EE 510 Computational Problems (GPP)
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EE 510 Computational Problems (GPP)
A = # -, b = # - , x!" = # - , x!2 = # -
Problem 1:
(i)Find A$"
(ii) Find detA
(iii)Find the characteristic polynomial of A
(iv)Find the eigenvalues and eigenvectors of A
(v)Bring A to Jordanform
(vi) Find the Singular Value Decomposition of A
(vii)Find ‖A‖, ‖A$" ‖, exp(At)
(viii)Solve A x = b using Gaussian elimination.
(ix)Solve A x = b using Jacobi, starting with initial conditions x!" and x!2 . (x)Solve A x = b using Gauss-Seidel, starting with initial conditions x!" and x!2 . (xi)Solve A x = b using SOR, starting with initial conditions x!" and x!2 .Try several w’s: -1,0,1, 2,3
(xii)Solve A x = b using Conjugate Gradient (A%A x = A% b), with initial conditions
x!" and x!2 .
and x!2 . Experiment with different choices of e.
Compare the results of the methods (viii)-(xiii).
(xiv)Do the QR decomposition of A
(xv) Try to find the eigenvalues of A by the following method: Do the QR decomposition of Yk'" where Yk'" = Rk Qk and where Qk Rk is the QR
decomposition of Yk .Start with Y! = A
(xiii)Solve A x = b using xk'" = xk + e(A%A xk − A% b), with initial conditions x!"
(xvii) Do the Choleskyfactorization of A%A and AA% .
Problem 2 (Optional)
Let B=A × # |
0 0 54 0 |
0 0 0 224 |
- |
$" |
(i) Find the eigenvalues of B and B% .
(ii) Find det(B) .
(iii) Find exp(Bt) .
(iv) Do the QR factorization of B.
(v) Find the singular value decomposition of B.
(vi) Do the Choleskyfactorization of B% B and BB% .
Comment on methods and results.
Problem 3
For
A = #
solve the problem:
(i)Subject to the condition: rankX ≤ 3,where X is a matrix 4 × 4 and the norm is the usual sup normfor matrices.
(ii) Subject to the condition: rankX ≤ 1,where X is a matrix 4 × 4 and the norm is the usual sup normfor matrices.
(iii) (Optional)Subject to the condition that X is a matrix 4 × 4 that has nonnegative elements and the norm is the usual sup normfor matrices.
Note: Use any software you prefer (Matlab, Mathematica, Python, C++, etc.) but mention what you use and any information on the method/program you use.
2022-11-30