(2424)3(44)2(343)(,)(,)(raarraarraar )3(4)4(3)4(2)4(14321)3(44)2(33)1(23)1(22131211000bbbbxxxxaaaaaaa 3)2(33)0(1313)2(33)1(232)(,)(raarraar )3(4)4(3)5(2)5(14321)3(44)2(33)1(22121100000bbbbxxxxaaaaa 2)1(22)0(121)(raar )3(4)4(3)5(2)6(14321)3(44)2(33)1(2211

9、bbbbxxxxaaaaiiiiabx/ 4321432144434241343332312423222114131211bbbbxxxxaaaaaaaaaaaaaaaa112121113131114141111212111313111414,)(,)(,)(aalaalaalraarraarraar 令令 )1(4)1(3)1(21)1()1(44)1(43)1(42)1(34)1(33)1(32)1(24)1(23)1(2214131211)1(,000bbbbbaaaaaaaaaaaaaA 10001000100014131211lllT)1(22)1(4242)1(22)1(32322

10、)1(22)1(4242)1(22)1(323,)(,)(aalaalraarraar 令令 )1(4)2(3)1(21)2()2(44)2(43)2(34)2(33)1(24)1(23)1(2214131211)2(,00000bbbbbaaaaaaaaaaaA 1000100010000142322llT )2(33)2(43433)2(33)2(434,)(aalraar令令 )3(4)2(3)1(21)3()3(44)2(34)2(33)1(24)1(23)1(2214131211)3(,000000bbbbbaaaaaaaaaaA 100010000100001433lT 10100

11、10001434241323121ttttttTTAA ATA1 UALT ,1 為單位上三角矩陣為單位上三角矩陣為下三角矩陣為下三角矩陣分解分解為上三角矩陣為上三角矩陣為單位下三角矩陣為單位下三角矩陣分解分解ULCourantULDoolittle,:,: 332322131211323121333231232221131211111uuuuuulllLUaaaaaaaaaA 11111111100001uuulanrrr ),.,1( ,11njaujj 1121211121112111221000001ualulululanrrr niualii,.,3 , 2,/1111 122122

12、222212212212211222200001ulauuuluululanrrr 1321232323132133231321132230001ulauuuluuululanrrr njulaujjj,.,3 , 2,12122 221231323222321231221232311233200001uulalululuullulanrrr 2212414242/ )(uulal 2212122/ )(uulaliii )(0012332133133333323321331332313323133ululauuululuuulla )(0124321431343434243214314434

13、2414323134ululauuululuuuulla 3323421341434333432342134133231343424143)(01uululalulululuuullla ),.,1,( ,11nkkjulaukrrjkrkjkj 14144343242141441313323213133121221143214321434241323121)()(1111jjjjjjylbylylylbyylbylylbyylbybybbbbyyyyllllll),.,1( ,/ )(11nkiuulalkkkrrkirikik 1141431321211224243232233431334

14、4444321432144313324232214131211)()(/ )(/uxuxuxuyxuxuxuyxuxuyxuyxyyyyxxxxuuuuuuuuuu ),. 3 , 2(1111niylbybyijjijii 52262342321321321xxxxxxxxx 332322131211323121111221231112uuuuuulllLUA )1 , 2, 1(/ )(/1nniuxuyxuyxiinijjijiinnnn 6 . 0005 . 15 . 2011216 . 05 . 0015 . 0001LUA 6 . 04456416 . 05 . 0015 . 00

15、01321321yyyyyy 1116 . 0446 . 0005 . 15 . 20112321321xxxxxx),.,1,( ,11nkkjulaukrrjkrkjkj ),.,1( ,/ )(11nkiuulalkkkrrkirikik ),. 3 , 2(1111niylbybyijjijii )1 , 2, 1(/ )(/1nniuxuyxuyxiinijjijiinnnnCourant分解法分解法LUuuulalalalbacbacbacbA 111132144332214433322211 11110001000bllb 111111/001000lcuulc 12222122

16、122200100uablluabulab 2222222222/01000lcuulculac ), 3 , 2()1, 2 , 1(111niniuabllcubliiiiiii 11/ )(jjjjiiiiixuyxlyady 022112111131124321xxxx 2110;0111;1132432143214321aaaaccccbbbb), 3 , 2()1, 2 , 1(111niniuabllcubliiiiiii 3/5/5/2/2/1/3/75/32/5233322211134442333122211 lculculcuuabluabluablbl 022113/51

17、5/212/113/725/312/5124321xxxx 23/75/32/14321yyyy 21104321xxxx 1111111134242314131244332211434241323121uuuuuuddddllllllA 1111434232413121llllllLT 1111111143423241312144332211434241323121llllllddddllllllALUdlddldlddldldlddllllllA 444333334222322222411131112111114342413231211111),.,1,( ,11nkkjulaukrrjk

18、rkjkj 11krrkkrkkkkulau 11211krrrkrkkkrrkkrkkkkdlaulad),.,1( ,/ )(11nkiuulalkkkrrkirikik nkidldlauulalkkkrrkrririkkkkrrkirikik,.,1,/ )(/ )(1111 332211332211332211dddddddddD 333222312111333231222111332211332211llllllllllllLLLddddddLLDLATTT ), 2;, 1()(), 2(), 2(1111211111111nknkilllalnklalnilalalkkkmkmimikikkmkmkkkkii nkkiddllaldladkkjjkjijikikkjjkjkkk,.,2, 1,/ )(11112 1111111),.,2(z*z/zzijjijiinilbblbbLiiidyDy/zz nijjijiinnTnixlyxyxyxL1)1 , 2,.,1(*一、理論基礎(chǔ)一、理論基礎(chǔ)