Here are 100 chapter titles for a computational mathematics curriculum, progressing from beginner to advanced levels:
I. Foundations & Numerical Representation (1-20)
- Introduction to Computational Thinking
- Number Representation in Computers: Floating-Point Arithmetic
- Errors in Numerical Computations: Rounding and Truncation
- Conditioning and Stability of Algorithms
- Big O Notation and Algorithm Analysis
- Computer Arithmetic and its Limitations
- Representation of Functions: Polynomials, Splines
- Interpolation: Lagrange, Newton, Spline
- Approximation Theory: Best Approximation, Least Squares
- Numerical Differentiation: Finite Differences
- Numerical Integration: Trapezoidal Rule, Simpson's Rule
- Introduction to Numerical Software (e.g., Python, MATLAB)
- Programming for Numerical Computations
- Data Structures for Numerical Algorithms
- Visualization of Numerical Results
- Debugging and Testing Numerical Code
- Introduction to Linear Algebra for Computation
- Vector and Matrix Operations
- Solving Linear Systems: Gaussian Elimination
- Practice Problems: Basic Numerical Computations
II. Linear Algebra Computations (21-40)
- LU Decomposition and its Applications
- Cholesky Decomposition for Symmetric Matrices
- QR Decomposition and Least Squares Problems
- Eigenvalues and Eigenvectors: Basic Computations
- Iterative Methods for Linear Systems: Jacobi, Gauss-Seidel
- Convergence Analysis of Iterative Methods
- Sparse Matrix Techniques
- Krylov Subspace Methods: Conjugate Gradient, GMRES
- Preconditioning for Linear Systems
- Singular Value Decomposition (SVD) and its Applications
- Eigenvalue Computation for Large Matrices
- The Power Method and Inverse Iteration
- QR Algorithm for Eigenvalues
- Applications: Solving Linear Systems in Practice
- Linear Programming: Introduction and Algorithms
- Optimization with Linear Constraints
- Integer Programming: Basic Concepts
- Network Flow Problems
- Applications: Linear Algebra in Data Analysis
- Practice Problems: Advanced Linear Algebra Computations
III. Nonlinear Equations and Optimization (41-60)
- Root Finding for Nonlinear Equations: Bisection, Newton
- Convergence Analysis of Root-Finding Methods
- Fixed-Point Iteration and its Convergence
- Systems of Nonlinear Equations: Newton-Raphson
- Optimization: Basic Concepts and Algorithms
- Gradient Descent and its Variants
- Conjugate Gradient Method for Optimization
- Newton's Method for Optimization
- Quasi-Newton Methods: BFGS, DFP
- Constrained Optimization: Lagrange Multipliers
- Linear Programming: Simplex Method
- Nonlinear Programming: Interior Point Methods
- Global Optimization: Simulated Annealing, Genetic Algorithms
- Applications: Optimization in Machine Learning
- Curve Fitting and Regression
- Least Squares Regression: Linear and Nonlinear
- Regularization Techniques: Ridge Regression, Lasso
- Model Selection and Cross-Validation
- Applications: Data Fitting and Model Building
- Practice Problems: Nonlinear Equations and Optimization
IV. Differential Equations (61-80)
- Numerical Methods for Ordinary Differential Equations (ODEs)
- Euler's Method and its Variants
- Runge-Kutta Methods: Explicit and Implicit
- Multistep Methods: Adams-Bashforth, Adams-Moulton
- Stability and Convergence of ODE Solvers
- Stiff ODEs and their Solution
- Boundary Value Problems for ODEs
- Finite Difference Methods for BVPs
- Finite Element Methods for BVPs (Introduction)
- Numerical Methods for Partial Differential Equations (PDEs)
- Finite Difference Methods for PDEs
- Finite Element Methods for PDEs (Advanced)
- Spectral Methods for PDEs
- Applications: Solving PDEs in Physics and Engineering
- Numerical Solution of the Heat Equation
- Numerical Solution of the Wave Equation
- Numerical Solution of Laplace's Equation
- Introduction to Computational Fluid Dynamics (CFD)
- Applications: Solving Differential Equations in Practice
- Practice Problems: Differential Equations
V. Advanced Topics and Applications (81-100)
- Numerical Linear Algebra: Advanced Topics
- Iterative Methods for Eigenvalue Problems
- Preconditioning Techniques for Large Systems
- Parallel Computing for Numerical Algorithms
- High-Performance Computing for Scientific Applications
- Numerical Methods for Integral Equations
- Approximation Theory: Advanced Topics
- Spline Interpolation and Approximation
- Numerical Methods for Optimization: Advanced Topics
- Stochastic Optimization Methods
- Computational Statistics: Monte Carlo Methods
- Random Number Generation
- Numerical Methods for Data Analysis
- Machine Learning Algorithms: Computational Aspects
- Deep Learning: Computational Challenges
- Image Processing: Computational Techniques
- Scientific Visualization: Advanced Techniques
- Symbolic Computation: Introduction to Computer Algebra Systems
- Applications: Computational Science and Engineering
- The Future of Computational Mathematics