{R}R 開発ノート

合計 8 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

7.4 Why QR Is Often Preferred

An in-depth, accessible explanation of why QR decomposition is the preferred method for solving least squares problems and ensuring numerical stability. Covers orthogonality, rank deficiency, Householder reflections, and the broader role of QR in scientific computing, with a smooth transition into eigenvalues and eigenvectors.
2025-10-05

1.4 A Brief Tour of Real-World Failures

A clear, accessible tour of real-world numerical failures in AI, ML, optimization, and simulation—showing how mathematically correct algorithms break inside real computers, and preparing the reader for Chapter 2 on floating-point reality.
2025-09-06

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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