x„ = d n
(4.6)
у
- 1
/
ga ega b o ia m iz . Bu y e rd a d n = -
7
^
7 7
.
Shunday qilib (4.1) tizim n in g x„. x„......... x ,, x, n o m a’lum larini G auss usuli
bo‘y ich a an iq lash u ch u n (4.2), (4.4), (4.5) va (4 .6 ) larga asosiangan quyidagi
yechish alg o ritm ig a eg a b o ia m iz :
x„ = d n,
= d„_t - c „ . u x„,
x,
= d
2
— c ,,x , - . . . - c
2
„x„,
x, = d } - c
13
x, - . . . - с |их ;1.
53
U shbu form ulani qisq ach a quyidagi k o 'rin ish d a yo zish m um kin
xk =dk-
k = n,n-
1— 1.
j= k + l
M isol. B erilg an ushbu
x, +
2
x
3
- x, =
- 1
2x, - 3x
2
+ 4x, = 13
- 3x, + x 2 — 2x, = - 6
ten g lam alar tizim ini G auss usulida yeching.
Y ech ish .
jf, +
2
x, — x, =
- 1
2
jc
,
— 3x
2
+ 4 x , = 13
<=> •
- 3x, + x
2
- 2 x
3
=
- 6
x, +
2
x, - X , = - ]
7x, -
6
x
3
= - 1 5
x
3
= 6
x, +
6 - 6
=
—1
x , = 3
X; = 6
x, +
2
x , - x , =
—1
7x, -
6
x, = - 1 5 о
— х , =
- 6
x, +
2
x , -
6
=
- 1
7 x
2
= 2 1
о
x, =
6
<=>
x, +
2
x, - x
3
=
- 1
7x, -
6
x, = - 1 5 <=>
7x, - 5x, = - 9
x, +
2
x, —
6
= - l
7x, - 3 6 = - 1 5
<=>
x , =
6
X, = - 1
x, = 3
X, = 6
J a v o b : x, = -1, x
2
= 3,
x , =
6
.
Paskal a lg o ritm ik tilid a n ta n o m a’lum li n ta ch iziq li alg eb raik tenglam alar
tizim ini G auss u su lid a yechish uchun tuzilgan d astu r m atni:
program gauss; uses crt;
const n=4;
{tizim dagi tenglam alar s o n i}
type
s t r o k a = a r r a y [ I ..n + l] o f real;
m a tr is a = a r r a y [l..n ] o f stroka;
vektor ^-array[ I n ] o f real;
a:m atrisa; x; vektor; max,с real;
ij.k , m'.Integer;
procedu re g a u ss / (b.m atrisa; va r у ; vektor);
begin
f o r i : = I to n do
h p & i n
o '
max: =abs(b[i, i]); j : =/;
f o r k; = i + l to n do i f a b s(b [k ,i])> m a x then begin
m ax; =abs(b[k, i]);
j: = k ;
end;
i f j < > i then f o r k: = i to n + 1 d o
54
end;
begin c:= b[i,k:]; b [i,k ]:= b [j,k ];
b [j,k ]:= c;
end:
c := b [i,i];
f o r k := i to n+1 do b [i,k ]:= b [i,k ]/c ;
f o r m : = i+ l to n do
begin
c:= b [m ,i]:
f o r k : = i + l to n+1 do
b[m, k]: =b[m, k]-b[i,k]*c;
end:
y[n ]:= b [n ,n + 1 ];
f o r i - n - 1 down!о 1 do
begin
y [ i ] : = b [ i,n + l] ;
f o r k : = i+ l to n d o у [i]: = y [i]-b [i,k ] *y[k]
end;
end;
begin clrscr;
f o r i: = / to n do f o r j : = l to n+ 1 do
begin
w rite('a[',i: 1, ' \j : l , ]= ') ; read(a[i,j]);
{tizim koeffitsiyentlarini kiritish}
end;
g a u ss 1 (a,x);
i v rite ln f 'Sistemaning y e c h im i= : ) ;
f o r i : = l to n do w riteln (’x[',i: 1, ]= ',x [i]:1 0 ;4 );
{tizim ye c h i mini chop etish }
end.
Misol:
Q uyidagi
2 x , - 5 * , +
8
x , + x ( = 1 3
3*, + x, — 1 Ox, -
6
x
4
= - 1 1
9x, - 2x, + x
4
=
8
x, + 3x, - 1 2x, + x
4
= 20
chiziqli alg eb raik tenglam alar tizim ini G auss usuliga tu zilg an dasturdan foydalanib
y ech in g .
Javob:
x, = 6 .2 1 6 7 ,
x , = 0 ,0 8 2 5 , x ,
=
-0,6278,
x
4
= 6.0018.
55
4.3. C hiziqli algebraik ten glam alar tizim ini yechishda
tesk ari m atritsa usuli
C h iziq li algebraik ten g lam alar tizim i (4.1) ni y ec h ish d a teskari m atritsa usuli
algoritm i bilan tanishaylik. B uning uchun (4.1)tizim dagi n o m a ’lum lar oldidagi
k o effitsiy en tlard an tuzilgan n- o 'lc h o v li
au a,,
.
■
a,„
ar
.
■
a2„
ам °»!
... am
kvadrat m atritsan i qaraym iz.
T a ’rif. A m atritsaga teskari m atritsa deb sh unday A '1 m atritsag a aytiladiki,
uning u ch u n
»
/ Г
1
A = E
ten g lik o 'rin li b o 'la d i. Bu y erd a E birlik m atritsa, y a ’ni
1
0
..
(
0
1
..
с
0
0 ...
1
T eo rem a . A gar A m atritsa elem entlaridan tu zilg an asosiy determ enant qi
ym ati n oldan farqli, y a 'n i d el A * 0 b o 'lsa , A m atritsag a teskari m atritsa m avjud.
A g a r A m atritsaga teskari m atritsa m avjud b o 'ls a , u quyidagi fo rm u la yor
dam id a hiso b lan ad i
A \
a
2I
..
A*
A"
1
~ A
A t
a
12
.. ■ A„,
A„
A2
ii
.
■
Alm
bu y erd a A = d e l A , At - a it elem entlarning algebraik to 'ld iru v c h ila ri, y a ’ni
a ,2
• ■■
■
a 22
.
a u «
■
a 2„
°,-U Ч-..2
-
a ,
• ■ a - , .
°,*u
oMJ.
■
а ы.„
a,„
an2
-
а.,,ы
■
a „n
M isol.
56
1
3
4
2
1
О
А =
m atritsag a teskari m atritsani toping.
Y ech ish .
5 1
3
A = d e t A = 0 4 2
-2 1 0
A m atritsan in g algebraik toM diruvchilarni hisoblaym iz:
Au = - 2 ;
An = - 4 ;
A „ =
8
, Л , = 3 ;
Aa = 6 ;
An = - 7 ; A„ = 10; .4,, = - 1 0 ;
A„ = 20.
U holda /Г ' teskari m atritsa
= - 4 + 2 4 - 1 0 = 10
1
' 5
2
’ 5
4
5
10
3
5
2 _
1 0
k o ‘rin ish g a ega b o ‘ladi.
C hiziqli alg eb raik te n g la m a la r tizim ini teskari m atritsa u su lid a yechish
uchun, (4 .1 ) ten g lam alar tizim ini m atritsa, y a ’ni
A X = B
(4.7)
k o 'rin ish d a yozib olam iz. B u yerda
A =
a n
..
a hi
b,
a 2\
a 22
.
a 2n
x 2
ь г
X =
;
b
=
a n2
.. a
mi
x n
b„
(4.7) tenglam aning h ar ikkala tom onini A '1 ga k o 'p ay tirib , А ' A = E v a EX - X
tengliklardan foydalansak, (4.1) chiziqli algebraik ten g lam alar tizim in in g yechim i
uchun X = A ' В k o 'rin ish d a g i form ulaga ega b o ’lam iz.
C hiziqli algebraik ten g lam alar tizim ini teskari m atritsa usulida yechishga
tuzilgan d a stu r m atni:
program o b ra t m atritsa; uses crt;
{tenglam alar soni}
const n = 3;
type v ecto r = a r r a y [ I n ] o f real;
type m a tr a rr a y ] I ..п. I ..n + 1 ] o f real;
var
a,c: m atr; b,x; vector;
i.j.m.k: integer;
procedure u m vfll;m a tr; 12.-vector; va r l3;vector);
57
va r i, к: integer;
begin
f o r i : = l to n do I3 [i]:= 0 .0 ;
f o r i : = l to n do f o r k: = l to n do
I 3 [ i]: = l3 [i]+ ll[ i,k ]* l2 [k ];
end;
p ro ced u re obrm at(ao: m atr; it: integer; var a lo : m atr);
label 1;
var lo: m a tr; xo,bo: vector; so: real;
begin
m := 0 ; b o [ I ] : = 1 ; f o r k := 2 to it d o b o [k ]:= 0 ;
f o r k : = l to it-1 d o f o r i:= k + 1 to it do
begin
Io[i,k]: = aofi,k]/ao[k, кJ;
f o r j : = k + l to it do a o [i,j]:= a o [i,j]-lo [i,k ]* a o [k ,j];
b o [i]:= b o [i]-lo [i,k ]* b o [k ]
end;
1: xo[itJ: = -b o [it]/a o [it,it]: m : = m + l;
f o r k : = it- l dow n to 1 do
begin
so := 0 ;
f o r j : = k + I to it do so: = so + a o [k ,j]* x o [/J ;
x o [k ]:= (b o [k ]-so )/a o [k ,k ]
end;
f o r k : = l to it do
i f m + 1 = k then b o [ k ] : = l else b o [k ]:~ 0 ;
f o r k: =1 to it-1 do f o r i: = k + l to it do
b o [i]: = b o [i]-lo [i, k ] *bo[k];
f o r j : = l to it d o a lo [ j,m ]:= x o [ j];
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