Numerical Analysis I

Department: 
MATH
Course Number: 
4640
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Every spring (MATH) and fall (CS)

Introduction to numerical algorithms for some basic problems in computational mathematics. Discussion of both implementation issues and error analysis. Crosslisted with CX 4640.

Prerequisites: 

MATH 2552 or MATH 2562 or MATH 2X52 or MATH 2603

Course Text: 

At the level of Atkinson, An Introduction to Numerical Analysis

Topic Outline: 
  • 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