Singular value decomposition

Screenshot 2025-02-20 at 15.12.40.pngScreenshot 2025-02-20 at 15.14.50.pngScreenshot 2025-02-20 at 15.15.29.pngScreenshot 2025-02-20 at 15.19.13.png

Singular Value Decomposition (SVD) is a powerful matrix factorization technique with numerous applications in numerical computing and data analysis.

Numerical Analysis Applications

In numerical analysis, SVD is particularly valuable for:

  • Solving ill-conditioned or rank-deficient linear systems
  • Data compression and dimensionality reduction
  • Noise filtering in various signal processing applications
  • Implementing Principal Component Analysis (PCA)
  • Computing pseudoinverses for non-square matrices

Numerical Algorithms

Several algorithms exist for computing the SVD:

  1. Golub-Reinsch Algorithm: The classical approach, generally stable and efficient
  2. Jacobi SVD: Highly accurate but slower for large matrices
  3. Randomized SVD: Approximates SVD for very large matrices where traditional algorithms are too costly

These numerical implementations balance computational efficiency, memory usage, and numerical stability.

See