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The exhaustive list of topics in Matrix Computations in which we provide Help with Homework Assignment and Help with Project is as follows:

Floating Point Computation:

  • Matrix computations.
  • Matrices and vectors, Norms, Singular value decompositions.
  • Floating point computations, IEEE floating point arithmetic, analysis of roundoff errors. Pitfalls of floating point computations.

Solution Of Linear Systems:

  • Concept of stability and ill-conditioning, Generic sensitivity analysis, Condition numbers, Backward and forward stability.
  • Solution of linear systems, triangular and banded systems, Gaussian elimination, Pivoting, Stability of Gaussian elimination, Cholesky factorization.

Solution Of Least Squares Of Problem:

  • Orthogonal matrices, Rotators and Reflectors, Gram-Schmidt process, QR factorizations, Stability of Householder QR factorization.
  • Solution of Least Squares Problem by QR method, The SVD and the Least Squares Problem, Sensitivity of the Least Squares Problem.
  • Solution Of Eigenvalue Problem And Computation Of SVD:
  • Eigenvalue problems, Overview of eigenvalue algorithms, Sensitivity analysis of eigenvalue problems.
  • Power, inverse power and Rayleigh quotient iterations; Reduction to Hessenberg  and Tridiagonal forms.
  • QR algorithm with and without shifts, convergence of the QR algorithm.
  • Implicit QR algorithm with Wilkinson’s shift.
  • Other eigevalue algoritms (Jacobi, bisection, divide and conquer).
  • Reduction to bi-diagonal form, Computing the SVD.

Iterative Methods For Solution Of Linear Systems:

  • Iterative methods, Jacobi, Gauss-Seidel and successive over relaxation methods. 
  • Generic Krylov process, Implicit restarts, The Arnoldi iteration, The Lanczos iteration.
  • Conjugate gradient method, Preconditioning.