Direct Linear System Solvers

Solve linear systems \( Ax = b \) using direct elimination methods. These methods provide exact solutions (within machine precision) in finite steps.

Gaussian Elimination

Classical elimination method with forward elimination and back substitution

Complexity:O(n³)
Stability:Stable with pivoting
Key Features:
  • Forward elimination
  • Back substitution
  • Partial pivoting
Process: Ax = b → Ux = c
Use Gaussian Elimination
Gauss Jordan

Modified Gaussian elimination that produces reduced row echelon form

Complexity:O(n³)
Stability:Stable
Key Features:
  • Complete elimination
  • Diagonal form
  • Direct solution
Process: Ax = b → Ix = x
Use Gauss Jordan
LU Decomposition

Factorizes matrix into lower and upper triangular matrices

Complexity:O(n³)
Stability:Very Stable
Key Features:
  • Matrix factorization
  • Multiple RHS
  • Efficient for repeated solving
Process: A = LU, then Ly = b, Ux = y
Use LU Decomposition
About Direct Methods

Direct methods solve linear systems \( Ax = b \) by transforming the coefficient matrix through a finite sequence of operations. These methods are characterized by:

  • Finite termination: Exact solution in finite steps
  • Predictable cost: Known computational complexity
  • Reliability: Work for most well-conditioned systems
  • Memory efficiency: Can operate in-place