{R}R 開発ノート

合計 17 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

7.3 Least Squares Problems

A clear, intuitive, book-length explanation of least squares problems, including the geometry, normal equations, QR decomposition, and SVD. Learn why least-squares solutions are central to ML and data science, and why QR provides a stable foundation for practical algorithms.
2025-10-04

7.2 Householder Reflections

A clear, intuitive, book-length explanation of Householder reflections and why they form the foundation of modern QR decomposition. Learn how reflections overcome the numerical instability of Gram–Schmidt and enable stable least-squares solutions across ML, statistics, and scientific computing.
2025-10-03

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

6.3 Applications in ML, Statistics, and Kernel Methods

A deep, intuitive explanation of how Cholesky decomposition powers real machine learning and statistical systems—from Gaussian processes and Bayesian inference to kernel methods, Kalman filters, covariance modeling, and quadratic optimization. Understand why Cholesky is essential for stability, speed, and large-scale computation.
2025-09-30

Chapter 5 — LU Decomposition

An in-depth, accessible introduction to LU decomposition—why it matters, how it improves on Gaussian elimination, where pivoting fits in, and what modern numerical libraries like NumPy and LAPACK do under the hood. Includes a guide to stability, practical applications, and a smooth transition into LU with and without pivoting.
2025-09-22

4.2 Row Operations and Elementary Matrices

A deep but intuitive explanation of row operations and elementary matrices, showing how Gaussian elimination is built from structured matrix transformations and how these transformations form the foundation of LU decomposition and numerical stability.
2025-09-19

4.1 Gaussian Elimination Revisited

A deep, intuitive exploration of Gaussian elimination as it actually behaves inside floating-point arithmetic. Learn why the textbook algorithm fails in practice, how instability emerges, why pivoting is essential, and how elimination becomes reliable through matrix transformations.
2025-09-18

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

1.3 Computation & Mathematical Systems

A clear explanation of how mathematical systems behave differently inside real computers. Learn why stability, conditioning, precision limits, and computational constraints matter for AI, ML, and numerical software.
2025-09-05

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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