Singular value decomposition

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

• Principal Component Analysis

• Advanced Topics (in Linear Algebra)

• Non-negative Matrix Factorization