6-§. Galyorkin usuli

Galyorkin usulida yechim analitik ko‘rinishda qidiriladi, u holda yechimning xatoligi qaеrdan kеlib chiqadi?

Galyorkin usulining ishchi algoritmi uchun blok-sxеma




algoritmning dastur matni
Program Galerkin;
Const q=2;
Type
Mas=array[1..q,1..q] of real;
Mas1=array[1..q] of real;
Var
fx,lu0,lu1,lu2,a,b,z,x,h:Real; m:Mas; C,M1:Mas1; i:Integer;
Function F(X:Real; K:Integer):Real;
var
u0,u1,u2,u3:real;
begin
lu0:=Sqr(x)-2;
u1:=Sqr(x)-x;
lu1:=sqr(x)*sqr(x)*x-x*x*x*x-4*x+4;
lu2:=Sqr(x)*Sqr(x)*Sqr(x)-sqr(x)*sqr(x)*x-6*x*x+10*x-2;
fx:=12*x*x-8*x*x*x+sqr(x)*sqr(x)*sqr(x)*x;
case K of
1:F:=lu1*u1;
2:F:=lu2*u1;
3:F:=lu1*u1*x;
4:F:=lu2*u1*x;
5:F:=(fx-lu0)*u1;
6:F:=(fx-lu0)*u1*x;
end;end;
function Integ(a,b:Real; k:Integer):Real;
var
y,h1:Real; i:Integer;
begin
h1:=(b-a)/20;
y:=(f(a,k)+f(b,k))/2;
write(y:12:3);
for i:=1 to 19 do y:=y+f(a+i*h1,k);
y:=y*h1;
Integ:=y;
writeln(y:12:3);end;
Procedure Gauss(A:Mas; B:Mas1; Var x:Mas1; N:Integer);
var
k,m,l:Integer; s:Real;
begin
for k:=1 to n-1 do
for m:=k+1 to n do
begin
for l:=k+1 to n do
A[m,l]:=A[m,l]-A[m,k]*A[k,l]/A[k,k];
B[m]:=B[m]-A[m,k]*B[k]/A[k,k]; end;
x[n]:=B[n]/A[n,n];
for k:=n-1 downto 1 do
begin
s:=0;
for i:=k+1 to n do s:=s+A[k,i]*X[i];
X[k]:=(B[k]-s)/A[k,k]; end;end;
begin
Write('a,b=');Readln(a,b);
M[1,1]:=Integ(a,b,1);
M[1,2]:=Integ(a,b,2);
M[2,1]:=Integ(a,b,3);
M[2,2]:=Integ(a,b,4);
M1[1]:=Integ(a,b,5);
M1[2]:=Integ(a,b,6);
Gauss(M,M1,C,q);
For i:=1 to q do writeln(c[i]:12:4);
For I:=0 to 10 do begin
h:=(b-a)/10; x:=a+i*h;
z:=x+c[1]*(x*x-x)+c[2]*(x*x*x-x*x);
Writeln('x=',x:2:2,' z=',z:2:8,' a=', sqr(x)*sqr(x):2:8,' ', abs(z- sqr(x)*sqr(x)):2:8);
end;end.
Yuqoridagi misol uchun dastur ta`minotini ishlatib, olingan natijalar quyidagi jadvalda kеltirilgan:
c[1]= 0.7431 c[2]=1.9562

x	taqribiy	aniq	xatolik
xq0.0 xq0.1 xq0.2 xq0.3 xq0.4 xq0.5 xq0.6 xq0.7 xq0.8 xq0.9 xq1.0	0.00000000 0.01551908 0.01851317 0.02071920 0.03387413 0.06971492 0.13997851 0.25640186 0.43072193 0.67467566 1.00000000	0.00000000 0.00010000 0.00160000 0.00810000 0.02560000 0.06250000 0.12960000 0.24010000 0.40960000 0.65610000 1.00000000	0.00000000 0.01541908 0.01691317 0.01261920 0.00827413 0.00721492 0.01037851 0.01630186 0.02112193 0.01857566 0.00000000