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| uses crt;
const
maxn = 3;
type
TMatr = array [1..maxn,1..maxn+1] of real;
TVekt = array [1..maxn] of real;
procedure WriteMatr(m:TMatr; k: integer);
var
i,j: integer;
begin
for i:= 1 to k do
begin
for j:= 1 to k do write(m[i,j]:9:4,' ');
writeln;
end;
end;
procedure UmnMatr(m1,m2: TMatr; var mr: TMatr; n: integer);
var
i,j,k: integer;
s: real;
begin
for i:= 1 to n do
for j:= 1 to n do
begin
s:= 0;
for k:= 1 to n do s:= s + m1[i,k]*m2[k,j];
mr[i,j]:= s;
end;
end;
procedure Krylov(m:TMatr; n: integer; var p: TVekt);
var
i,j,k: integer;
y1,y2: TVekt;
m1: TMatr;
a: real;
begin
y1[1]:= 1;
for i:= 2 to n do y1[i]:= 0;
for k:= 1 to n do
begin
for i:= 1 to n do m1[i,n-k+1]:= y1[i];
for i:= 1 to n do
begin
y2[i]:= 0;
for j:= 1 to n do y2[i]:= y2[i] + m[i,j]*y1[j];
end;
y1:= y2;
end;
for i:= 1 to n do m1[i,n+1]:= -y1[i];
for k:= 1 to n do
begin
a:= m1[k,k];
for j:= k to n+1 do m1[k,j]:= m1[k,j]/a;
for i:= k+1 to n do
begin
a:= m1[i,k]/m1[k,k];
for j:= k to n+1 do
m1[i,j]:= m1[i,j]-m1[k,j]*a;
end;
end;
for k:= n downto 1 do
begin
p[k]:= m1[k,n+1];
for j:= k+1 to n do p[k]:= p[k] - m1[k,j]*p[j];
end;
end;
procedure Leverje(m:TMatr; n: integer; var p: TVekt);
var
i,j,k: integer;
s: TVekt;
m1,m2: TMatr;
begin
m1:= m;
for k:= 1 to n do
begin
s[k]:= 0;
for i:= 1 to n do s[k]:= s[k] + m1[i,i];
if k<n then
begin
UmnMatr(m1,m,m2,n);
m1:= m2;
end;
end;
for k:= 1 to n do
begin
p[k]:= s[k];
for i:= 1 to k-1 do p[k]:= p[k] + p[i]*s[k-i];
p[k]:= - p[k]/k;
end;
end;
procedure WriteHarakt(p: TVekt; n: integer);
var
i: integer;
begin
for i:= 1 to n do write(' p',i,' = ',p[i]:6:4);
writeln;
end;
procedure WriteSb(a: TVekt; n: integer);
var
i: integer;
begin
for i:= 1 to n do write(' a',i,' = ',a[i]:6:4);
writeln;
end;
function step(x,y: real): real;
begin
step:= exp(y*ln(x));
end;
function arccos(x: real): real;
begin
arccos:= pi/2-arctan(x/sqrt(1-sqr(x)));
end;
function sign(x: real): real;
begin
if x<0 then sign:= -1
else sign:= 1;
end;
function KubUr(p: TVekt; var x: TVekt): integer;
var
q,r,t,aa,bb: real;
a2,q3,r2: real;
begin
q:= (p[1]*p[1]-3*p[2])/9;
r:= (p[1]*(2*p[1]*p[1] - 9*p[2])+27*p[3])/54;
q3:= q*q*q;
r2:= r*r;
if abs(r2-q3)<0.000000001 then r2:= q3;
if (r2<q3) then
begin
t:= arccos(r/sqrt(q3))/3;
x[1]:= -2*sqrt(q)*cos(t)-p[1]/3;
x[2]:= -2*sqrt(q)*cos(t+2*pi/3)-p[1]/3;
x[3]:= -2*sqrt(q)*cos(t-2*pi/3)-p[1]/3;
KubUr:= 3;
end
else
begin
aa:= -sign(r)*step(abs(r)+sqrt(r2-q3),1/3);
if aa=0 then bb:= 0 else bb:= q/aa;
if abs(aa-bb)<0.000000001 then aa:= bb;
x[1]:= (aa+bb)-p[1]/3;
KubUr:= 1;
if aa=bb then
begin
x[2]:= -aa-p[1]/3;
KubUr:= 2;
end;
end;
end;
var
m1: TMatr;
p,x: TVekt;
k: integer;
begin
clrscr;
m1[1,1]:= 1; m1[1,2]:= -3; m1[1,3]:= 4;
m1[2,1]:= 4; m1[2,2]:= -7; m1[2,3]:= 8;
m1[3,1]:= 6; m1[3,2]:= -7; m1[3,3]:= 7;
writeln('Матрица:');
WriteMatr(m1,3);
writeln;
writeln('Метод Крылова:');
writeln;
Krylov(m1,3,p); {┬н┬а─║┬о┬ж┬д─Д┬н┬и─Д ┼Ю┬о├н├д├д┬и─З┬и─Д┬н├в┬о╦Ш
─║┬а┼Х┬а┼Ю├в─Д┼Х┬и├б├в┬и├з─Д├б┼Ю┬о┼Б┬о ─Г┼Х┬о╦Ш┬н─Д┬н┬и─П ┬м─Д├в┬о┬д┬о┬м ┼а┼Х├л┬л┬о╦Ш┬а}
writeln('Коэффициенты характеристического уравнения:');
WriteHarakt(p,3);
writeln;
writeln('Собственные значения:');
k:= KubUr(p,x); {┬н┬а─║┬о┬ж┬д─Д┬н┬и─Д ├б┬о╦З├б├в╦Ш. ┬з┬н┬а├з. ┼╗┬о
─║┬а┼Х┬а┼Ю-┬м─Г ─Г┼Х-├о}
WriteSb(x,k);
writeln;
{----------}
writeln('Метод Леверье:');
Leverje(m1,3,p); {┬н┬а─║┬о┬ж┬д─Д┬н┬и─Д ┼Ю┬о├н├д├д┬и─З┬и─Д┬н├в┬о╦Ш
─║┬а┼Х┬а┼Ю├в─Д┼Х┬и├б├в┬и├з─Д├б┼Ю┬о┼Б┬о ─Г┼Х┬о╦Ш┬н─Д┬н┬и─П ┬м─Д├в┬о┬д┬о┬м тА╣─Д╦Ш─Д┼Х─Ы─Д}
writeln;
writeln('Коэффициенты характеристического уравнения:');
WriteHarakt(p,3);
writeln;
writeln('Собственные значения:');
k:= KubUr(p,x); {┬н┬а─║┬о┬ж┬д─Д┬н┬и─Д ├б┬о╦З├б├в╦Ш. ┬з┬н┬а├з. ┼╗┬о
─║┬а┼Х┬а┼Ю-┬м─Г ─Г┼Х-├о}
WriteSb(x,k);
writeln;
readln;
end. |
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