Question

Please Urgent help me!!!(QR decomposition queastion)

You have not to solve all parts of question!!!

The QR decomposition can be used to solve a linear system. Let A be an n x n matrix, with A system Axb can be written as QR. Then, the linear QRx = b The process goes as follows Solve Qy b for y Solve Rx-y for x a. It is very easy to solve for y without using Gaussian elimination. Why? b. The solution to Rxy can be done quickly why? c. Develop a function qr solve that solves an n × n system using the MATLAB reduced QR decomposition. If the matrix is singular, the QR decomposition will complete with no error. Make the return values [x, resid], where resid is the residual |Ax -bl. A large residual will indicate that the matrix is singular or ill-conditioned. d. Apply the function to solve each system 1-10 21 3 -4 8 1 3 59 0 1 2-22 35*- 3 78 100 3

Answer 1

save below code as qrsolve.m

function qrsolve(A,B)
[a,b]=size(A);
R=A; %Start with R=A
Q=eye(a); %Set Q as the identity matrix
for k=1:a-1
X=zeros(a,1);
X(k:a,1)=R(k:a,k);
F=norm(X);
v=X; v(k)=X(k)+F;

s=norm(v);
if s~=0, w=v/s; u=2*R'*w;
R=R-w*u';
Q=Q-2*Q*w*w';
end
end


Xsolved = inv(R)*(Q'*B)
residual=abs(A*Xsolved-B)
end

OUTPUT---------------------TEST-------------CODE-------------------------------------------------------

>>qrsolve([1 -1 0;2 4 5;-7 1 3], [1; -1;8])
Xsolved =

  -0.75000
  -1.75000
   1.50000

residual =

   8.8818e-16
   1.7764e-15
   0.0000e+00

>> qrsolve([21 3 -4 8;1 3 59 0;1 2 -22 35;3 78 100 3],  [1; -1; 1; 2])
Xsolved =

   0.032058
   0.049597
  -0.020014
   0.012241

residual =

   4.4409e-16
   4.4409e-16
   2.2204e-16
   4.4409e-16