Numerical Analysis

About This Series

Numerical analysis is the branch of mathematics concerned with algorithms for approximating solutions to mathematical problems and with the study of their errors and convergence behavior. Even when an analytical solution is unavailable, numerical methods can yield reliable approximate solutions.

Numerical analysis underpins scientific computing and finds applications in physics simulations, machine learning, financial engineering, image processing, and many other fields. This series provides a systematic treatment of numerical analysis in five levels, from foundational numerical computation to verified (interval) arithmetic.

Levels

Introduction

High-school mathematics level

  • What is numerical computation?
  • Fundamentals of computational errors
  • Bisection method
  • Introduction to numerical differentiation and integration

Basics

Undergraduate level (Years 1–2)

  • Floating-point numbers
  • Newton's method and the secant method
  • Interpolation
  • Numerical integration (trapezoidal rule, Simpson's rule)

Intermediate

Undergraduate level (Years 3–4)

  • Direct methods for linear systems
  • Approximation theory, QR / Cholesky / SVD decompositions
  • Numerical optimization and multivariate nonlinear problems
  • Numerical methods for ordinary differential equations

Advanced

Graduate level

  • Partial differential equations and the finite element method
  • Integral equations and the boundary element method
  • Inverse problems and regularization
  • Monte Carlo and quasi-Monte Carlo methods

Special Topics

Advanced topics

  • Interval arithmetic and affine arithmetic
  • Verified numerical computation
  • Random number generation algorithms

Learning Path

Five-level learning flow for Numerical Analysis: from Introduction to Special Topics Introduction (high school), Basics (undergraduate years 1-2), Intermediate (undergraduate years 3-4), Advanced (graduate), Special Topics (verified computation and interval arithmetic) Intro High School Basics Years 1-2 Intermediate Years 3-4 Advanced Graduate Special Verified Comp. *Post-Advanced elective topics Introduction: concept of errors, bisection method, intuition for differentiation & integration Basics: floating-point numbers, Newton's method, interpolation, numerical integration Intermediate: linear systems, eigenvalue problems, ODEs Advanced: PDEs, finite element method, FFT Special Topics: interval arithmetic, verified computation, random number generation

Key Topics

Error Analysis

Understanding errors in numerical computation: rounding errors, truncation errors, and condition numbers.

Root Finding

Numerical methods for nonlinear equations, including the bisection method and Newton's method.

Numerical Linear Algebra

Numerical algorithms for linear systems, eigenvalue problems, and matrix decompositions.

Differential Equations

Numerical methods for ordinary and partial differential equations, and stability analysis.

Verified Computation & Interval Arithmetic

Error-bound enclosure via interval arithmetic, verified numerical computation, and random number generation.

Key Formulas

Newton's Method

$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$

Lagrange Interpolation

$$L(x) = \sum_{i=0}^{n} y_i \prod_{j \neq i} \frac{x - x_j}{x_i - x_j}$$

Simpson's Rule

$$\int_a^b f(x)\,dx \approx \frac{h}{3}(f_0 + 4f_1 + f_2)$$

Euler's Method

$$y_{n+1} = y_n + h f(t_n, y_n)$$

LU Decomposition

$$A = LU \quad \Rightarrow \quad Ly = b,\; Ux = y$$

Prerequisites

  • Introduction: High-school mathematics (basic calculus)
  • Basics: Calculus and basic linear algebra
  • Intermediate: Linear algebra, basic differential equations
  • Advanced: Partial differential equations, basic functional analysis
  • Special Topics: Fundamentals of computer architecture

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