Lecture Notes: Condition Number in Numerical Analysis

Introduction

• Topic: Condition number in relation to matrices.
• Lecturer: Dr. Harishkar
• The lecture is part of a series on numerical analysis available on the lecturer's YouTube channel.

Solving Systems of Equations

• Systems of equations can be solved via two main methods:
  • Direct Method
  • Iterative Method (Indirect Method)
• Important to know that some systems are sensitive to perturbations in the values of A or B.

Stability of Solutions

• A system may have unstable solutions if small changes in A or B lead to large changes in the results.
  • Example: Perturbing B values leads to significant differences in solutions.
• Rounding or truncating numbers can drastically change the solution, indicating instability.

Condition Numbers

• Condition number measures how sensitive a system is to perturbations.
• Systems are classified as:
  • Ill-Conditioned: Significant changes in the solution for small changes in A or B.
  • Well-Conditioned: Small changes in A or B lead to small changes in the solution.
• Measuring ill-condition requires understanding the condition number:
  • Large condition number indicates ill-conditioning.
  • Small condition number indicates well-conditioning.

Definitions and Properties

• For a system Ax = B:
  • A must be invertible (det(A) ≠ 0).
• Perturbations:
  • Change in B: B + ΔB
  • Change in A: A + ΔA
• Relative error in the solution can be bounded by:
  • ||ΔX|| ≤ K * ||ΔB||
  • ||ΔX|| ≤ K * ||ΔA||
• Condition number (K) relates relative changes in the solution to the changes in A and B.

Calculation of Condition Numbers

• The condition number is calculated using different norms (e.g., infinity norm, spectral norm).
• Spectral norm involves eigenvalues:
  • K(A) = (λ_max / λ_min) where λ_max and λ_min are the maximum and minimum eigenvalues of A.

Practical Examples

1. Using Absolute Row Sum Norm:
   • Find the absolute sum of rows in A and A inverse to compute K.
2. Using Spectral Norm:
   • For symmetric matrices, the condition number can be derived from eigenvalues.

Conclusion

• Understanding and calculating the condition number is essential for determining the stability of solutions in numerical analysis.
• Next topics will include Gaussian elimination methods and further applications of condition numbers.

• Encouragement to like, share, and comment on the lecture video.