Question

Implement Gaussian elimination(with partial pivoting) and backward substitu- tion in MATLAB. You need to submit your code on moodle page.

Answer 1

MATLAB code for the Gaussian elimination with partial pivoting and backward substitution to find the roots are provided below, please comment if any doubts:

Code:

%set the value of A anf b
A = [1 1 -2; 2 -3 1; 3 1 4]
B = [1; -8; 7]

%%Call the function
Guass_partial_back(A,B)

%%The function to peform the guassian eliminatio
%by partial pivoting and back substitution
function [myroots,nrow,AugmentedMat]= Guass_partial_back(A,b);
   %create the augmented matrix
   augMat=[A b];
  
   %find the rank of the matrx
   ra=rank(A);
  
   %create a row vector of size rank
   for itr=1:ra
       nrow(itr)=itr;
   end
  
   %fibnd the transverse
   nrow=nrow';
   for i=1:ra-1;
   largest=0;
   numIndx=0;
   %find maximum in a row and return it
   for k=i:ra
       if abs(augMat(nrow(k),i))>largest
           largest=abs(augMat(nrow(k),i));
           numIndx=k;
       end
   end
  
   %row exchange procedure
   if nrow(i)~=nrow(numIndx)
       tempRow=nrow(i);
       nrow(i)=nrow(numIndx);
       nrow(numIndx)=tempRow;
   end
  
   %perform the Gaussian elimination
   for itr=(i+1):ra
       n(nrow(itr),i)=augMat(nrow(itr),i)/augMat(nrow(i),i);
     
       for k=i:ra+1
           augMat(nrow(itr),k)=augMat(nrow(itr),k)-n(nrow(itr),i)*augMat(nrow(i),k);
       end
          
   end
   end
  
   %the backward subsitution to find the roots
   x(ra)=0;
   x=x';
   x(ra)=augMat(nrow(ra),ra+1)/augMat(nrow(ra),ra);
   itr=ra-1;
  
   %find all x values
   while itr>0
       x(itr)=(augMat(nrow(itr),ra+1)-augMat(nrow(itr),itr+1:ra)*x(itr+1:ra))/(augMat(nrow(itr),itr));
   itr=itr-1;
   end
  
   %return the solution
   myroots=x;
  
   %augmented matrix
   AugmentedMat=augMat;

end

Output: