Question

Linear equation question

Use Matlab for b) and c) and show all the work

Oct 3,ao16 MTH 301: Matrix Theory and Applications Project 1 on Linear System of Equations, and LU factorization This project studies a problem on heat transfer, where a steady-state temperature distribution of a thin plate is sought when the temperature around the boundary is known. Assume the plate shown in the figure represents a cross-section of a metal beam with very negligible heat flow in the direction perpendicular to the plate. Let T,T2, T denote the temperatures at the 8 interior nodes of the mesh in the figure. The temperature at a node is approximately equal to the average of 4 nearest nodes -to the left, above, right, and below. a). Write the system of equations in the matrix form Ax b. IThe matrix A should be a band matrix. Such matrices occur in a variety of applications, and often are large with many rows and columns, but relatively narrow bands. Finding A-1 and then computing x A-1b is not an efficient way of solving this problem, as you will find in the next two steps.) wkdb). Find an LU factorization of A, and then use it to solve this system. (You will notice that both factor matrices are band matrices with two nonzero diagonals below or above the main diagonal.) c) obtain A-1 and note that A-1 is a dense matrix with no band structure. When A is large, L and U can be stored in much less space than A This is another reason for preferring LU factorization of Ato A-1 itself. 00 06 00 00 5 7 30° 10 10 10% 100

Answer 1

Part A)

Solve Axeb 5-5 Eliminate x 0.3 in ea ue get

411 e., a) 다 @ み551 d s

The o Intin hang 忧Moto 1o o 0 0 6 2-1 Mull Mthpt) o.on in een ⓐ and add "thus 0 2.5 : 5 2 e l

牛 与6 丿 0-3

Part B) solution: Mathlab code to estimate LU for above solved matrix A: A-10 -70; 0 2.5 5; 0 0 6.2] B-17 2.5 6.2 Main function: Am 10 -7 0; 0 2.5 5; 0 0 6.2] Bm= [ 7 ; 2 . 5 ; 6.21; LUEstimate (Am, Bnm)

Supporting function function [L, U, x] =LUEs timate (Arn, Bm) [a bl size (Am) if (a b) disp 'Error' ); return; end % Seperate A matrix into L an U L-zeros (a, a) U-zeros (a, a) for i=1 : a % Find L for k=1:i-1 L(i,k)-Arn (1, k) ; for j=1: k-1 し(i,k)= L(i,k)-L (i,j)"U (j , k); end

し(i,k) L(1,k) /U(k,k); = end % Find U for k-i:a U(i,k) = Arn (i,k); for j=1 : i-1 end end end for i-1:a end % Display L and U b-zeros (a, 1); % initiation for B

B (1) =Bn (1) /L (1,1) ; for i-2:a for k-2:i-1 end; B (i)-(Brn (i) +B (i) ) /l. (i, i) ; end; x= zeros (a, 1) ; x (a) =B (a) /U(a, a) ; 1-a-1 ; 9-0 while (i0)

while (q -a) end; end;

Sample Output 10.00000 -7.00000 0.00000 0.00000 2.50000 5.0000e 6.2000e 0.0000e , 00000 10.00000 -7.00000 .00000 2.50000 e.0000e 0.0000e . 00000 5.0000e 6.2000e 1 0 0 7.0000 2.5000 6.2000 ans - 010

Program code:

Main function:

Am=[ 10 -7 0; 0 2.5 5; 0 0 6.2]

Bm=[7;2.5;6.2];

LUEstimate(Am,Bm)

Supporting function:

function[L,U,X]=LUEstimate(Am,Bm)

[a b]=size(Am);

if (a ~= b )

disp ( 'Error' );

return;

end;

% Seperate A matrix into L and U

L=zeros(a,a);

U=zeros(a,a);

for i=1:a

% Find L

for k=1:i-1

L(i,k)=Am(i,k);

for j=1:k-1

L(i,k)= L(i,k)-L(i,j)*U(j,k);

end

L(i,k) = L(i,k)/U(k,k);

end

% Find U

for k=i:a

U(i,k) = Am(i,k);

for j=1:i-1

U(i,k)= U(i,k)-L(i,j)*U(j,k);

end

end

end

for i=1:a

L(i,i)=1;

end

% Display L and U

U

L

b=zeros(a,1); % initiation for B

B(1)=Bm(1)/L(1,1);

for i=2:a

   %B(i)=Bm(i)-L(i,1)*B(1)-L(i,2)*B(2)-L(i,3)*B(3);

    B(i)=-L(i,1)*B(1);

      for k=2:i-1

          B(i)=B(i)-L(i,k)*B(k);

      end;

      B(i)=(Bm(i)+B(i))/L(i,i);

end;

B

x=zeros(a,1);

x(a)=B(a)/U(a,a);

i=a-1;

q=0;

while (i~= 0)

x(i)=-U(i,a)*x(a);

   q=i+1;

      while (q~=a)

          x(i)=x(i)-U(i,q)*x(q);

          q=q+1;

      end;

    x(i)=(B(i)+x(i))/U(i,i);

    i=i-1;

end;