ЧМ ЛР6

>> A = rand(4, 4)

A =

0.6787 0.6555 0.2769 0.6948

0.7577 0.1712 0.0462 0.3171

0.7431 0.7060 0.0971 0.9502

0.3922 0.0318 0.8235 0.0344

>> f = rand(4, 1)

f =

0.4387

0.3816

0.7655

0.7952

>> det(A)

ans = 0.0562

Определитель не равен нулю, значит, матрица А не вырожденная

>> A1 = inv(A)

A1 =

0.8947 1.9082 -1.2817 -0.2567

6.9823 -0.6046 -4.8407 -1.7432

-0.4517 -0.8455 0.5636 1.3472

-5.8416 -0.9566 5.5940 1.3583

А1 – обратная матрица

>> X = A1*f

X =

-0.0647

-2.2591

0.9820

2.4344

for i=1:1:4

B = A;

B(:,i) = f;

x(i) = det(B)/det(A);

end

>> x'

ans =

-0.0647

-2.2591

0.9820

2.4344

B = [A,f];

for j = 1:1:3

[m, p] = max(B(j:4,j));

buf = B(p+j-1,:);

B(p+j-1,:) = B(j,:);

B(j,:) = buf;

for i=4:-1:j+1

B(i,:) = B(i,:)-B(i-1,:)*(B(i,j)/B(i-1,j));

end

end

for i = 1:1:3

for j = i+1:1:4

B(i,:) = B(i,:)-B(j,:)*B(i,j)/B(j,j);

end

end

for i = 1:1:4

B(i,:) = B(i,:)/B(i,i);

end

X = B(:,5)

X =

-0.0647

-2.2591

0.9820

2.4344

>> rref([A f])

ans =

Columns 1 through 4

1.0000 0 0 0

0 1.0000 0 0

0 0 1.0000 0

0 0 0 1.0000

Column 5

-0.0647

-2.2591

0.9820

2.4344

>> X = A\f

X =

-0.0647

-2.2591

0.9820

2.4344

>> A = rand(20, 20)

A =

>> X = rand(20,1)

X =

0.3411

0.6074

0.1917

0.7384

0.2428

0.9174

0.2691

0.7655

0.1887

0.2875

0.0911

0.5762

0.6834

0.5466

0.4257

0.6444

0.6476

0.6790

0.6358

0.9452

>> cond(A)

ans = 132.2375

>> A(12, 15) = 1000;

>> cond(A)

ans = 5.1522e+04

>> f = A*X

f =

4.8630

4.9108

5.1919

5.6667

4.6585

5.2881

5.2380

5.2048

6.1637

6.0354

5.1088

430.2967

5.2014

5.3065

4.8302

3.6557

3.9953

5.2577

4.9717

5.2846

>> A1 = inv(A)

>> x = A1*f

>> e = abs(X-x)

>> e'

ans =

1.0e-13 *

0.0216 0.0722 0.0883 0.0056 0.0230 0.0722 0.0105 0.0211 0.0083 0.0333 0.0125 0.0067 0.0111 0.0322 0.0006 0.0178 0.0111 0.3653 0.0977 0.0666

for i=1:1:20

B = A;

B(:,i) = f;

x(i) = det(B)/det(A);

end

>> e = abs(X-x)

>> e'

ans =

1.0e-14 *

0.0999 0.0222 0.5690 0.1776 0.0916 0.1998 0.1443 0 0.1388 0.2109 0.3622 0.0555 0.0222 0.7327 0.0444 0.6328 0.1998 0.7661 0.8549 0.8771

>> x = A\f

>> e = abs(X-x)

>> e'

ans =

1.0e-14 *

0.0722 0.0555 0.0583 0.1665 0.0749 0.1776 0.0833 0.0777 0.2359 0.0722 0.1679 0.1998 0.2109 0 0.0056 0.1221 0.0999 0.1554 0.0999 0.3331

B = [A,f];

for j = 1:1:19

[m, p] = max(B(j:20,j));

buf = B(p+j-1,:);

B(p+j-1,:) = B(j,:);

B(j,:) = buf;

for i=20:-1:j+1

B(i,:) = B(i,:)-B(i-1,:)*(B(i,j)/B(i-1,j));

end

end

for i = 1:1:19

for j = i+1:1:20

B(i,:) = B(i,:)-B(j,:)*B(i,j)/B(j,j);

end

end

for i = 1:1:20

B(i,:) = B(i,:)/B(i,i);

end

x = B(:,21)

>> e = abs(X-x)

>> e'

ans =

1.0e-09 *

0.0336 0.1283 0.0320 0.0236 0.0047 0.0622 0.0046 0.0048 0.0293 0.0328 0.0557 0.0703 0.0015 0.0320 0.0000 0.1303 0.0379 0.0326 0.0317 0.0366