Numerical Analysis

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• Exact vs. approximate solutions ~10 min
• Algorithms, convergence, and rate of convergence ~11 min
• Sources of error: round-off and truncation ~10 min
• Absolute error, relative error, and significant digits ~12 min
• Floating-point representation and machine epsilon ~13 min
• Overflow, underflow, and rounding modes ~11 min
• Catastrophic cancellation and loss of significance ~12 min
• Conditioning: how sensitive is the problem? ~13 min
• Stability: how sensitive is the algorithm? ~13 min

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• Bracketing a root: the Intermediate Value Theorem in practice ~12 min
• The Bisection Method ~16 min
• Fixed-point iteration and the cobweb diagram ~14 min
• Convergence of fixed-point iteration: the contraction condition ~14 min
• Newton's Method: derivation from the tangent line ~16 min
• Newton's Method: convergence order and failure modes ~15 min
• The Secant Method ~14 min
• The Method of False Position ~13 min
• Müller's Method and roots of polynomials ~15 min
• Choosing a root-finding method ~12 min

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• The interpolation problem: existence and uniqueness ~13 min
• The Lagrange form of the interpolating polynomial ~15 min
• Newton's divided-difference form ~18 min
• Neville's method ~14 min
• The interpolation error formula ~15 min
• Equally spaced nodes and the Runge phenomenon ~16 min
• Chebyshev nodes: taming the Runge phenomenon ~15 min
• Hermite interpolation ~15 min
• Extrapolation: why it's riskier than interpolation ~13 min

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• Why one high-degree polynomial can go wrong ~12 min
• Piecewise linear and piecewise quadratic splines ~15 min
• Cubic splines: setting up the system ~20 min
• Natural boundary conditions ~22 min
• Clamped boundary conditions ~18 min
• Continuity of position, slope, and curvature ~16 min
• Solving the spline system: tridiagonal solvers ~17 min
• Parametric curves from splines ~15 min

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• Forward and backward difference formulas ~14 min
• The central difference formula ~14 min
• Deriving difference formulas from Taylor series ~18 min
• Truncation error vs. round-off error in differentiation ~16 min
• Choosing the step size h: the U-shaped error curve ~18 min
• Richardson extrapolation ~20 min

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• The midpoint and trapezoidal rules: error bounds ~14 min
• Simpson's rule and degree of precision ~20 min
• Composite rules and error bounds ~18 min
• Romberg integration ~18 min
• Adaptive quadrature ~16 min
• Gaussian quadrature: nodes and weights ~18 min
• Improper integrals, numerically ~14 min
• Multiple integrals, briefly ~12 min
• Trigonometric approximation and the idea of the FFT ~12 min
• Choosing a quadrature method ~12 min

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• Gaussian elimination, revisited numerically ~10 min
• Partial pivoting and why it's necessary ~20 min
• Operation counts: why elimination is $O(n^3)$ ~12 min
• LU factorization from elimination ~15 min
• Cholesky factorization for symmetric positive-definite systems ~15 min
• Vector and matrix norms ~14 min
• The condition number of a matrix ~16 min
• Iterative refinement ~14 min
• Special matrix structures: banded and tridiagonal systems ~14 min

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• Why iterate instead of solving directly? ~11 min
• The Jacobi method ~18 min
• The Gauss-Seidel method ~15 min
• Successive over-relaxation (SOR) ~16 min
• Convergence: diagonal dominance and spectral radius ~17 min
• The Conjugate Gradient method ~17 min
• Direct vs. iterative: choosing a method ~12 min

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• Overdetermined systems and the least-squares problem ~14 min
• The normal equations ~14 min
• Why the normal equations can be numerically dangerous ~18 min
• Least squares via QR factorization ~20 min
• Polynomial and nonlinear curve fitting ~16 min
• Orthogonal polynomials as a stable fitting basis ~16 min
• Residuals and goodness of fit ~15 min

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• Why not just use the characteristic polynomial? ~12 min
• The Power Method ~16 min
• The Rayleigh quotient and accelerated convergence ~20 min
• The Inverse Power Method and Shifted Inverse Iteration ~18 min
• Deflation ~16 min
• Householder Reflections and Reduction to Hessenberg Form ~18 min
• The QR Algorithm ~20 min
• Computing the SVD Numerically ~18 min
• Choosing an Eigenvalue Method ~14 min

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• Fixed Points for Functions of Several Variables ~16 min
• Newton's Method for Systems: the Jacobian ~22 min
• Quasi-Newton (Broyden's) Methods ~18 min
• Steepest Descent as a Globalization Strategy ~18 min
• Homotopy and Continuation Methods ~18 min
• Golden-Section Search: the One-Dimensional Case ~16 min

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• Well-posedness of the initial-value problem, revisited ~14 min
• Euler's method, revisited numerically ~15 min
• Local vs. global truncation error, made formal ~20 min
• Higher-order Taylor methods ~16 min
• Runge-Kutta methods: RK2 and the midpoint method ~18 min
• RK4 ~22 min
• Adaptive step size: the Runge-Kutta-Fehlberg method ~20 min
• Multistep methods: Adams-Bashforth and Adams-Moulton ~20 min
• Predictor-corrector methods ~16 min
• Absolute stability and stiff equations ~24 min
• Systems and higher-order equations ~18 min

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• What makes a BVP different from an IVP ~16 min
• The Linear Shooting Method ~22 min
• The Shooting Method for Nonlinear Problems ~20 min
• Finite-Difference Methods for BVPs ~20 min
• The Rayleigh-Ritz Method, Briefly ~14 min

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• Classifying PDEs for numerical purposes ~16 min
• Finite differences for the heat equation, revisited ~14 min
• Finite differences for the wave equation, revisited ~14 min
• Finite differences for Laplace's equation, revisited ~16 min
• Choosing a numerical PDE method ~14 min
• Capstone: building a numerical solution end to end ~22 min