The course is paced for approximately a 12 week program, with 3 lectures per week.
Week 1: Truncation and Round-off Error
Day 1: Course Introduction
Day 2: Round-off error and finite precision representations
Day 3: True and approximate error, and truncation error
Day 4: Solvability (Over/Under-determined systems & the Pseudo-Inverse), Conditioning, and Complexity
Day 5: Direct methods: Gaussian elimination, Diagonal dominance, and pivoting
Day 6: Direct methods: LU decomposition, Matrix types, and sparsity
Week 3: Linear Systems (Continued)
Day 7: Iterative methods: Stationary methods
Day 8: Iterative methods: Krylov subspace methods
Day 9: Iterative methods: Preconditioners
Week 4: Interpolation and Curve Fitting
Day 10: Polynomial interpolation
Day 11: Splines, Radial basis functions, and Summary of interpolation
Day 12: Curve fitting (Linear least squares, Weighted, Polyfit, Best fit RBF)
Day 13: Polynomial roots and Closed methods
Day 14: Open methods (Newton-Raphson)
Day 15: Basins of attraction and Global convergence methods
Day 16: Closed methods and Open methods (Gradient-based optimizers)
Day 17: Indefinite systems and Trust region methods
Day 18: Equality constraints, Nonlinear least squares, and Neural networks
Midterm Break
Week 7: Numerical Differentiation
Day 19: Finite difference (First-order derivatives)
Day 20: Finite difference (Second-order and higher dimensions)
Day 21: Even vs. uneven spacing, and Uncertainty amplification
Day 22: Newton-Cotes formulae (Riemann’s integrals, Trapezoid rule, Simpson’s rules)
Day 23: Romberg rule
Day 24: Gaussian quadrature
Week 9: Differential Equations (Initial Value Problems)
Day 25: Explicit methods (Runge-Kutta) and Adaptive time stepping
Day 26: Implicit methods, Stiffness, and Implicit Runge-Kutta
Day 27: Multistep methods, Systems of equations, and Reduction of order
Week 10: Boundary Value Problems and PDEs
Day 28: Shooting method and Collocation methods
Day 29: Finite difference method and Finite volume
Day 30: Partial differential equations
Week 11: Finite Element Method
Day 31: Motivating example and Theoretical background
Day 32: Elements, Discretization, and Assembly
Day 33: Solution methods
Day 34: Eigenfunctions and Powers method
Day 35: Matrix decomposition methods
Day 36: Package tools