Ο αλγόριθμος της Απαλοιφής του Γκάους

1. Grammik 'Algebra
Apaloif tou Gkˆouc
Tm ma Hlektrolìgwn Mhqanik¸n kai Mhqanik¸n Upologist¸n
Panepist mio JessalÐac
2 OktwbrÐou 2014

2. Parˆdeigma
2x1 + 2x2 + 0x3 + 6x4 = 8
2x1 + x2 x3 + x4 = 1
3x1 x2 x3 + 2x4 = 3
x1 + 2x2 + 6x3 x4 = 4
Epauxhmènoc pÐnakac:
eA
=
2
664
2 2 0 6 8
2 1 1 1 1
3 1 1 2 3
1 2 6 1 4
3
775

3. Genikì sÔsthma
a11x1 + a12x2 + ::: + a1nxn = b1
a21x1 + a22x2 + ::: + a2nxn = b2
: : : : : : : : : : : : : : : : : : : : : :
an1x1 + an2x2 + ::: + annxn = bn
Se morf pÐnaka
Ax = b
ìpou A =
2
664
a11 a12 ::: a1n
a21 a22 ::: a2n
::: ::: ::: :::
an1 an2 ::: ann
3
775
kai b =
2
664
b1
b2
:::
bn
3
775
eÐnai o pÐnakac twn suntelest¸n kai to dexiì mèroc
Epauxhmènoc pÐnakac:
eA
(1) = eA
= [A b]

4. Apaloif tou 1ou agn¸stou
eA
2
66666666666664
(1)=
a(1)
11 a(1)
12 a(1)
13 ::: a(1)
1;i1 a(1)
1i ::: a(1)
1n a(1)
1;n+1
a(1)
21 a(1)
22 a(1)
23 ::: a(1)
2;i1 a(1)
2i a(1)
2n a(1)
2;n+1
::: ::: ::: ::: ::: ::: ::: ::: :::
a(1)
i1;1 a(1)
i1;2 a(1)
i1;3 ::: a(1)
i1;i1 a(1)
i1;i ::: a(1)
i1;n a(1)
i1;n+1
a(1)
i1 a(1)
i2 a(1)
i3 ::: a(1)
i;i1 a(1)
ii ::: a(1)
in a(1)
i;n+1
a(1)
i+1;1 a(1)
i+1;2 a(1)
i+1;3 ::: a(1)
i+1;i1 a(1)
i+1;i ::: a(1)
i+1;n a(1)
i+1;n+1
::: ::: ::: ::: ::: ::: ::: ::: :::
a(1)
n1 a(1)
n2 a(1)
n3 ::: a(1)
n;i1 a(1)
ni ::: a(1)
nn a(1)
n;n+1
3
77777777777775
E2
a(1)
21
a(1)
11
E1
!
! E2 Gia j = 2; :::; n
E3
a(1)
31
a(1)
11
E1
!
! E3 mj1 = a(1)
j1 =a(1)
11
. . . . . . . . . .
Ej mj1E1
! Ej
En
a(1)
n1
a(1)
E1
!
! En tèloc

5. Apaloif tou 2ou agn¸stou
eA
2
66666666666664
(2)=
a(1)
11 a(1)
12 a(1)
13 ::: a(1)
1;i1 a(1)
1i ::: a(1)
1n a(1)
1;n+1
0 a(2)
22 a(2)
23 ::: a(2)
2;i1 a(2)
2i a(2)
2n a(2)
2;n+1
::: ::: ::: ::: ::: ::: ::: ::: :::
0 a(2)
i1;2 a(2)
i1;3 ::: a(2)
i1;i1 a(2)
i1;i ::: a(2)
i1;n a(2)
i1;n+1
0 a(2)
i2 a(2)
i3 ::: a(2)
i;i1 a(2)
ii ::: a(2)
in a(2)
i;n+1
0 a(2)
i+1;2 a(2)
i+1;3 ::: a(2)
i+1;i1 a(2)
i+1;i ::: a(2)
i+1;n a(2)
i+1;n+1
::: ::: ::: ::: ::: ::: ::: ::: :::
0 a(2)
n2 a(2)
n3 ::: a(2)
n;i1 a(2)
ni ::: a(2)
nn a(2)
n;n+1
3
77777777777775
gia j = 3; :::; n
mj2 = a(2)
j2 =a(2)
22
Ej mj2E2
! Ej
tèloc

6. Apaloif tou istou agn¸stou
eA
2
66666666666664
(i)=
12 a(1)
13 ::: a(1)
a(1)
11 a(1)
1;i1 a(1)
1i ::: a(1)
1n a(1)
1;n+1
0 a(2)
2;i1 a(2)
2i a(2)
22 a(2)
23 ::: a(2)
2n a(2)
2;n+1
::: ::: ::: ::: ::: ::: ::: ::: :::
0 0 0 ::: a(i1)
i1;i1 a(i1)
i1;i ::: a(i1)
i1;n a(i1)
i1;n+1
0 0 0 ::: 0 a(i)
ii ::: a(i)
in a(i)
i;n+1
0 0 0 ::: 0 a(i)
i+1;i ::: a(i)
i+1;n a(i)
i+1;n+1
::: ::: ::: ::: ::: ::: ::: ::: :::
0 0 0 ::: 0 a(i)
ni ::: a(i)
nn a(i)
n;n+1
3
77777777777775
gia j = i+1; :::; n
mji = a(i)
ji =a(i)
ii
Ej mjiEi
! Ej
tèloc
Shmei¸ste: An a(i)
ii = 0 tìte h isth exÐswsh ennalˆsetai me mia apo tic
epìmenec exis¸seic wc ex c: Ei $ Ep ìpou p i eÐnai o (mikrìteroc)
akèraioc gia ton opoÐo èqoume a(i)
pi6= 0

7. eA
2
66666666666664
(n)=
a(1)
11 a(1)
12 a(1)
13 ::: a(1)
1;i1 a(1)
1i ::: a(1)
1n a(1)
1;n+1
0 a(2)
22 a(2)
23 ::: a(2)
2;i1 a(2)
2i a(2)
2n a(2)
2;n+1
::: ::: ::: ::: ::: ::: ::: ::: :::
0 0 0 ::: a(i1)
i1;i1 a(i1)
i1;i ::: a(i1)
i1;n a(i1)
i1;n+1
0 0 0 ::: 0 a(i)
ii ::: a(i)
in a(i)
i;n+1
0 0 0 ::: 0 0 ::: a(i+1)
i+1;n a(i+1)
i+1;n+1
::: ::: ::: ::: ::: ::: ::: ::: :::
0 0 0 ::: 0 0 ::: a(n)
nn a(n)
n;n+1
3
77777777777775
EpÐlush ˆnw trigwnikoÔ sust matoc
xn = a(n)
n;n+1=a(n)
nn
Gia i = n 1; ::; 1
xi =
0
@a(i)
i;n+1
Xn
j=i+1
a(i)
ij xj
1
A=a(i)
ii
tèloc

8. O algìrijmoc thc apaloif c (1o B ma)
Apaloif 1ou agn¸stou apo i = 2; : : : ; n exÐswsh

9. O algìrijmoc thc apaloif c (1o B ma)
Apaloif 1ou agn¸stou apo i = 2; : : : ; n exÐswsh
a1;1x1 + + a1;jxj + + a1;nxn = b1
ai;1x1 + + ai;jxj + + ai;nxn = bi

10. O algìrijmoc thc apaloif c (1o B ma)
Apaloif 1ou agn¸stou apo i = 2; : : : ; n exÐswsh
a1;1x1 + + a1;jxj + + a1;nxn = b1
ai;1x1 + + ai;jxj + + ai;nxn = bi
pollaplasiast c p = ai;1
a1;1
, odhgìc = a1;1,

11. O algìrijmoc thc apaloif c (1o B ma)
Apaloif 1ou agn¸stou apo i = 2; : : : ; n exÐswsh
a1;1x1 + + a1;jxj + + a1;nxn = b1
ai;1x1 + + ai;jxj + + ai;nxn = bi
pollaplasiast c p = ai;1
a1;1
, odhgìc = a1;1,
ai;j ai;j p a1;j; j = 2; : : : ; n

12. O algìrijmoc thc apaloif c (k = 1; : : : ; n 1 B ma)
Apaloif k agn¸stou apo i = k+1; : : : ; n exÐswsh

13. O algìrijmoc thc apaloif c (k = 1; : : : ; n 1 B ma)
Apaloif k agn¸stou apo i = k+1; : : : ; n exÐswsh
ak;kxk + + ak;jxj + + ak;nxn = bk
ai;kxk + + ai;jxj + + ai;nxn = bi

14. O algìrijmoc thc apaloif c (k = 1; : : : ; n 1 B ma)
Apaloif k agn¸stou apo i = k+1; : : : ; n exÐswsh
ak;kxk + + ak;jxj + + ak;nxn = bk
ai;kxk + + ai;jxj + + ai;nxn = bi
pollaplasiast c p = ai;k
ak;k
, odhgìc = ak;k,

15. O algìrijmoc thc apaloif c (k = 1; : : : ; n 1 B ma)
Apaloif k agn¸stou apo i = k+1; : : : ; n exÐswsh
ak;kxk + + ak;jxj + + ak;nxn = bk
ai;kxk + + ai;jxj + + ai;nxn = bi
pollaplasiast c p = ai;k
ak;k
, odhgìc = ak;k,
ai;j ai;j p ak;j; j = k + 1; : : : ; n

16. O algìrijmoc thc apaloif c (qwrÐc enallagèc gramm¸n)
Gia k = 1; : : : ; n 1 (ta b mata thc apaloif c)
. Gia i = k + 1; : : : ; n (oi upìloipec exis¸seic)
. p = ai;k=ak;k
. Gia j = k + 1; : : : ; n (sunt. upoloÐpwn agn¸stwn)
. ai;j = ai;j p ak;j
. bi = bi p bk

17. Erwt seic
1. Pìsec perÐpou prˆxeic ekteloÔntai sthn apaloif enìc
sust matoc n exis¸sewn me n agn¸stouc.

18. Erwt seic
1. Pìsec perÐpou prˆxeic ekteloÔntai sthn apaloif enìc
sust matoc n exis¸sewn me n agn¸stouc.
2. Metatrèyte ton parapˆnw algìrijmo ètsi ¸ste autìc na
ekmetaleÔetai katˆllhla thn mh-mhdenikh dom enìc
sust matoc tou opoÐou oi suntelestèc twn agn¸stwn
ikanopoioÔn thn parakˆtw sunj kh ìpou k akèraioc jetikìc
arijmìc mikrìteroc tou n.
ai;j = 0; 8i; j : ji jj k

19. O algìrijmoc thc apaloif c (qwrÐc enallagèc gramm¸n)
Gia k = 1; : : : ; n 1 (ta b mata thc apaloif c)
. Gia i = k + 1; : : : ; n (oi upìloipec exis¸seic)
. p = ai;k=ak;k
. Gia j = k + 1; : : : ; n (sunt. upoloÐpwn agn¸stwn)
. ai;j = ai;j p ak;j
. bi = bi p bk

20. Od ghsh
prin xekin seic kˆpoio (èstw k) b ma thc apaloif c
1. an ak;k6= 0 suneqÐzeic thn apaloif kanonikˆ

21. Od ghsh
prin xekin seic kˆpoio (èstw k) b ma thc apaloif c
1. an ak;k6= 0 suneqÐzeic thn apaloif kanonikˆ
2. an ak;k = 0 yˆqneic gia èna stoiqeÐo sthn upost lh kˆtw apo
to odhgì stoiqeÐo

22. Od ghsh
prin xekin seic kˆpoio (èstw k) b ma thc apaloif c
1. an ak;k6= 0 suneqÐzeic thn apaloif kanonikˆ
2. an ak;k = 0 yˆqneic gia èna stoiqeÐo sthn upost lh kˆtw apo
to odhgì stoiqeÐo
2.1 an upˆrqei tètoio stoiqeÐo (èstw to as;k) to kˆneic odhgì
stoiqeÐo allˆzontac thn seirˆ twn exis¸sewn kai suneqÐzeic
thn apaloif kanonikˆ

23. Od ghsh
prin xekin seic kˆpoio (èstw k) b ma thc apaloif c
1. an ak;k6= 0 suneqÐzeic thn apaloif kanonikˆ
2. an ak;k = 0 yˆqneic gia èna stoiqeÐo sthn upost lh kˆtw apo
to odhgì stoiqeÐo
2.1 an upˆrqei tètoio stoiqeÐo (èstw to as;k) to kˆneic odhgì
stoiqeÐo allˆzontac thn seirˆ twn exis¸sewn kai suneqÐzeic
thn apaloif kanonikˆ
2.2 an den upˆrqei tìte proqwrˆc sto epìmeno b ma thc
apaloif c

24. Parˆdeigma me mhdenikì odhgì
2
664
0 0 6 0 12
2 2 1 6 4
4 4 1 10 13
8 8 1 26 23
3
775

25. Parˆdeigma me mhdenikì odhgì
2
664
0 0 6 0 12
2 2 1 6 4
4 4 1 10 13
8 8 1 26 23
3
775
!
2
664 2 2 1 6 4
4 4 1 10 13
8 8 1 26 23
0 0 6 0 12
3
775

26. Parˆdeigma me mhdenikì odhgì
2
664
0 0 6 0 12
2 2 1 6 4
4 4 1 10 13
8 8 1 26 23
3
775
!
2
664 2 2 1 6 4
4 4 1 10 13
8 8 1 26 23
0 0 6 0 12
3
775
!

27. Parˆdeigma me mhdenikì odhgì
2
664
0 0 6 0 12
2 2 1 6 4
4 4 1 10 13
8 8 1 26 23
3
775
!
2
664 2 2 1 6 4
4 4 1 10 13
8 8 1 26 23
0 0 6 0 12
3
775
!
2
664
2 2 1 6 4
0 0 3 2 5
0 0 3 2 7
0 0 6 0 12
3
775

28. Parˆdeigma me mhdenikì odhgì
2
664
0 0 6 0 12
2 2 1 6 4
4 4 1 10 13
8 8 1 26 23
3
775
!
2
664 2 2 1 6 4
4 4 1 10 13
8 8 1 26 23
0 0 6 0 12
3
775
!
2
664
2 2 1 6 4
0 0 3 2 5
0 0 3 2 7
0 0 6 0 12
3
775
!
2
664
2 2 1 6 4
0 0 3 2 5
0 0 0 4 2
0 0 0 4 2
3
775

29. Parˆdeigma me mhdenikì odhgì
2
664
0 0 6 0 12
2 2 1 6 4
4 4 1 10 13
8 8 1 26 23
3
775
!
2
664 2 2 1 6 4
4 4 1 10 13
8 8 1 26 23
0 0 6 0 12
3
775
!
2
664
2 2 1 6 4
0 0 3 2 5
0 0 3 2 7
0 0 6 0 12
3
775
!
2
664
2 2 1 6 4
0 0 3 2 5
0 0 0 4 2
0 0 0 4 2
3
775
odhgˆ stoiqeÐa ta 2; 0; 0; 4,

30. Parˆdeigma me mhdenikì odhgì
2
664
0 0 6 0 12
2 2 1 6 4
4 4 1 10 13
8 8 1 26 23
3
775
!
2
664 2 2 1 6 4
4 4 1 10 13
8 8 1 26 23
0 0 6 0 12
3
775
!
2
664
2 2 1 6 4
0 0 3 2 5
0 0 3 2 7
0 0 6 0 12
3
775
!
2
664
2 2 1 6 4
0 0 3 2 5
0 0 0 4 2
0 0 0 4 2
3
775
odhgˆ stoiqeÐa ta 2; 0; 0; 4,
lÔsh h x4 = 1=2; x3 = s; x2 = t; x1 = 2 t + s=2 3=2.

31. Erwt seic
'Allaxe mia perissìterec timèc ston arqikì pÐnaka ètsi ¸ste to
sÔsthma na
1. èqei mìnon mia lÔsh.
2. mhn èqei kamÐa lÔsh.