{R}R 開発ノート

合計 24 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

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

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

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

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

5.2 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

4.4 When Elimination Fails

An in-depth, practical explanation of why Gaussian elimination fails in real numerical systems—covering zero pivots, instability, ill-conditioning, catastrophic cancellation, and singular matrices—and how these failures motivate the move to LU decomposition.
2025-09-21

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

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

3.4 Exact Algorithms vs Implemented Algorithms

Learn why textbook algorithms differ from the versions that actually run on computers. This chapter explains rounding, floating-point errors, instability, algorithmic reformulation, and why mathematically equivalent methods behave differently in AI, ML, and scientific computing.
2025-09-16

3.3 Conditioning of Problems vs Stability of Algorithms

Learn the critical difference between problem conditioning and algorithmic stability in numerical computing. Understand why some systems fail even with correct code, and how sensitivity, condition numbers, and numerical stability determine the reliability of AI, ML, and scientific algorithms.
2025-09-15

3.2 Measuring Errors

A clear and intuitive guide to absolute error, relative error, backward error, and how numerical errors propagate in real systems. Essential for understanding stability, trustworthiness, and reliability in scientific computing, AI, and machine learning.
2025-09-14

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

2.4 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

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

1.3 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

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

1.1 What Breaks Real AI Systems

Many AI failures come from numerical instability, not algorithms. This guide explains what actually breaks AI systems and why numerical linear algebra matters.
2025-09-03

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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