. , , ,

,,,

, —

̳

̔

()+-1(-1)++11+0=

-4-97

1998


-

-


/=f(x,y)

y(x0)=y0

x0, x0+a


h, h/2


k:=0


xk+1/2:=xk+h/2

yk+1/2:=yk+f(xk, yk)h/2

αk:= f(xk+1/2, yk+1/2)

xk+1:=xk+h

yk+1:=ykkh


k:=n

x0, y0,

x1, y1

xn, yn


/=f(x,y) - 0, 1, n 0, =F(x), 1, 2,, n, i=F(xi)(i=1,2,, n) F(x0)=y0.

,

=F(x) . h=xk-xk-1 .

, (). . , , .

y/=f(x,y) (1)

x=x0, y(x0)=y0 (2)

(1) [,b].

[a, b] n 0, 1, 2,, n, xi=x0+ih (i=0,1,, n), h=(b-a)/n- .

(i)yi i+hf(xi, yi) (i=0,1,2).

=(), 0(0, 0), 012 i(xi, yi) (i=0,1,2,); iMi+1 , , , (1), i.

(1) R{|x-x0|£a, |y-y0|£b} :


|f(x, y1)- f(x, y2)| £ N|y1-y2| (N=const),

|df/dx|=|df/dx+f(df/dy)| £ M (M=const),

:

|y(xn)-yn| £ hM/2N[(1+hN)n-1], (3)

(n)- (1) =n, n- , n- .

(3) . : h, h/2. n*

|yn-y(xn)||yn*-yn|.

. .

.

(1) y/=f(x,y)

y(x0)=y0. n

. [x0,x0+h]

󠠠

Nk/ y=y(x) . (,).

/

yk+1

yk

1/2 xk+h=xk1

: ==f(xk,yk)(x-xk).

(,1) :

xNk/=xk+h/2=xk+1/2

yNk/=yk+f(xk,yk)h/2=yk+yk+1/2

Nk/. :

y(xk+1/2)=f(xk+1/2, yk+1/2)=αk

α 1. /. +1 /. :

+1=h

xk+1=xk+h

(4) αk=f(xk+h/2, yk+f(xk,Yk)h/2)

yk=yk-1+f(xk-1,yk-1)h

(4)- .

+1/2 +1/2, (1) y/k+1/2=f(xk+1/2, yk+1/2) +1.

h, 2h 1/3 :

| *-()|=1/3(yk*-yk),

()- .

, . , y//=f(y/,y,x) c y/(x0)=y/0, y(x0)=y0, :

y/=z

z/=f(x,y,z)

: y(x0)=y0, z(x0)=z0, z0=y/0.


,

1. :

y/=2x-y

[0,1] c h=(1-0)/5=0,2

: 0=1;

(4), :

1). x1=0,2; 1/2=0,1; y(x1)=y(x0)+α0h; y(x1/2)=y(x0)+f(x0,y0)h/2;

f(x0,y0)=2*0-1=-1

y(x1/2)=1-1*0,1=0,9

α0=2*0,1-0,9=-0,7

y1=1-0,1*0,2=0,86

2). y(x2)=y(x1)+α1h; x2=0,2+0,2=0,4; x1+1/2=x1+h/2=0,2+0,1=0,3

y(x1+1/2)=y(x1)+f(x1,y(x1))h/2

f(x1,y1)=2*0,2-0,86=-0,46

y(x1+1/2)=0,86-0,46*0,1=0,814

α1=2*0,3-0,814=-0,214

y2=0,86-0,214*0,2=0,8172

3). x3=0,4+0,2=0,6; x2+1/2=x2+h/2=0,4+0,1=0,5

f(x2,y2)=2*0,4-0,8172=-0,0172

y2+1/2=0,8172-0,0172*0,1=0,81548

α2=2*0,5-0,81548=0,18452

y3=0,8172+0,18452*0,2=0,854104

4).x4=0,8; x3+1/2=x3+h/2=0,6+0,1=0,7

f(x3,y3)=2*0,6-0,854104=0,345896

y3+1/2=0,854104+0,345896*0,1=0,8886936

α3=2*0,7-0,89=0,5113064

y4=0,854104+0,5113064*0,2=0,95636528

 

5).x5=1; x4+1/2=0,8+0,1=0,9

f(x4,y4)=2*0,8-0,956=0,64363472

y4+1/2=0,956+0,643*0,1=1,020728752;

α4=2*0,9-1,02=0,779271248

y5=0,956+0,7792*0,2=1,11221953

 

2. :

y//=2x-y+y/

[0,1] c h=0,2;

: y/=z

z/=2x-y+z

: 0=1

z0=1

1).x1=0,2; x1/2=0,1

y(z1)=y(z0)+α0h z(x1,y1)=z(x0,y0)+β0h

y(z1/2)=y(z0)+f(z0,y0)h/2 z(x1/2,y1/2)=z(x0,y0)+f(x0,y0,z0)h/2

f(z0,y0)=f10=1 f(x0,y0,z0)=f20=2*0-1+1=0

y1/2=1+1*0,1=1,1 z1/2=1+0*0,1=1

α0=z0=1 β0=2*0,1-1,1+1=0,1

y1=1+0,2*1=1,2 z1=1+0,2*0,1=1,02

 

2).x2+0,4; x1+1/2=0,3

f11=z1=1,02 f21=2*0,2-1,2+1,02=0,22

y1+1/2=1,2+1,02*0,1=1,1 z1+1/2=1,02+0,22*0,1=1,042

α1=z1+1/2=1,042 β1=2*0,3-1,302+1,042=0,34

y2=1,2+1,042*0,2=1,4084 z2=1.02+0,34*0,2=1,088

3).x3=0,6; x2+1/2=0,5

f12=z2=1,088 f22=2*0,4-1,4084+1,088=0,4796

y2+1/2=1,4084+1,088*0,1=1,5172 z2+1/2=1,088+0,4796*0,1=1,13596

α2=z2+1/2=1,13596 β2=2*0,5-1,5172+1,13596=0,61876

y3=1,4084+1,136*0,2=1,635592 z3=1,088+0,61876*0,2=1,211752

4).x4=0,8; x3+1/2=0,7

f13=z3=1,211752 f23=2*0,6-1,636+1,212=0,77616

y3+1/2=1,636+1,212*0,1=1,7567672 z3+1/2=1,212+0,776*0,1=1,289368

α3=z3+1/2=1,289368 β3=2*0,7-1,7568+1,289=0,9326008

y4=1,6+1,289*0,2=1,8934656 z4=1,212+0,93*0,2=1,39827216

5).x5=1; y4+1/2=0,9

f14=z4=1,39827216 f24=2*0,8-1,893+1,398=1,10480656

y4+1/2=1,893+1,398*0,1=2,0332928 z4+1/2=1,398+1,105*0,1=1,508752816

α4=z4+1/2=1,508752816 β4=2*0,9-2,03+1,5=1,27546

y5=1,893+1,5*0,2=2,195216163 z5=1,398+1,275*0,2=1,65336416

3.

y///=2x-y-y/+y//

[0,1], h=0,2

y0//=1

y0/=1

y0=1

3 : y/=a y0/=a0=1

y//=a/=b y0//=b0=1

b/=2x-y-a+b

1).x1=0,2; x1/2=0,1

y(a1)=y(a0)+a0h y(a1/2)=y(a0)+f10h/2

a(b1)=a(b0)+β0h a(b1/2)=a(b0)+f20h/2

b(x1,y1,a1)=b(x0,y0,a0)+γ0h b(x1/2,y1/2,a1/2)=b(x0,y0,a0)+f30h/2

f10=f(a0,y(a0))=1 y1/2=1+1*0,1=1,1

f20=f(b0,a(b0))=1 a1/2=1+1*0,1=1,1

f30=f(x0,y0,a0,b0)=-1 b1/2=1-1*0,1=0,9

α0=a1/2=1,1 y(a1)=1+1,1*0,2=1,22

β0=b1/2=0,9 a(b1)=1+0,9*0,2=1,18

γ0=2*0,1-1,1-1,1+0,9=-1,1 b(x1,y1,a1)=1-1,1*0,2=0,78

2).x2=0,4; x1+1/2=x1+h/2=0,3

f11=a1=1,18 y1+1/2=1,22+1,18*0,1=1.338

f21=b1=0,78 a1+1/2=1,18+0,78*0,1=1,258

f31=2*0,2-1,22-1,18+0,78=-1,22 b1+1/2=-1,22*0,1+0,78=0,658

α1=a1+1/2=1,258 y2=1,22+1,258*0,2=1,4716

β1=b1+1/2=0,658 a2=1,18+0,658*0,2=1,3116

γ1=2*0,3-1,338-1,258+0,658=-1,338 b2=0,78-1,338*0,2=0,5124

3).x3=0,6; x2+1/2=0,5

f12=a2=1,3116 y2+1/2=1,47+1,3*0,1=1,60276

f22=b2=0,5124 a2+1/2=1,3116+0,5*0,1=1.36284

f32=2*0,4-1,47-1,31+0,512=-1,4708 b2+1/2=0,4-1,4*0,1=0,36542

α2=1,36284 y3=1,4716+1,3116*0,2=1,744168

β2=0,36542 a3=1,3116+0,3654*0,2=1,384664

γ2=2*0,5-1,6-1,36+0,365=-1,60018 b3= 0,51-1,60018*0,2=0,192364

4).x4=0,8; x3+1/2=0,7

f13=1,384664 y3+1/2=1,74+1,38*0,1=1,8826364

f23=0,192364 a3+1/2=1,38+0,19*0,1=1,4039204

f33=2*0,6-1,7-1,38+0,19=-1,736488 b3+1/2=0,19-1,7*0,1=0,0187152

α3=1,4039204 y4=1,74+1,4*0,2=2,0249477

β3=0,0187152 a4=1,38+0,9187*0,2=1,388403

γ3=2*0,7-1,88-1,4+0,0187=-1,8678416 b4=0,192-1,87*0,2=-0,1812235

5).x4=1; x4+1/2=0,9

f14=1,388403 y4+1/2=2,02+1,388*0,1=2,16379478

f24=-0,1812235 a4+1/2=1,4-0.181*0,1=1,370306608

f34=2*0,8-2,02-1,388-0,18=-1,9945834 b4+1/2=-0,18-1,99*0,1=-0,38066266

α4=1,3703 y5=2,02+1,37*0,2=2,2990038

β4=-0,38066 a5=1,388-0,38*0,2=1,3122669

γ4=2*0,9-2,16-1,37-0,38=-2,114764056 b5=-0,181-2,1*0,2=-0,6041734

 

Turbo Pascal

uses crt,pram,kurs1_1;

var

yx,xy,l,v,p,ff,ay,by,x:array [0..10] of real;

y,a,b:array[0..10,0..1] of real;

i,n,o:integer;

c,d,h,k:real;

label

lap1;

begin

screen1;

clrscr;

writeln(' ');

readln(n);

if n=0 then begin

writeln('  ');

goto lap1;end;

writeln(' {a0,a1}');

for i:=0 to n do

readln(l[i]);

if (n=1) and (l[1]=0) or (n=2) and (l[2]=0) or (n=3) and (l[3]=0) then begin

writeln(' ');

goto lap1;

end;

writeln(' x');

readln(k);

writeln(' ');

readln(c,d);

o:=5;

h:=abs(d-c)/o;

writeln('=',h:1:1);

writeln(' y(x)= ');

for i:=0 to n-1 do

readln(v[i]);

if n=3 then begin

yx[0]:=v[0];

ay[0]:=v[1];

by[0]:=v[2];

p[0]:=(k*c-l[0]*v[0]-l[1]*v[1]-l[2]*v[2])/l[3];

x[0]:=c;

gotoxy(32,1);

write(' ');

gotoxy(32,2);

write(' x y a b ');

gotoxy(32,3);

write(' ',c:7:7,' ',yx[0]:7:7,' ',ay[0]:7:7,' ',by[0]:7:7,' ');

for i:=0 to o-1 do begin

x[i]:=x[i]+h/2;

y[i,1]:=yx[i]+(h/2)*ay[i];

a[i,1]:=ay[i]+(h/2)*by[i];

b[i,1]:=by[i]+(h/2)*p[i];

ff[i]:=(k*x[i]-l[0]*y[i,1]-l[1]*a[i,1]-l[2]*b[i,1])/l[3];

xy[i]:=x[i]+h/2;

yx[i+1]:=yx[i]+h*a[i,1];

ay[i+1]:=ay[i]+h*b[i,1];

by[i+1]:=by[i]+h*ff[i];

x[i+1]:=x[i]+h/2;

p[i+1]:=(k*xy[i]-l[0]*yx[i+1]-l[1]*ay[i+1]-l[2]*by[i+1])/l[3];

end;

for i:=0 to o-1 do begin

gotoxy(32,4+i);

write(' ',xy[i]:7:7,' ',yx[i+1]:7:7,' ',ay[i+1]:7:7,' ',by[i+1]:7:7,' ');

end;

gotoxy(32,4+o);

write(' ');

end;

if n=2 then begin

x[0]:=c;

yx[0]:=v[0];

ay[0]:=v[1];

p[0]:=(k*c-l[0]*yx[0]-l[1]*v[1])/l[2];

gotoxy(32,1);

write(' ');

gotoxy(32,2);

write(' x y a ');

gotoxy(32,3);

write(' ',c:7:7,' ',yx[0]:7:7,' ',ay[0]:7:7,' ');

for i:=0 to o-1 do begin

x[i]:=x[i]+h/2;

y[i,1]:=yx[i]+(h/2)*ay[i];

a[i,1]:=ay[i]+(h/2)*p[i];

ff[i]:=(k*x[i]-l[0]*y[i,1]-l[1]*a[i,1])/l[2];

xy[i]:=x[i]+h/2;

yx[i+1]:=yx[i]+h*a[i,1];

ay[i+1]:=ay[i]+h*ff[i];

x[i+1]:=x[i]+h/2;

p[i+1]:=(k*xy[i]-l[0]*yx[i+1]-l[1]*ay[i+1])/l[2];

end;

for i:=0 to o-1 do begin

gotoxy(32,4+i);

write(' ',xy[i]:7:7,' ',yx[i+1]:7:7,' ',ay[I+1]:7:7,' ');

end;

gotoxy(32,4+o);

write(' ');

end;

if n=1 then begin

x[0]:=c;

yx[0]:=v[0];

p[0]:=(k*x[0]-l[0]*yx[0])/l[1];

for i:=0 to o-1 do begin

x[i]:=x[i]+h/2;

y[i,1]:=yx[i]+(h/2)*p[i];

xy[i]:=x[i]+h/2;

ff[i]:=(k*x[i]-l[0]*y[i,1])/l[1];

yx[i+1]:=yx[i]+h*ff[i];

x[i+1]:=x[i]+h/2;

p[i+1]:=(k*xy[i]-l[0]*yx[i+1])/l[1];

end;

gotoxy(32,1);

write(' ');

gotoxy(32,2);

write(' x y ');

gotoxy(32,3);

write(' ',c:7:7,' ',yx[0]:7:7,' ');

for i:=0 to o-1 do begin

gotoxy(32,4+i);

write(' ',xy[i]:7:7,' ',yx[i+1]:7:7,' ');

end;

gotoxy(32,o+4);

write(' ');

end;

lap1:readln;

pramo;

delay(10000);

clrscr;

end.

kursova1.pas, 2 , . pram.tpu kurs1_1.tpu.

kursova1.pas Turbo Pascal F9. , enter : , , (0). . enter , .

1 ,

2 , , ,

3

4

5

6

7

8

9

10

11

12 , ,

, .

̳

 

 

 

! , , , .
. , :