{R}R 開発ノート

合計 10 件の記事が見つかりました。

8.4 PCA and Spectral Methods

An intuitive, in-depth explanation of PCA, spectral clustering, and eigenvector-based data analysis. Covers covariance matrices, graph Laplacians, and why eigenvalues reveal hidden structure in data. Concludes Chapter 8 and leads naturally into SVD in Chapter 9.
2025-10-10

8.3 The QR Algorithm (High-Level Intuition)

A clear, intuitive, and comprehensive explanation of the QR algorithm—how repeated QR factorizations reveal eigenvalues, why orthogonal transformations provide stability, and how shifts and Hessenberg reductions make the method efficient. Ends with a smooth bridge to PCA and spectral methods.
2025-10-09

8.2 Rayleigh Quotient

An intuitive and comprehensive explanation of the Rayleigh quotient, why it estimates eigenvalues so accurately, how it connects to the power method and inverse iteration, and why it forms the foundation of modern eigenvalue algorithms. Ends with a natural transition to the QR algorithm.
2025-10-08

8.1 Power Method and Inverse Iteration

A clear, practical, and intuitive explanation of the power method and inverse iteration for computing eigenvalues. Covers dominance, repeated multiplication, shifted inverse iteration, and real applications in ML, PCA, and large-scale systems. Smoothly introduces the Rayleigh quotient.
2025-10-07

Chapter 8 — Eigenvalues and Eigenvectors

A deep, intuitive introduction to eigenvalues and eigenvectors for engineers and practitioners. Explains why spectral methods matter, where they appear in real systems, and how modern numerical algorithms compute eigenvalues efficiently. Leads naturally into the power method and inverse iteration.
2025-10-06

7.1 Gram–Schmidt and Modified GS

A clear, practical, book-length explanation of Gram–Schmidt and Modified Gram–Schmidt, why classical GS fails in floating-point arithmetic, how MGS improves stability, and why real numerical systems eventually rely on Householder reflections. Ideal for ML engineers, data scientists, and numerical computing practitioners.
2025-10-02

Chapter 7 — QR Decomposition

A deep, intuitive introduction to QR decomposition, explaining why orthogonality and numerical stability make QR essential for least squares, regression, kernel methods, and large-scale computation. Covers Gram–Schmidt, Modified GS, Householder reflections, and why QR is often preferred over LU and normal equations.
2025-10-01

5.4 Practical Examples

Hands-on LU decomposition examples using NumPy and LAPACK. Learn how pivoting, numerical stability, singular matrices, and performance optimization work in real systems, with clear Python code and practical insights.
2025-09-26

2.4 Vector and Matrix Storage in Memory

A clear, practical guide to how vectors and matrices are stored in computer memory. Learn row-major vs column-major layout, strides, contiguity, tiling, cache behavior, and why memory layout affects both speed and numerical stability in real systems.
2025-09-11

Numerical Linear Algebra: Understanding Matrices and Vectors Through Computation

Learn how linear algebra actually works inside real computers. A practical guide to LU, QR, SVD, stability, conditioning, and the numerical foundations behind modern AI and machine learning.
2025-09-01

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