线性方程组的直接解法(16)分析

1、汪远征汪远征汪远征汪远征汪远征汪远征汪远征汪远征 nnnnnnaaaaaaaaaA212222111211 nxxxx21 nbbbb21汪远征汪远征汪远征汪远征汪远征汪远征直接法的基础直接法的基础 116122231111|bA 2156700320111 1156140320111汪远征汪远征 )1()1(2)1(1)1()1(2)1(1)1(2)1(22)1(21)1(1)1(12)1(11)1()1(|nnnnnnnbbbaaaaaaaaabA|00)2()2()2()2(2)1(1)2()2(2)2(2)2(22)1(1)1(12)1(11bAbbbaaaaaaannnnnn 汪远征

2、汪远征 )1()1(2)1(1)1()1(2)1(1)1(2)1(22)1(21)1(1)1(12)1(11)1()1(|nnnnnnnbbbaaaaaaaaabAniaabbaabbaanjaaaaaiiiiijiijij,.,3 , 2,.,3 , 2)1(11)1(1)1(1)1(1)1(11)1(1)1()2()1(11)1(1)1(1)1(11)1(1)1()2( 倍的减去倍行的减去第1|00)2()2()2()2(2)1(1)2()2(2)2(2)2(22)1(1)1(12)1(11bAbbbaaaaaaannnnnn 注：若注：若a11(1) = 0, 则则在第在第1列中至列中至

3、少有一个元素不为少有一个元素不为0, 可可交换行交换行后再消元后再消元njaaaaajjj,.3 , 2)1(1)1(11)1(21)1(2)2(2 汪远征汪远征 )()()1(1)()()()()1(1)1(1)1(11)()(0|knkkknnknkkknkkknkkkbbbaaaaaaabA|00)1()1()1()1(1)()1(1)1()1(1,)1(, 1)1(1, 1)()(1,)()1(1)1(11)1(1)1(11 kkknkkkkknnkknknkkkkkknkkkkkknkkbAbbbbaaaaaaaaaaankkiaabbaabbaaknkkjaaaaakkkkikkk

4、kkkkkkikkikikkkkikkkjkkkkikkijkij,.,.,)()()()()()()()()()()()()()()(212111 倍的减去倍行的减去第汪远征汪远征nkkiblbbnkkjalaakkikkikikkjikkijkij,.,.,)()()()()()(212111 )()(kkkkikikaal nkkiaabbaabbaaknkkjaaaaakkkkikkkkkkkkkikkikikkkkikkkjkkkkikkijkij,.,.,)()()()()()()()()()()()()()()(212111 倍的减去倍行的减去第汪远征汪远征 )()2(2)1(1

5、)()2(2)2(22)1(1)1(12)1(11)()(|nnnnnnnnnbbbaaaaaabA汪远征汪远征 1,.,2, 1)(1)()()()(nniaxabxabxiiinijjiijiiinnnnnn )()2(2)1(121)()2(2)2(22)1(1)1(12)1(11)()(,nnnnnnnnnnbbbxxxaaaaaabxA汪远征汪远征输入输入A, n, epsFor (k = 1 to n - 1)for (i = k+1 to n)A(i, k) = A(i, k)/ A(k, k)for (j = k+1 to n+1)A(i,j) = A(i,j) - A(i,k

6、)* A(k,j)A(n, n+1) = A(n, n+1) / A(n, n)for (k = n 1 to 1 step -1)w = 0for (j = k+1 to n)w = w + A(k, j)* A(j, n+1)A(k, n+1) = A(k, n+1) wA(k, n+1) = A(k, n+1)/A(k, k) ),.,(/ )(/)()()()()(12211nnkaxabxabxkkknkjjkkjkkknnnnnn)()(kkkkikikaal ), 1( ,)()()1(nkjalaakkjikkijkij )()()1(kkikkikiblbb 汪远征汪远征fu

7、nction x = gauss(a)m=size(a);n=m(1);for k=1:n-1 for i=k+1:n a(i,:)=a(i,:)-a(k,:)*a(i,k)/a(k,k); endendfor j=n:-1:1 a(j,n+1)=(a(j,n+1)-a(j,j+1:n)*a(j+1:n,n+1)/a(j,j)endx=a(1:n,n+1);输入输入A, n, epsFor (k = 1 to n - 1)for (i = k+1 to n)A(i, k) = A(i, k)/ A(k, k)for (j = k+1 to n+1)A(i,j) = A(i,j) - A(i,k

8、)* A(k,j)A(n, n+1) = A(n, n+1) / A(n, n)for (k = n 1 to 1 step -1)w = 0for (j = k+1 to n)w = w + A(k, j)* A(j, n+1)A(k, n+1) = A(k, n+1) wA(k, n+1) = A(k, n+1)/A(k, k)汪远征汪远征 )1 , 2,.,2, 1(/ )(/)(1)()()()(nniaxabxabxiiinijjiijiiinnnnnn)()(kkkkikikaal ), 1( ,)()()1(nkjalaakkjikkijkij )()()1(kkikkikibl

9、bb 汪远征汪远征汪远征汪远征汪远征汪远征nkkiblbbnkkjalaakkikkikikijikkijkij,.,2, 1,.,2, 1)()()1()()()1( nnnknknnk6523)2)(2311 汪远征汪远征 1,.,2, 1)(1)()()()(nniaxabxabxiiinijjiijiiinnnnnn22)1()1(12111nnininnini nnnknknnk6523)2)(2311 汪远征汪远征22)1()1(12111nnininnini nnnknknnk6523)2)(2311 )(3333323nOnnnnMD 汪远征汪远征汪远征汪远征|max)(kik

10、nikkac )()()1(1)()()()()1(1)1(1)1(110knkkknnknkkknkkknkbbbaaaaaaa汪远征汪远征 0 .4710. 631. 52 .599 .580300. 02121xxxx汪远征汪远征 25510212116xxxx 55102211621xxxx0.0000000000000001551120.0000000000000001550-50000000000000000.0-50000000000000000.011120.0000000000000001551121551汪远征汪远征汪远征汪远征)()(|maxkikiknikksac .,

11、.,1,|,|max)()(nkkiaskijnjkki 汪远征汪远征汪远征汪远征function x=gaussa(a)m=size(a); n=m(1); x=zeros(n,1);for k=1:n-1 c,i=max(abs(a(k:n,k); q=i+k-1; if q=k d=a(q,:);a(q,:)=a(k,:);a(k,:)=d end for i=k+1:n a(i,:)=a(i,:)-a(k,:)*a(i,k)/a(k,k) endendfor j=n:-1:1 x(j)=(a(j,n+1)-a(j,j+1:n)*x(j+1:n)/a(j,j)end汪远征汪远征汪远征汪远

12、征 yUxbLy汪远征汪远征 1*1*1L汪远征汪远征汪远征汪远征 nnnnnnnnnnnnuuuuuulllaaaaaaaaa222112112121212222111211111 kmmjkmnmmjkmkjulula11汪远征汪远征 kmmkimnmmkimikulula11 nnnnnnnnnnnnuuuuuulllaaaaaaaaa222112112121212222111211111汪远征汪远征欲求欲求lik和和ukj kmmjkmnmmjkmkjulula11 kmmkimnmmkimikulula11汪远征汪远征1,.3 , 2,.,2, 1,.,1,1111 nknkkiuu

13、lalnkkjulaukkkmmkimikikkmmjkmkjkj 11nmmnnmnnnnulau kmmkimnmmkimikulula11 kmmjkmnmmjkmkjulula11汪远征汪远征 niylbybyijjijii,.,321111 1211,.,/ /nniuxuyxuyxiinijjijiinnnn nnnnbbbyyylll21212121111 nnnnnnyyyxxxuuuuuu212122211211汪远征汪远征 725101391444321131243301024321xxxx 24212330102A),.,3 , 2(),.,2 , 1(111111niu

14、alniauiiii 汪远征汪远征 1162113212171211233010224212330102A )4 , 3()4 , 3 , 2(22121222121222iuulaliulaukkikiikkikii 725101391444321131243301024321xxxx汪远征汪远征 11621132121712112330102A )4()4 , 3(33131333131333iuulaliulaukkikiikkikii 911621121131132121712112330102 725101391444321131243301024321xxxx汪远征汪远征 9116

15、21121131132121712112330102A.141444444 kkkulau 4911621121131132121712112330102 725101391444321131243301024321xxxx汪远征汪远征)()(161117201091162113212334324214144232131331212211ylylylbyylylbyylbybyy 4 , 3 , 21111iylbybyrkkikii 725101391444321131243301024321xxxx汪远征汪远征444334343223234242112123134141/ )(/ )(/

16、)(432116111720104112113217121130102uyuxuyuxuxuyuxuxuxuyxy )1 , 2 , 3(/ )(/414444kuxuyxuyxkkkiikikk 725101391444321131243301024321xxxx汪远征汪远征 nnnnnnnnnnnnnnnnnnyyyyullluulluuuluuuubbbbaaaaaaaaaaaaaaaa321321333323122322211131211321321333323122322211131211 nkylbybykmmkmkk,.,3 , 21111 nkkjulauaukmmjkmkjk

17、jjj,.,1,1111汪远征汪远征function x=lua(a,b)m=size(a);n=m(1);x=zeros(n,1);y=zeros(n,1);for k=1:n a(k,k:n) = a(k,k:n)-a(k,1:k-1)*a(1:k-1,k:n) a(k+1:n,k)=(a(k+1:n,k)-a(k+1:n,1:k-1)*a(1:k-1,k)/a(k,k)endU = triu(a,0)L =eye(n)+ tril(a,-1)for i=1:n y(i) = b(i)-L(i,1:i-1)*y(1:i-1)endfor i=n:-1:1 x(i)=(y(i)-U(i,i+

18、1:n)*x(i+1:n)/U(i,i)end1,.3 , 2,.,2, 1,.,1,1111 nknkkiuulalnkkjulaukkkmmkimikikkmmjkmkjkj niylbybyijjijii,.,3 , 21111 1,.,2, 1/ /1nniuxuyxuyxiinijjijiinnnn汪远征汪远征function x=luj(a)m=size(a);n=m(1);x=zeros(n,1)for k=1:n a(k,k:n+1) = a(k,k:n+1)-a(k,1:k-1)*a(1:k-1,k:n+1) a(k+1:n,k)=(a(k+1:n,k)-a(k+1:n,1:

19、k-1)*a(1:k-1,k)/a(k,k)endU = triu(a,0)for j=n:-1:1 x(j)=(a(j,n+1)-U(j,j+1:n)*x(j+1:n)/U(j,j)end汪远征汪远征57910266810918710892257659=A1=B1=C1=D1=E1=(E5-(B5*F6+C5*F7+D5*F8)/A5=A2/A$5=B2-$A6*B5=C2-$A6*C5=D2-$A6*D5=E2-$A6*E5=(E6-(C6*F7+D6*F8)/B6=A3/A$5=(B3-A7*B$5)/B$6=C3-($A7*C5+$B7*C6)=D3-($A7*D5+$B7*D6)=E

20、3-($A7*E5+$B7*E6)=(E7-D7*F8)/C7=A4/A$5=(B4-A8*B$5)/B$6=(C4-A8*C$5-B8*C$6)/C$7=D4-($A8*D5+$B8*D6+$C8*D7)=E4-($A8*E5+$B8*E6+$C8*E7)=E8/D8. 1,.,3 , 2),.,2, 1(),.,1,(1111 njnjjiuulalnjjiulaujjjkkjikijijjkkijkjiji 11nkknnknnnnulau汪远征汪远征汪远征汪远征 nnnnuuuuuu*22111122111汪远征汪远征汪远征汪远征汪远征汪远征),.,(212/1ndddDdiag 汪远

21、征汪远征 nnnnnnnnggggggggggggGGA2221211121222111T.,.,njjiggggggggajjijjkjkikjinkjkiknkkjikij11111 .,.,2, 1,.,1/ )()(1121112njnjigggaggagjjjkjkikijijjkjkjjjj 汪远征汪远征.,.,2, 1,.,1/ )()(1121112njnjigggaggagjjjkjkikijijjkjkjjjj jkjkjjga12jjjkag |汪远征汪远征)1,.,1,(/),.,2 , 1(/111 nnigxgyxnigygbyiinimmmiiiiiimmimii

22、汪远征汪远征.,.,2 , 1,.,1/ )()(1121112njnjigggaggagjjjkjkikijijjkjkjjjj 25. 7645 . 375. 2175. 275. 41114321xxx 5 . 375. 2175. 414 25. 122/15 . 42/12 5 . 375. 22/175. 42/12 5 . 322/15 . 42/12G 118034. 141421. 15 . 012132. 25 . 02汪远征汪远征function x=choleskey(a,b)m=size(a);n=m(1);x=zeros(n,1);y=zeros(n,1);G=ze

23、ros(m)for j=1:n G(j,j)=sqrt(a(j,j)-G(j,1:j-1)*G(j,1:j-1) G(j+1:n,j)=(a(j+1:n,j)-G(j+1:n,1:j-1)*G(j,1:j-1)/G(j,j)endfor i=1:n y(i)=(b(i)-G(i,1:i-1)*y(1:i-1)/G(i,i)endfor i=n:-1:1 x(i)=(y(i)-G(i+1:n,i)*x(i+1:n)/G(i,i)end),.,2, 1(/11nigygbyiiimmimii )1,.,1,(/1 nnigxgyxiinimmmiii.,.,2 , 1,.,1/ )()(11211

24、12njnjigggaggagjjjkjkikijijjkjkjjjj 汪远征汪远征汪远征汪远征 nnnnnnnnnddddxxxxbacbacbacbdAx12112111122211汪远征汪远征 111112122111122211nnnnnnnnuuulvlvlbacbacbacb )1,., 2, 1(),.,3, 2(),.,3, 2(111niulclbniluvbnivaiiiiiiiii 111112122111122211nnnnnnnnuuulvlvlbacbacbacb汪远征汪远征 111112122111122211nnnnnnnnuuulvlvlbacbacbacb

25、)1,., 2, 1(),.,3, 2(),.,3, 2(111niulclbniluvbnivaiiiiiiiii.)1,., 2, 1(/),.,3, 2(,),.,3, 2(111 nilcuniuvblblniaviiiiiiiii汪远征汪远征 nnnnnnnyyyxxxuudddyyylalal2121112121221111, ).,.,3, 2(1111nidylyadyliiiii ).,.,3, 2(/1111nilyadyldyiiiii汪远征汪远征 nnnnnnnyyyxxxuudddyyylalal2121112121221111,).1,.,2, 1(1 niyxyx

26、uxnniiii )1 , 2,.,2, 1(1nnixuyxyxiiiinn汪远征汪远征.)1,., 2, 1(/),.,3, 2(,),.,3, 2(111 nilcuniuvblblniaviiiiiiiii ).,.,3, 2(/1111nilyadyldyiiiii汪远征汪远征.)1,., 2, 1(/),.,3, 2(,),.,3, 2(111 nilcuniuvblblniaviiiiiiiii ).,.,3, 2(/1111nilyadyldyiiiii 111111111111/ )(1,.,2, 1/iiiiiiiiiiiilyadyniuabllculdybl汪远征汪远征

27、.)1,., 2, 1(/),.,3, 2(,),.,3, 2(111 nilcuniuvblblniaviiiiiiiii )1 , 2,.,2, 1(1nnixuyxyxiiiinn汪远征汪远征 )1 , 2,.,2, 1(1nnixuyxyxiiiinn 111111111111/ )(1,.,2, 1/iiiiiiiiiiiilyadyniuabllculdybl汪远征汪远征.2146321141141124321 xxxx 99994. 2,7308. 11923. 3,2692. 0,7143. 31429. 2,2857. 0, 5 . 35 . 1, 5 . 0, 243444

28、344432333333233321222222122211111111lyadyuabllyadylcuuabllyadylcuuablldylcubl 111111111111/ )(1,.,2 , 1/iiiiiiiiiiiilyadyniuabllculdybl汪远征汪远征 99994. 2,7308. 11923. 3,2692. 0,7143. 31429. 2,2857. 0, 5 . 35 . 1, 5 . 0, 243444344432333333233321222222122211111111lyadyuabllyadylcuuabllyadylcuuablldylcubl

29、. 2, 1, 4, 321113222433344 xuyxxuyxxuyxyx )1 , 2,.,2, 1(1nnixuyxyxiiiinn.2146321141141124321 xxxx汪远征汪远征 )1 , 2,.,2, 1(1nnixuyxyxiiiinn 111111111111/ )(1,.,2, 1/iiiiiiiiiiiilyadyniuabllculdybl汪远征汪远征function x=chase(a,b,c,d)m=size(b);n=m(2);L=linspace(0,0,n);U=L;x=L;y=L;L(1)=b(1);y(1)=d(1)/L(1);for k=

30、1:n-1 U(k)=c(k)/L(k) L(k+1)=b(k+1)-a(k)*U(k) y(k+1)=(d(k+1)-a(k)*y(k)/L(k+1)endx(n)=y(n)for i=n-1:-1:1 x(i)=y(i)-U(i)*x(i+1)end汪远征汪远征 iiiiiiiyuldFcCbBaA 汪远征汪远征function d=chase(a,b,c,d)m=size(b);n=m(2);d(1)=d(1)/b(1);for k=1:n-1 c(k)=c(k)/b(k); b(k+1)=b(k+1)-a(k)*c(k); d(k+1)=(d(k+1)-a(k)*d(k)/b(k+1)

31、;endfor i=n-1:-1:1 d(i)=d(i)-c(i)*d(i+1);end iiiiiiiyuldFcCbBaA 汪远征汪远征 )1 , 2,.,2, 1(1nnixuyxyxiiiinn 111111111111/ )(1,.,2, 1/iiiiiiiiiiiilyadyniuabllculdybl汪远征汪远征汪远征汪远征|,1|iiiiiiablabu 汪远征汪远征QybRxy汪远征汪远征汪远征汪远征jiPij 101101汪远征汪远征 csscG )sin()cos(sincoscossinsincos rrrrGw汪远征汪远征jijiG 1cossinsincos1),(

32、 汪远征汪远征 jkikxycxsxysxcxykkjijjii,)(cos22jiixxxc )(sin22jijxxxs jijiG 1cossinsincos1),( 汪远征汪远征 6 . 00084. 18 . 10296. 392. 46 . 3528. 256. 28 . 4A 6 . 06 . 38 . 46 . 3sin, 8 . 0)6 . 38 . 4(8 . 4cos2222 )(cos22jiixxxc )(sin22jijxxxs 汪远征汪远征 6 . 06 . 38 . 46 . 3sin, 8 . 0)6 . 38 . 4(8 . 4cos2222 118 . 0

33、6 . 06 . 08 . 011cssc 6 . 00084. 18 . 10296. 392. 46 . 3528. 256. 28 . 4A汪远征汪远征 118 . 06 . 06 . 08 . 011cssc 6 . 00084. 18 . 1012. 14 . 20456 6 . 00084. 18 . 10296. 392. 46 . 3528. 256. 28 . 4A汪远征汪远征 6 . 00084. 18 . 1012. 14 . 204561A 6 . 08 . 14 . 28 . 1sin, 8 . 0)8 . 14 . 2(4 . 2cos2222 6 . 00084. 18 . 10296. 392. 46 . 3528. 256. 28 . 4A汪远征汪远征 18 . 06 . 06 . 08 . 0111cssc 6 . 0008 . 000230456 6 . 00084. 18 . 1012. 14 . 2045