Scientific Computing

  1. Core Numerical Methods
    1. Computer Arithmetic and Error Analysis
      1. Number Representation
        1. Binary and Hexadecimal Systems
          1. Fixed-Point and Floating-Point
            1. Sources of Error in Computation
              1. Round-off Error
                1. Floating-Point Representation
                  1. Loss of Significance
                    1. Truncation Error
                      1. Taylor Series Approximation
                        1. Discretization Error
                          1. Error Propagation
                            1. Numerical Stability and Conditioning
                              1. Well-posed vs Ill-posed Problems
                                1. Sensitivity to Input Data
                                  1. Condition Numbers
                                  2. Solving Systems of Linear Equations
                                    1. Problem Formulation
                                      1. Matrix Properties and Classification
                                        1. Direct Methods
                                          1. Gaussian Elimination
                                            1. Forward and Backward Substitution
                                              1. Pivoting Strategies
                                                1. LU Decomposition
                                                  1. Existence and Uniqueness
                                                    1. Computational Complexity
                                                      1. Cholesky Decomposition
                                                        1. Application to Symmetric Positive Definite Matrices
                                                          1. QR Decomposition
                                                            1. Singular Value Decomposition
                                                            2. Iterative Methods
                                                              1. Jacobi Method
                                                                1. Convergence Criteria
                                                                  1. Gauss-Seidel Method
                                                                    1. Successive Over-Relaxation
                                                                      1. Conjugate Gradient Method
                                                                        1. Application to Sparse Systems
                                                                          1. Preconditioning Techniques
                                                                            1. Krylov Subspace Methods
                                                                          2. Finding Roots of Nonlinear Equations
                                                                            1. Single Variable Methods
                                                                              1. Bisection Method
                                                                                1. Bracketing and Convergence
                                                                                  1. Newton-Raphson Method
                                                                                    1. Derivative Calculation
                                                                                      1. Convergence and Divergence Issues
                                                                                        1. Secant Method
                                                                                          1. Comparison with Newton-Raphson
                                                                                            1. Fixed-Point Iteration
                                                                                            2. Multivariable Methods
                                                                                              1. Newton's Method for Systems
                                                                                                1. Quasi-Newton Methods
                                                                                                  1. Broyden's Method
                                                                                                2. Interpolation and Approximation
                                                                                                  1. Polynomial Interpolation
                                                                                                    1. Lagrange Polynomials
                                                                                                      1. Newton's Divided Differences
                                                                                                        1. Runge's Phenomenon
                                                                                                          1. Chebyshev Polynomials
                                                                                                          2. Spline Interpolation
                                                                                                            1. Linear Splines
                                                                                                              1. Cubic Splines
                                                                                                                1. Boundary Conditions
                                                                                                                  1. B-Splines
                                                                                                                  2. Least-Squares Fitting
                                                                                                                    1. Linear Regression
                                                                                                                      1. Nonlinear Regression
                                                                                                                        1. Overfitting and Regularization
                                                                                                                          1. Cross-Validation
                                                                                                                          2. Rational Function Approximation
                                                                                                                            1. Trigonometric Interpolation
                                                                                                                            2. Numerical Differentiation
                                                                                                                              1. Finite Difference Formulas
                                                                                                                                1. Forward Difference
                                                                                                                                  1. Backward Difference
                                                                                                                                    1. Central Difference
                                                                                                                                    2. Higher-Order Derivatives
                                                                                                                                      1. Error Analysis
                                                                                                                                        1. Stability Considerations
                                                                                                                                        2. Automatic Differentiation
                                                                                                                                          1. Symbolic Differentiation
                                                                                                                                          2. Numerical Integration
                                                                                                                                            1. Newton-Cotes Formulas
                                                                                                                                              1. Trapezoidal Rule
                                                                                                                                                1. Simpson's Rule
                                                                                                                                                  1. Error Estimation
                                                                                                                                                    1. Composite Rules
                                                                                                                                                    2. Gaussian Quadrature
                                                                                                                                                      1. Weight Functions and Nodes
                                                                                                                                                        1. Application to Polynomials
                                                                                                                                                          1. Gauss-Legendre Quadrature
                                                                                                                                                            1. Gauss-Hermite Quadrature
                                                                                                                                                            2. Adaptive Quadrature
                                                                                                                                                              1. Monte Carlo Integration
                                                                                                                                                                1. Random Sampling
                                                                                                                                                                  1. Convergence Properties
                                                                                                                                                                    1. Importance Sampling
                                                                                                                                                                      1. Quasi-Monte Carlo Methods
                                                                                                                                                                      2. Multidimensional Integration
                                                                                                                                                                      3. Solving Ordinary Differential Equations
                                                                                                                                                                        1. Initial Value Problems
                                                                                                                                                                          1. Euler's Method
                                                                                                                                                                            1. Stability and Accuracy
                                                                                                                                                                              1. Runge-Kutta Methods
                                                                                                                                                                                1. Classical RK4
                                                                                                                                                                                  1. Higher-Order Methods
                                                                                                                                                                                    1. Embedded Methods
                                                                                                                                                                                      1. Adaptive Step-Size Control
                                                                                                                                                                                        1. Error Estimation
                                                                                                                                                                                          1. Step-Size Selection Algorithms
                                                                                                                                                                                            1. Multistep Methods
                                                                                                                                                                                              1. Adams-Bashforth Methods
                                                                                                                                                                                                1. Adams-Moulton Methods
                                                                                                                                                                                                  1. Stiff Equations
                                                                                                                                                                                                    1. Implicit Methods
                                                                                                                                                                                                      1. Backward Differentiation Formulas
                                                                                                                                                                                                      2. Boundary Value Problems
                                                                                                                                                                                                        1. Shooting Method
                                                                                                                                                                                                          1. Implementation Steps
                                                                                                                                                                                                            1. Finite Difference Method
                                                                                                                                                                                                              1. Discretization of Boundary Conditions
                                                                                                                                                                                                                1. Collocation Methods
                                                                                                                                                                                                                2. Systems of ODEs
                                                                                                                                                                                                                  1. Higher-Order Equations
                                                                                                                                                                                                                  2. Solving Partial Differential Equations
                                                                                                                                                                                                                    1. Classification of PDEs
                                                                                                                                                                                                                      1. Elliptic Equations
                                                                                                                                                                                                                        1. Laplace's Equation
                                                                                                                                                                                                                          1. Poisson's Equation
                                                                                                                                                                                                                            1. Parabolic Equations
                                                                                                                                                                                                                              1. Heat Equation
                                                                                                                                                                                                                                1. Diffusion Equations
                                                                                                                                                                                                                                  1. Hyperbolic Equations
                                                                                                                                                                                                                                    1. Wave Equation
                                                                                                                                                                                                                                      1. Transport Equations
                                                                                                                                                                                                                                      2. Discretization Methods
                                                                                                                                                                                                                                        1. Finite Difference Method
                                                                                                                                                                                                                                          1. Grid Construction
                                                                                                                                                                                                                                            1. Stability and Consistency
                                                                                                                                                                                                                                              1. Von Neumann Stability Analysis
                                                                                                                                                                                                                                                1. Finite Volume Method
                                                                                                                                                                                                                                                  1. Conservation Laws
                                                                                                                                                                                                                                                    1. Flux Calculation
                                                                                                                                                                                                                                                      1. Finite Element Method
                                                                                                                                                                                                                                                        1. Mesh Generation
                                                                                                                                                                                                                                                          1. Basis Functions
                                                                                                                                                                                                                                                            1. Weak Formulation
                                                                                                                                                                                                                                                              1. Assembly Process
                                                                                                                                                                                                                                                                1. Spectral Methods
                                                                                                                                                                                                                                                                  1. Fourier and Chebyshev Bases
                                                                                                                                                                                                                                                                    1. Convergence Properties
                                                                                                                                                                                                                                                                    2. Boundary Conditions
                                                                                                                                                                                                                                                                      1. Initial Conditions
                                                                                                                                                                                                                                                                        1. Time-Stepping Schemes
                                                                                                                                                                                                                                                                        2. Eigenvalue Problems
                                                                                                                                                                                                                                                                          1. Standard Eigenvalue Problems
                                                                                                                                                                                                                                                                            1. Generalized Eigenvalue Problems
                                                                                                                                                                                                                                                                              1. Power Iteration Method
                                                                                                                                                                                                                                                                                1. Dominant Eigenvalue Calculation
                                                                                                                                                                                                                                                                                  1. Inverse Power Method
                                                                                                                                                                                                                                                                                    1. QR Algorithm
                                                                                                                                                                                                                                                                                      1. Matrix Decomposition
                                                                                                                                                                                                                                                                                        1. Convergence Analysis
                                                                                                                                                                                                                                                                                          1. Arnoldi Method
                                                                                                                                                                                                                                                                                            1. Lanczos Method
                                                                                                                                                                                                                                                                                            2. Fourier Analysis
                                                                                                                                                                                                                                                                                              1. Continuous Fourier Transform
                                                                                                                                                                                                                                                                                                1. Discrete Fourier Transform
                                                                                                                                                                                                                                                                                                  1. Mathematical Formulation
                                                                                                                                                                                                                                                                                                    1. Applications in Signal Processing
                                                                                                                                                                                                                                                                                                      1. Fast Fourier Transform Algorithm
                                                                                                                                                                                                                                                                                                        1. Computational Efficiency
                                                                                                                                                                                                                                                                                                          1. Common Implementations
                                                                                                                                                                                                                                                                                                            1. Windowing Functions
                                                                                                                                                                                                                                                                                                              1. Spectral Analysis
                                                                                                                                                                                                                                                                                                              2. Optimization Methods
                                                                                                                                                                                                                                                                                                                1. Problem Classification
                                                                                                                                                                                                                                                                                                                  1. Unconstrained Optimization
                                                                                                                                                                                                                                                                                                                    1. Gradient Descent
                                                                                                                                                                                                                                                                                                                      1. Step-Size Selection
                                                                                                                                                                                                                                                                                                                        1. Convergence Criteria
                                                                                                                                                                                                                                                                                                                          1. Newton's Method
                                                                                                                                                                                                                                                                                                                            1. Quasi-Newton Methods
                                                                                                                                                                                                                                                                                                                              1. Conjugate Gradient
                                                                                                                                                                                                                                                                                                                                1. Application to Quadratic Functions
                                                                                                                                                                                                                                                                                                                                2. Constrained Optimization
                                                                                                                                                                                                                                                                                                                                  1. Linear Programming
                                                                                                                                                                                                                                                                                                                                    1. Simplex Method
                                                                                                                                                                                                                                                                                                                                      1. Nonlinear Programming
                                                                                                                                                                                                                                                                                                                                        1. Lagrange Multipliers
                                                                                                                                                                                                                                                                                                                                          1. Karush-Kuhn-Tucker Conditions
                                                                                                                                                                                                                                                                                                                                            1. Penalty Methods
                                                                                                                                                                                                                                                                                                                                              1. Barrier Methods
                                                                                                                                                                                                                                                                                                                                              2. Global Optimization
                                                                                                                                                                                                                                                                                                                                                1. Stochastic Optimization

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