{R}R 開発ノート

合計 20 件の記事が見つかりました。

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

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

7.2 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

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

6.2 Memory Advantages

A detailed, intuitive explanation of why Cholesky decomposition uses half the memory of LU decomposition, how memory locality accelerates computation, and why this efficiency makes Cholesky essential for large-scale machine learning, kernel methods, and statistical modeling.
2025-09-29

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

Chapter 6 — Cholesky Decomposition

A deep, narrative-driven introduction to Cholesky decomposition explaining why symmetric positive definite matrices dominate real computation. Covers structure, stability, performance, and the role of Cholesky in ML, statistics, and optimization.
2025-09-27

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

5.3 LU in NumPy and LAPACK

A practical, in-depth guide to how LU decomposition is implemented in NumPy and LAPACK. Learn about partial pivoting, blocked algorithms, BLAS optimization, error handling, and how modern numerical libraries achieve both speed and stability.
2025-09-25

5.2 Numerical Pitfalls

A deep, accessible explanation of the numerical pitfalls in LU decomposition. Learn about growth factors, tiny pivots, rounding errors, catastrophic cancellation, ill-conditioning, and why LU may silently produce incorrect results without proper pivoting and numerical care.
2025-09-24

5.1 LU with and without Pivoting

A clear and practical explanation of LU decomposition with and without pivoting. Learn why pivoting is essential, how partial and complete pivoting work, where no-pivot LU fails, and why modern numerical libraries rely on pivoted LU for stability.
2025-09-23

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

4.3 Pivoting Strategies

A practical and intuitive guide to pivoting strategies in numerical linear algebra, explaining partial, complete, and scaled pivoting and why pivoting is essential for stable Gaussian elimination and reliable LU decomposition.
2025-09-20

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

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

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

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

1.2 Floating-Point Reality vs. Textbook Math

Floating-point numbers don’t behave like real numbers. This article explains how rounding, cancellation, and machine precision break AI systems—and why it matters.
2025-09-04

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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