{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

7.1 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

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

6.1 SPD Matrices and Why They Matter

A deep, intuitive explanation of symmetric positive definite (SPD) matrices and why they are essential in machine learning, statistics, optimization, and numerical computation. Covers geometry, stability, covariance, kernels, Hessians, and how SPD structure enables efficient Cholesky decomposition.
2025-09-28

4.0 Solving Ax = b

A deep, accessible introduction to solving linear systems in numerical computing. Learn why Ax = b sits at the center of AI, ML, optimization, and simulation, and explore Gaussian elimination, pivoting, row operations, and failure modes through intuitive explanations.
2025-09-17

3.1 Norms and Why They Matter

A deep yet accessible exploration of vector and matrix norms, why they matter in numerical computation, and how they influence stability, conditioning, error growth, and algorithm design. Essential reading for AI, ML, and scientific computing engineers.
2025-09-13

Chapter 3 — Computation & Mathematical Systems

A clear, insightful introduction to numerical computation—covering norms, error measurement, conditioning vs stability, and the gap between mathematical algorithms and real implementations. Essential reading for anyone building AI, optimization, or scientific computing systems.
2025-09-12

2.3 Overflow, Underflow, Loss of Significance

A clear and practical guide to overflow, underflow, and loss of significance in floating-point arithmetic. Learn how numerical computations break, why these failures occur, and how they impact AI, optimization, and scientific computing.
2025-09-10

Chapter 2 — The Computational Model

An introduction to the computational model behind numerical linear algebra. Explains why mathematical algorithms fail inside real computers, how floating-point arithmetic shapes computation, and why understanding precision, rounding, overflow, and memory layout is essential for AI, ML, and scientific computing.
2025-09-07

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

1.0 Why Numerical Linear Algebra Matters

A deep, practical introduction to why numerical linear algebra matters in real AI, ML, and optimization systems. Learn how stability, conditioning, and floating-point behavior impact models.
2025-09-02

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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