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Session 4
Numerical Methods

Direct Methods for Linear Systems: Gaussian Elimination

4 hours
Duration
8
Materials
7
Objectives
Session Overview

Comprehensive treatment of direct methods for solving linear systems including Gaussian elimination with partial and complete pivoting, LU decomposition, and computational complexity analysis.

Learning Objectives
By the end of this session, you should be able to:
  • Master Gaussian elimination with partial pivoting for numerical stability
  • Understand and implement complete pivoting strategies
  • Derive and implement LU decomposition using Doolittle and Crout methods
  • Analyze computational complexity: O(n³) operations for elimination
  • Handle special matrix structures: tridiagonal, symmetric, and banded matrices
  • Understand forward and backward substitution algorithms
  • Evaluate numerical stability and condition number effects
Course Materials
Download materials for offline study and reference
Comprehensive Linear Algebra Review (40 pages)
Available material
Pivoting Strategy Analysis and Implementation
Available material
LU Decomposition Algorithms with Examples
Available material
Computational Complexity Analysis
Available material
Special Matrix Solvers (Tridiagonal, Symmetric)
Available material
Numerical Stability Case Studies
Available material
Programming Assignment: Linear System Solver Package
Available material
Performance Benchmarking Tools
Available material