Rozwiązywanie układów równań liniowych algebraicznych

a) c)

b) d)

Metoda Cramera

a)

program:

%n - licznik wierszy

%m - licznik kolumn

tic

clear all;

clc;

format long e

b=[11

12

13

14];

A=[1 2 3 4; 2 3 4 1; 3 4 1 2; 4 1 2 3];

W=det(A);

Wx=det([11 2 3 4; 12 3 4 1; 13 4 1 2; 14 1 2 3]);

Wy=det([1 11 3 4; 2 12 4 1; 3 13 1 2; 4 14 2 3]);

Wz=det([1 2 11 4; 2 3 12 1; 3 4 13 2; 4 1 14 3]);

Ww=det([1 2 3 11; 2 3 4 12; 3 4 1 13; 4 1 2 14]);

xA=[Wx/W

Wy/W

Wz/W

Ww/W];

xA

toc

wynik:

xA =

2.000000000000000e+00

1.000000000000000e+00

1.000000000000000e+00

9.999999999999998e-01

Elapsed time is 0.010930 seconds.

b)

program:

%n - licznik wierszy

%m - licznik kolumn

tic

clear all;

clc;

format long e

b=[1

2

-3

-6];

A=[2 -1 1 -1; 2 -1 0 -3; 3 0 -1 1; 2 2 -2 5];

W=det(A);

Wx=det([1 -1 1 -1; 2 -1 0 -3; -3 0 -1 1; -6 2 -2 5]);

Wy=det([2 1 1 -1 ; 2 2 0 -3 ; 3 0 -3 1 ; 2 -6 -2 5]);

Wz=det([2 -1 1 -1; 2 -1 2 -3; 3 0 -3 1 ; 2 2 -6 5]);

Ww=det([2 -1 1 1 ; 2 -1 0 2 ; 3 0 -1 -3; 2 2 -2 6]);

xB=[Wx/W

Wy/W

Wz/W

Ww/W];

xB

toc

wynik:

xB =

-1.973729821555835e-16

-9.555555555555559e+00

1.666666666666667e+00

-5.333333333333334e+00

Elapsed time is 0.010858 seconds.

c)

program:

%n - licznik wierszy

%m - licznik kolumn

tic

clear all;

clc;

format long e

b=[7

-2

10

0];

A=[2 2 -1 1; 1 -1 1 -1; 1 1 1 1; 4 3 -2 -1];

W=det(A);

Wx=det([7 2 -1 1; -2 -1 1 -1; 10 1 1 1; 0 3 -2 -1]);

Wy=det([2 7 -1 1; 1 -2 1 -1; 1 10 1 1; 4 0 -2 -1]);

Wz=det([2 2 7 1; 1 -1 -2 -1; 1 1 10 1; 4 3 0 -1]);

Ww=det([2 2 -1 7; 1 -1 1 -2; 1 1 1 10; 4 3 -2 0]);

xC=[Wx/W

Wy/W

Wz/W

Ww/W];

xC

toc

wynik:

xC =

1.000000000000001e+00

1.999999999999999e+00

3.000000000000000e+00

4.000000000000000e+00

Elapsed time is 0.010612 seconds.

d)

program:

%n - licznik wierszy

%m - licznik kolumn

tic

clear all;

clc;

format long e

b=[3

0

-2

-2];

A=[3 1 1 2; 3 2 3 5; 6 5 1 5; 3 2 3 7];

W=det(A);

Wx=det([3 1 1 2; 0 2 3 5; -2 5 1 5; -2 2 3 7]);

Wy=det([3 3 1 2; 3 0 3 5; 6 -2 1 5; 3 -2 3 7]);

Wz=det([3 1 3 2; 3 2 0 5; 6 5 -2 5; 3 2 -2 7]);

Ww=det([3 1 1 3; 3 2 3 0; 6 5 1 -2; 3 2 3 -2]);

xD=[Wx/W

Wy/W

Wz/W

Ww/W];

xD

toc

wynik:

xD =

2.000000000000000e+00

-1.999999999999999e+00

9.999999999999997e-01

-9.999999999999997e-01

Elapsed time is 0.010689 seconds.

Metoda dzielenia lewostronnego A\b

a)

program:

%n - licznik wierszy

%m - licznik kolumn

tic

clear all;

clc;

format long e

b=[11

12

13

14];

A=[1 2 3 4; 2 3 4 1; 3 4 1 2; 4 1 2 3];

xA=A\b;

xA

toc

wynik:

xA =

2.000000000000000e+00

1.000000000000000e+00

1.000000000000000e+00

1.000000000000000e+00

Elapsed time is 0.009719 seconds.

b)

program:

%n - licznik wierszy

%m - licznik kolumn

tic

clear all;

clc;

format long e

b=[1

2

-3

-6];

A=[2 -1 1 -1; 2 -1 0 -3; 3 0 -1 1; 2 2 -2 5];

xB=A\b;

xB

toc

wynik:

xB =

7.401486830834377e-17

2.000000000000000e+00

1.666666666666667e+00

-1.333333333333333e+00

Elapsed time is 0.085826 seconds.

c)

program:

%n - licznik wierszy

%m - licznik kolumn

tic

clear all;

clc;

format long e

b=[7

-2

10

0];

A=[2 2 -1 1; 1 -1 1 -1; 1 1 1 1; 4 3 -2 -1];

xC=A\b;

xC

toc

wynik:

xC =

1

2

3

4

Elapsed time is 0.009957 seconds.

d)

program:

%n - licznik wierszy

%m - licznik kolumn

tic

clear all;

clc;

format long e

b=[3

0

-2

-2];

A=[3 1 1 2; 3 2 3 5; 6 5 1 5; 3 2 3 7];

xD=A\b;

xD

toc

wynik:

xD =

2.000000000000000e+00

-2.000000000000000e+00

9.999999999999994e-01

-9.999999999999997e-01

Elapsed time is 0.009966 seconds.

Metoda inv(A)*b

a)

program:

%n - licznik wierszy

%m - licznik kolumn

tic

clear all;

clc;

format long e

b=[11

12

13

14];

A=[1 2 3 4; 2 3 4 1; 3 4 1 2; 4 1 2 3];

xA=inv(A)*b;

xA

toc

wynik:

xA =

1.999999999999999e+00

1.000000000000000e+00

9.999999999999999e-01

1.000000000000000e+00

Elapsed time is 0.009778 seconds.

b)

program:

%n - licznik wierszy

%m - licznik kolumn

tic

clear all;

clc;

format long e

b=[1

2

-3

-6];

A=[2 -1 1 -1; 2 -1 0 -3; 3 0 -1 1; 2 2 -2 5];

xB=inv(A)*b;

xB

toc

wynik:

xB =

0

2.000000000000000e+00

1.666666666666667e+00

-1.333333333333333e+00

Elapsed time is 0.010263 seconds.

c)

program:

%n - licznik wierszy

%m - licznik kolumn

tic

clear all;

clc;

format long e

b=[7

-2

10

0];

A=[2 2 -1 1; 1 -1 1 -1; 1 1 1 1; 4 3 -2 -1];

xC=inv(A)*b;

xC

toc

wynik:

xC =

1.000000000000000e+00

2.000000000000000e+00

3.000000000000001e+00

4.000000000000000e+00

Elapsed time is 0.009957 seconds.

d)

program:

%n - licznik wierszy

%m - licznik kolumn

tic

clear all;

clc;

format long e

b=[3

0

-2

-2];

A=[3 1 1 2; 3 2 3 5; 6 5 1 5; 3 2 3 7];

xD=inv(A)*b;

xD

toc

wynik:

xD =

2.000000000000000e+00

-2.000000000000000e+00

9.999999999999998e-01

-9.999999999999998e-01

Elapsed time is 0.010038 seconds.

Metoda eliminacji Gaussa

a)

program:

tic

clear all;

clc;

format long e

b=[ 11

12

13

14];

A=[1 2 3 4; 2 3 4 1; 3 4 1 2; 4 1 2 3];

[nb mb]= size (b);

[n m]= size (A);

L= eye (n);

U= [A b];

for k = 1 :m -1

L(k+1:n,k)=U(k+1:n,k)/U(k,k);

U(k+1:n,:)=U(k+1:n,:)-L(k+1:n,k)*U(k,:);

end

L2=eye(n);

for k = m : -1 : 2

L2(1:k-1,k)=U(1:k-1,k)/U(k,k);

U(1:k-1,:)=U(1:k-1,:)-L2(1:k-1,k)*U(k,:);

end

U1 = U(:,1:m);

B=U(:,m+1:m+mb);

for k=1 : n

x(k,:)=B(k,:)/U1(k,k);

end

x

toc

wynik:

x =

2

1

1

1

Elapsed time is 0.014379 seconds.

b)

program:

tic

clear all;

clc;

format long e

b=[ 1

2

-3

-6];

A=[2 -1 1 -1; 2 -1 0 -3; 3 0 -1 1; 2 2 -2 5];

[nb mb]= size (b);

[n m]= size (A);

L= eye (n);

U= [A b];

for k = 1 :m -1

L(k+1:n,k)=U(k+1:n,k)/U(k,k);

U(k+1:n,:)=U(k+1:n,:)-L(k+1:n,k)*U(k,:);

end

L2=eye(n);

for k = m : -1 : 2

L2(1:k-1,k)=U(1:k-1,k)/U(k,k);

U(1:k-1,:)=U(1:k-1,:)-L2(1:k-1,k)*U(k,:);

end

U1 = U(:,1:m);

B=U(:,m+1:m+mb);

for k=1 : n

x(k,:)=B(k,:)/U1(k,k);

end

x

toc

wynik:

x =

NaN

NaN

NaN

NaN

Elapsed time is 0.014151 seconds.

c)

program:

tic

clear all

clc

format long e

b=[7

-2

10

0];

A=[2 2 -1 1;1 -1 1 -1;1 1 1 1;4 3 -2 -1];

[nb mb] = size(b);

[n m] = size(A);

L=eye(n);

U=[A b];

for k = 1 : m-1

L(k+1:n,k) = U(k+1:n,k)/U(k,k);

U(k+1:n,:) = U(k+1:n,:)-L(k+1:n,k)*U(k,:);

end

L2 = eye(n);

for k=m : -1 : 2

L2(1:k-1,k) = U(1:k-1,k)/U(k,k);

U(1:k-1,:) = U(1:k-1,:)-L2(1:k-1,k)*U(k,:);

end

U1 = U(:,1:m);

B = U(:,m+1:m+mb);

for k = 1 : n

x(k,:) = B(k,:)/U1(k,k);

end

x

toc

wynik:

x =

1

2

3

4

Elapsed time is 0.013919 seconds.

d)

program:

tic

clear all

clc

format long e

b=[3

0

-2

-2];

A=[3 1 1 2;3 2 3 5;6 5 1 5;3 2 3 7];

[nb mb] = size(b);

[n m] = size(A);

L=eye(n);

U=[A b];

for k = 1 : m-1

L(k+1:n,k) = U(k+1:n,k)/U(k,k);

U(k+1:n,:) = U(k+1:n,:)-L(k+1:n,k)*U(k,:);

end

L2 = eye(n);

for k=m : -1 : 2

L2(1:k-1,k) = U(1:k-1,k)/U(k,k);

U(1:k-1,:) = U(1:k-1,:)-L2(1:k-1,k)*U(k,:);

end

U1 = U(:,1:m);

B = U(:,m+1:m+mb);

for k = 1 : n

x(k,:) = B(k,:)/U1(k,k);

end

x

toc

wynik:

x =

2

-2

1

-1

Elapsed time is 0.015360 seconds.

Wnioski

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