Question

I need the Matlab code and the Solution to the problem.

Answer 1

Note: I have modified the GEpivot.m and saved it in the file GaussWithOutPivot.m

Answer: Code %File Name: GEpivot %Define function function [x, lu,piv] - GEpivot (A, b) %Find value [m, nsize (A) %Check condition &Prin t error 'The matrix is not square.' %End %Find value mlength (b); %Check condition %print eFror ( The atrix and the vector do not match in size.

%End end %Create variable piv- (1:n)'; %Loop for k- 1:n-1 %Find value [colmax index] max (abs (A (k: n, k) ) ) ; = - %Increase value index - index k-1; %Check condition index ~= k %swap rows k and index tempAA (k, k:n); %Copy value A(k, k : n) A(index, k: n); = %Copy value A (index, k:n) tempA;

%swap b %Copy value tempbb (k) %Copy value b (k) = b (index); &Copy value b (index) = tempb; %swap pivot %Copy value temp piv (k) ; %Copy value piv(k) -piv(index); &Copy value piv (index) temp ; = %End end %Find value %Loop

for i k+1 : n = %Find value %End end %Find value b(k+1 : n) b(kt1:n) -A(k+1:n, k) *b ( k ) ; = %End end &Create x = zeros (n, 1) ; %set value x (n) b (n) /A (n, n) ; = %Loop for in-1:-1:1 %Find value %End

end %set value lu = A; %End end %File Name: GausswithoutPivot.n %Define function function x = GausswithOutpivot (A, b) %Find size [n, n] = size (A) %Find size [n, k] = size (b); %create variable tx = zeros (n, k) ; %Loop for i1:n-1 %Find value

%Find value %Find value %End end %Find value tx (n, :) %Loop for i = n-1:-1:1 %Find value %End end set value %End

%File Name: MysampleRun.n %Sample Run nAry[2 5 10 20 : SPrint &Print message %Print stars fprintf(* %Loop for kk= 1: length (nary) %Get current n value nnAry (kk); Define A A - z eros (n, n); %Define b b- ones (n, 1) %Loop for i-1:n

%Loop for j = 1:n %Find value A(i,j) max (i,j); %End end %End end %Find value xDinv (A) *bi %Call function (xP, lu, piv] = GEpivot (A, b); %Call function [xNP] Gaus sWithoutpivot (A,b); = %Find norm between xD and XP err1 = norm (xD - XP, Inf); %Find norm between xD and xNP err2 = norm (xD- xNP, inf);

Print value fprintf("ti\t%f\t%f\n", n, err!, err2); %End end Print Sample Output XD- xP 0.000000 0.000000 0.000000 0.000000 0.000000 10 20 0.000000

Copyable code:

%File Name: GEpivot.m

%Define function

function [x,lu,piv] = GEpivot(A,b)

     %Find value

     [m,n] = size(A);

     %Check condition

     if m ~= n

     %Print

     error('The matrix is not square.')

     %End

     end

     %Find value

     m = length(b);

     %Check condition

     if m ~= n

     %Print

     error('The matrix and the vector do not match in size.')

     %End

     end

     %Create variable

     piv = (1:n)';

     %Loop

     for k = 1:n-1

     %Find value  

     [col_max index] = max(abs(A(k:n,k)));

     %Increase value

     index = index + k-1;

     %Check condition

     if index ~= k

     %Swap rows k and index

     tempA = A(k,k:n);

     %Copy value

     A(k,k:n) = A(index,k:n);

     %Copy value

     A(index,k:n) = tempA;

     %Swap b

     %Copy value

     tempb = b(k);

     %Copy value

     b(k) = b(index);

     %Copy value

     b(index) = tempb;

     %Swap pivot

     %Copy value

     temp = piv(k);

     %Copy value

     piv(k) = piv(index);

     %Copy value

     piv(index) = temp;

     %End

     end

     %Find value

     A(k+1:n,k) = A(k+1:n,k)/A(k,k);

     %Loop

     for i = k+1:n

     %Find value

     A(i,k+1:n) = A(i,k+1:n) - A(i,k)*A(k,k+1:n);

     %End

     end

     %Find value

     b(k+1:n) = b(k+1:n) - A(k+1:n,k)*b(k);

     %End

     end

     %Create

     x = zeros(n,1);

     %Set value

     x(n) = b(n)/A(n,n);

     %Loop

     for i = n-1:-1:1

     %Find value

     x(i)=(b(i)-A(i,i+1:n)*x(i+1:n))/A(i,i);

     %End

     end

     %Set value

     lu = A;

%End

end

%File Name: GaussWithOutPivot.m

%Define function

function x = GaussWithOutPivot(A, b)

     %Find size

     [n, n] = size(A);

     %Find size   

     [n, k] = size(b);

     %Create variable   

     tx = zeros(n,k);

     %Loop   

     for i = 1:n-1

     %Find value

     tp = -A(i+1:n,i)/A(i,i);

     %Find value

     A(i+1:n,:) = A(i+1:n,:) + tp*A(i,:);

     %Find value

     b(i+1:n,:) = b(i+1:n,:) + tp*b(i,:);

     %End

     end

     %Find value

     tx(n,:) = b(n,:)/A(n,n);

     %Loop

     for i = n-1:-1:1

     %Find value

     tx(i,:) = (b(i,:) - A(i,i+1:n)*tx(i+1:n,:))/A(i,i);

     %End

     end

     %Set value

     x = tx;

%End

end

%File Name: MySampleRun.m

%Sample Run

nAry = [2 5 10 20 ];

%Print

fprintf("\n************************************\n");

%Print message

fprintf("n\t||xD - xP||\t||xD - xNP||\n");

%Print stars

fprintf("************************************\n");

%Loop

for kk= 1: length(nAry)

     %Get current n value

     n = nAry (kk);

     %Define A

     A = zeros(n,n);

     %Define b

     b = ones(n, 1);

     %Loop

     for i=1:n

          %Loop

          for j = 1:n

          %Find value

          A(i,j) = max(i,j);

          %End

          end

     %End

     end

     %Find value

     xD = inv(A)*b;

     %Call function

     [xP, lu, piv] = GEpivot(A, b);

     %Call function

     [xNP] = GaussWithOutPivot(A,b);

     %Find norm between xD and xP    

     err1 = norm(xD - xP, inf);

     %Find norm between xD and xNP

     err2 = norm(xD - xNP, inf);

     %Print value

     fprintf("%i\t%f\t%f\n", n, err1, err2);

%End

end

%Print

fprintf("************************************\n");