Math 5610 - Computational Linear Algebra

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Direct Methods Limitations

Below is a summary of some of the limitations of direct methods for solving linear systems of equations. Direct methods include Gauss Elimination with back substitution and factorization.

• Limitations of Gauss Elimination and Back Substitution:
  • Algorithm fails if there are any zeros on the diagonals, unless partial pivoting is employed.
  • Complexity grows with order n^3.
  • Space complexity can become an issue.
  • Error grows linearly with the size of the problem.
• Limitations of LU Decomposition:
  • Could easily be unstable.
  • Does not find approximate solutions.
• Limitations of LU Decomposition with partial pivoting:
  • Reduces instability, but in many cases still far from optimal stability
• Disadvantages of QR compared to LU:
  • More complicated to find QR decomposition
  • Slower than LU

Source 1 , Source 2