Verification and Exploration of Linear Algebra Techniques

2 min read

📅 2024-08-20 — Session: Verification and Exploration of Linear Algebra Techniques

🕒 03:40–06:30
🏷️ Labels: LAPACK, Linear Algebra, SVD, Gaussian Elimination, Jordan Form
📂 Project: Dev
⭐ Priority: MEDIUM

Session Goal

The session aimed to verify LAPACK routine solutions and explore advanced linear algebra techniques, including Gaussian elimination, Jordan block structures, and Singular Value Decomposition (SVD).

Key Activities

  • Verified the solution of the LAPACK routine DGETRS using LU factorization to ensure it satisfies the equation A · X = B.
  • Conducted Gaussian elimination with partial pivoting on a 4x4 matrix, covering matrix setup, elimination, and back substitution.
  • Provided an in-depth guide on Jordan block structures and Jordan canonical forms, explaining their significance and computation methods.
  • Reflected on the evolution and impact of LAPACK in numerical linear algebra, highlighting its innovations and relevance.
  • Implemented a Singular Value Decomposition (SVD) routine for complex matrices, detailing initialization, decomposition, and QR iteration.
  • Explained how to compute the null space of a matrix using SVD, providing an intuitive understanding of the concept.
  • Explored methods for computing the kernel of a matrix, focusing on SVD as an efficient technique.

Achievements

  • Successfully verified LAPACK routine solutions and deepened understanding of linear algebra techniques.
  • Developed comprehensive guides and reflections on key matrix operations and their applications.

Pending Tasks

  • Further exploration of alternative methods for computing the kernel of a matrix beyond SVD.

Tags

LAPACK, Linear Algebra, SVD, Gaussian Elimination, Jordan Form

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