Givens QR to find Symmetric Tridiagonal Eigenpairs Octave Implementation

John Bryan

• Givens QR to find Symmetric Tridiagonal Eigenpairs.
  
  
• n = number of rows = number of columns
  
  Initial conditions:
  
  $$ \text{initial matrix} = \mathbf{A} $$ $$\mathbf{Q0} = \mathbf{I_n} $$ $$\mathbf{V0} = \mathbf{I_n} $$ $$\mathbf{P} = \mathbf{I_n} $$ $$\mathbf{P0} = \mathbf{A}. $$
• while any nondiagonal entry absolute value is greater than chosen small number loop
  
  
  • for loop from k=1:n-1
    
    
    1. $$\mathbf{Q0}=\mathbf{I_n} $$ $$\mathbf{P}=\mathbf{I_n} $$
    2. $$[c,s]=\text{Givens Rotation}(\mathbf{P0}(k,k),\mathbf{P0}(k+1,k)) $$ $$ \mathbf{P}(k,k) = c $$ $$ \mathbf{P}(k,k+1) = -s $$ $$ \mathbf{P}(k+1,k) = s $$ $$ \mathbf{P}(k+1,k+1) = c $$
    3. $$ \mathbf{P0}=\mathbf{P}*\mathbf{A} $$
    4. $$ \mathbf{Q0}=\mathbf{Q0}*\mathbf{P}^\intercal $$
    5. $$ \mathbf{R0}=\mathbf{P0} $$
    6. $$ \mathbf{A}=\mathbf{R0*Q0} $$
    7. $$ \mathbf{V0}=\mathbf{V0*Q0} $$
    end for loop
    
    
  end while loop
  
  
• Octave implementation.
  
  
• Results.
  
  

  Check of n=5 computed eigenpairs by inserting into eigenvalue equation.
  using eigenpair 1	using eigenpair 2	using eigenpair 3	using eigenpair 4	using eigenpair 5
  -3.45e-14	7.76e-14	-5.56e-14	-1.25e-14	8.88e-16
  -6.46e-14	6.25e-14	-4.62e-15	1.45e-14	5.16e-16
  -6.38e-14	1.36e-14	4.41e-14	-4.48e-14	8.88e-16
  -5.60e-14	-7.64e-14	1.77e-15	1.20e-14	8.88e-16
  -3.11e-14	-6.15e-14	-5.41e-14	-1.51e-14	1.01e-15