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//Eg-3.5
//pg-78
clear
clc
// Matrices A and B (AX=B)
A=[-4 1 0 0 0 1;3 -7 2 0 2 0;0 4 -7 3 0 0;0 0 1 -5 4 0;0 2 0 6 -11 3;3 0 0 0 3 -8];
[n,n]=size(A);
B=[-130;0;0;0;0;0];
//Initialising Permutation matrix(which keeps note of column changes in this code) as Identity matrix
P=eye(n);
//Algorithm of gauss elimination with complete pivoting
//COMPLETE PIVOTING(both rows and columns)
//loop from 1 to n-1 pivot elements
for k=1:n-1
u=k;
l=k;
big=abs(A(k,k));
//Obtaining max absolute element and its position/index in column
for t=k+1:n
dummy=abs(A(t,k));
if dummy>big
big=dummy;
u=t;
end
end
//Obtaining max absolute element and its position/index in row
for t=k+1:n
dummy=abs(A(k,t));
if dummy>big
big=dummy;
l=t;
end
end
//Row and Column interchanges according to index of maximum absolute element obtained
if l~=k
for j=1:n//column interchanges
dummy=A(j,l);
A(j,l)=A(j,k);
A(j,k)=dummy;
temp=P(j,l);//column interchanges are noted with the help of a permutation matrix
P(j,l)=P(j,k);
P(j,k)=temp;
end
elseif u~=k
for j=1:n//row interchanges
dummy=A(u,j);
A(u,j)=A(k,j);
A(k,j)=dummy;
end
temp=B(u);//B rows are also interchanged according to A row interchanges
B(u)=B(k);
B(k)=temp;
end
//Forward Elimination
for i=(k+1):n
facotr=A(i,k)/A(k,k);
for j=1:n
A(i,j)=A(i,j)-facotr*A(k,j);
end
B(i)=B(i)-facotr*B(k);
end
end
//Backward Substitution
X=zeros(n,1);
X(n)=B(n)/A(n,n);
for i=(n-1):-1:1
summ=B(i);
for j=(i+1):n
summ=summ-A(i,j)*X(j);
end
X(i)=summ/A(i,i);
end
//Column interchanges in A correspond to row interchanges in X which can be cancelled out by multiplying with the permutation matrix defined above
X=P*X;
//Result
v1=X(1);
v2=X(2);
v3=X(3);
v4=X(4);
v5=X(5);
v6=X(6);
//Displaying Results
disp(v1,"v1=")
disp(v2,"v2=")
disp(v3,"v3=")
disp(v4,"v4=")
disp(v5,"v5=")
disp(v6,"v6=")
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