Introduction to numerical algorithms for some basic problems in computational mathematics. Discussion of both implementation issues and error analysis. Crosslisted with CX 4640 (formerly CS 4642).
At the level of Atkinson, An Introduction to Numerical Analysis
- Finite precision and accumulation of round-off errors; Introduction to the solution of linear systems of equations by direct and iterative methods; Gaussian elimination and pivoting: PLU factorization, norms, condition numbers and errors; the Jacobi and Gauss-Seidel iterative methods, convergence of the Jacobi method; QR factorization
- Introduction to the solution of nonlinear systems of equations; Bisection and secant method; General fixed point methods, convergence; Newton and quasi-Newton methods; Newton's method for systems
- Introduction to eigenvalue problems: power-method based algorithms; Location of eigenvalues, Gerschgorin circle theorem; Power and inverse power methods, convergence; Acceleration and the Rayleigh-Ritz quotient; Similarity transformations and deflation
- Introduction to approximation theory; Function norms and errors; Polynomial and piecewise polynomial interpolation; Bases for polynomial spaces, Lagrange formula; Least squares approximation, the L2 projection; Approximation error bounds; Polynomial evaluation via Horner rule; Approximation by Fourier series, DFT and FFT; Solution of discrete least squares problems; Chebychev polynomials; Splines
- Introduction to numerical integration; Trapezoidal, midpoint, and Simpson's rules; General Newton-Cotes formulas; Error and convergence; Composite rules; Orthogonal polynomials, Gauss quadrature rules, error and convergence; Change of intervals, singular integrals; Multiple integrals