b. Solve system (1.2) using your code from part a. Also, determine whether or no

Question

b. Solve system (1.2) using your code from part a. Also, determine whether or not the matrices L and U that you found approximately satisfy A= LUusing norm(A-L*U)/norm(A) in Matlab, which finds the matrix 2-norm of the error matrix A - L * Uand divides it by norm(A).

1. A set of experiments has found that the specific volume v of saturated steam in m3/Kg at six dimensionless temperature T values of 1, 2, 3, 4, 5, and 6 (where 1 represents 10°C in physical units) is respectively 106.4, 57.79, 32.9, 19.52, 12.03, and 7.67. A fifth-degree polynomial of the form: can be used to represent volume v as a function of temperature T. The unknown coefficients x1, X2, X3, x4, xs, and x6 can be found by solving the following system of linear algebraic equations obtained by substituting the given data in equation (1.1): X1 + X2 + X3 +X,+X5 + X6 = 106.4 x1 2x2 22x3 23x424xs +25x6 57.7 x1 3x2 +32x3 +33x4+34xs +35x6 32.9 X1 + 4X2 + 42X3 + 43X4 + 44X5 + 45X6-1952 x1 5x2 52x3 +53x4+5*xs +55x6 12.03 X1 + 6X2 + 62X3 + 63X4 + 64X5 + 65X6-7.67

Explanation / Answer

The code is as follows:

function[x]=LU_Decompos(A,B)
clc
[m,n]=size(A);
if (m ~= n )
disp ( 'LR2 error: Matrix must be square' );
return;
end;
    % Part 2 : Decomposition of matrix into L and U
L=zeros(m,m);
U=zeros(m,m);
for i=1:m
% Finding L
for k=1:i-1
L(i,k)=A(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
% Finding U
for k=i:m
U(i,k) = A(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:m
L(i,i)=1;
end
% Program shows U and L
U
L
Ly=zeros(m,1); % initiation for y
y(1)=B(1)/L(1,1);
for i=2:m
   %y(i)=B(i)-L(i,1)*y(1)-L(i,2)*y(2)-L(i,3)*y(3);
    y(i)=-L(i,1)*y(1);
      for k=2:i-1
          y(i)=y(i)-L(i,k)*y(k);
      end;
      y(i)=(B(i)+y(i))/L(i,i);
end;
% Now we use this y to solve Ux = y
x=zeros(m,1);
x(m)=y(m)/U(m,m);
i=m-1;
q=0;
while (i~= 0)
x(i)=-U(i,m)*x(m);
   q=i+1;
      while (q~=m)
          x(i)=x(i)-U(i,q)*x(q);
          q=q+1;
      end;
    x(i)=(y(i)+x(i))/U(i,i);
    i=i-1;
end;
   end

For part (b)

clear all
clc
for i=1:6
    for j=1:6
        A(i,j)=i^(j-1);
    end
end
A;
b=[106.4,57.7,32.9,19.52,12.03,7.67]';
LU_Decompos(A,b)

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