Respublikasi oliy va o‘rta maxsus ta’lim vazirligi samarqand davlat universiteti

3-Misol. Faraz qilaylik, ushbu f1^x, y^ xy  y^31  0; ^ f x, y  x^2y^2 y^3 5  0 ^ 2 ^ nochiqli tenglamalar sistemasining aniq yechimi (x,y)=(2;1) bo‘lib, uni dastlab analitik usulda Maple dasturi yordamida, keyin esa uning taqribiy yechimini Nyuton usulida topaylik. Yechish. Dastlab berilgan nochiqli tenglama- lar sistemasining yechimi mavjudligini Maple dasturi yordamida grafik usulda aniqlaylik (3.14- rasm): plots[implicitplot]({x*y-y^3-1=0,x^2*y^2+y^3- 5=0},x=-5..5,y=0..2); Berilgan nochiqli tenglamalar sistemasining aniq yechimi analitik usulda Maple dasturi yordamida quyidagicha topiladi: solve({x*y-y^3-1=0,x^2*y^2+y^3-5=0},{x,y}); { x  2, y  1 }

3.14-rasm. 3-misolda berilgan tenglamalar sistemasi ildizining boshlang‘ich yaqinlashishini grafik usul bilan aniqlash.

Shu yechimni Nyuton usuli yordamida Maple dasturida taqribiy hisoblaymiz:
Avvalo Yakob matritsasini linalg paketining jacobian funksiyasi yordamida hisoblaymiz, keyin esa uning teskarisini linalg paketining inverse funksiyasidan foy-
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dalanib hisoblaymiz. eval funksiyasi ifodaning son qiymatini beradi. evalm funksiyasi esa matritsa va vektorlar ustida amal bajarib, son natija beradi. Bosh- lang‘ich vektorni xx:=[0.5;1.5] va eps:=0.001 aniqlik darajasi deb, Nyuton usuli bo‘yicha taqribiy hisoblashlarni bajaramiz:

• 
  with(linalg):
  

F:=(x,y)->[x*y-y^3-1,x^2*y^2+y^3-5];
FP:=jacobian(F(x,y),[x,y]); FPINV:=inverse(FP);
xx:=[0.5,1.5]; eps:=0.001; Err:=1000; v:=xx; v1:=[1e10,1e10]; j:=0;
for i while Err>eps do
v1:=eval(v); M:=eval(eval(FPINV),[x=v[1],y=v[2]]):
v:=evalm(v-M&*F(v[1],v[2])); Err:=max(abs(v1[1]-v[1]),abs(v1[2]-v[2])); j:=j+1;
end do;
Natijalar quyidagicha:

F := ( x, y )  [ x y  y^3  1, x^2 y^2  y^3  5 ]  y x  3 y^2  ^F^P^:^=^2 x y^2 2 x^2 y  3 y^2^  2 x^2  3 y  x  3 y^2  FPINV :=  3 y^2 ( 1  2 x y ) 3 y^3 ( 1  2 x y )  ^2 x 1 ^ ^^3 y ( 1  2 x y ) ^  3 y^2 ( 1  2 x y )  xx := [ 0.5, 1.5 ] eps := 0.001 Err := 1000 v := [ 0.5, 1.5 ] v1 := [ 0.1 10 ^11, 0.1 10 ^11 ] j := 0 v1 := [ 0.5, 1.5 ] M :=  0.2962962963 0.2469135803  ^-0.08888888887 0.05925925927^ v := [ 1.836419753, 1.240740741] Err := 1.336419753 j := 1 v1 := [ 1.836419753, 1.240740741] M :=  0.4078488757 0.08736397583 -0.1775644435 0.03896485017^

v := [ 1.910372475, 1.046712441] Err := 0.194028300 j := 2 v1 := [ 1.910372475, 1.046712441] M :=  0.6353135777 0.08003024373 -0.2433868149 0.06085855173^ v := [ 1.992251965, 1.002053670] Err := 0.081879490 j := 3 v1 := [ 1.992251965, 1.002053670] M :=  0.7276965560 0.06768724860 -0.2654774883 0.06649093770^ v := [ 1.999976663, 1.000005095] Err := 0.007724698 j := 4 v1 := [ 1.999976663, 1.000005095] M :=  0.7333182897 0.06666959200 -0.2666635987 0.06666633793^ v := [ 2.000000000, 1.000000000] Err := 0.000023337 j := 5

Iteratsion jarayonning 5-qadamida berilgan aniqlikdagi yechimga erishildi.