Unit · year 3
MU-308 · Numerical Analysis
Threads change24 lectures4 theorems
Approximating the continuous with the computable, and controlling the error.
Lectures
| L01 | Floating-Point Arithmetic and Machine Epsilon — |
| L02 | Conditioning and Stability — |
| L03 | Error Analysis: Forward and Backward — |
| L04 | Root Finding: Bisection — |
| L05 | Fixed-Point Iteration |
| L06 | Convergence of Fixed-Point Iteration |
| L07 | Newton's Method |
| L08 | Quadratic Convergence of Newton's Method |
| L09 | Secant and Quasi-Newton Methods |
| L10 | Systems of Nonlinear Equations — |
| L11 | Direct Solvers: LU and Pivoting — |
| L12 | Norms, Condition Numbers, and Error Bounds — |
| L13 | Iterative Solvers: Jacobi and Gauss–Seidel |
| L14 | Krylov Methods and Conjugate Gradient — |
| L15 | Polynomial Interpolation |
| L16 | The Lagrange Interpolation Error |
| L17 | Runge's Phenomenon and Chebyshev Nodes |
| L18 | Splines and Piecewise Interpolation — |
| L19 | Numerical Differentiation — |
| L20 | Newton–Cotes Quadrature |
| L21 | Gaussian Quadrature |
| L22 | Adaptive Quadrature and Richardson Extrapolation — |
| L23 | Numerical ODEs: Euler and Runge–Kutta — |
| L24 | Synthesis: Accuracy, Cost, and Stability |
Theorems in this unit
T-114
Convergence of fixed-point iteration
Contraction guarantees convergence of iterative schemes.
T-115
Convergence of Newton's method
Quadratic convergence near a simple root.
T-116
Lagrange interpolation and its error
The unique interpolating polynomial and its remainder.
T-117
Gaussian quadrature
Optimal node placement integrates high-degree polynomials exactly.