Most problems cannot be computed exactly, and even those that can might be very expensive. Numerical methods try to find good enough approximations efficiently. ================ Tentative Schedule: 1. The derivative, fixed point iteration, Newton’s method, Euler, Runge-Kutta, and multistep methods. 2. Quadrature a.k.a. numerical integration, finite difference methods and correctors. 3. Linear differential equations, matrix fundamentals, the QR algorithm, Newton-like methods, stiffness. 4. Finite element methods, various bases, Fourier/DCT transforms, Fourier analysis (faster solvers), Fourier analysis (stability). 5. Optimization: Binary and golden-section search, the simplex method, gradient descent, conjugate gradient descent, and Adam. ================ Class website: https://programjames.github.io/hssp-spring-numerical-methods/

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Solving integrals is hard, wouldn't it be better to get a computer to do that for you? Here we explore numerical methods, beginning at "What is a derivative? (20 min. edition)" and using that answer to solve impossible problems quickly and accurately. Lecture schedule (tentative): 1. The derivative, fixed point iteration, Newton's, Euler & Runge-Kutta methods. 2. Gauss-Legendre quadrature, solving PDE's (e.g. heat equation). Finite difference methods & correctors. 3. Matrices: finding eigenvalues/vectors, Broyden's method, condition number & stiffness. Common bases and their condition numbers (i.e. why polynomials are bad). 4. Finite element methods w/ Chebyshev polynomials or sines/cosines. Fourier/DCT transforms, Fourier analysis (faster solvers) & Fourier analysis (determining convergence rate). 5. Optimization: golden section, simplex, gradient descent, conjugate gradient method, Adam. 6. Different ideas, maybe a combination of them: Image recognition w/ Bayes' theorem, the Lagrangian w/ the simplex method, image compression w/ principal component analysis.