Numerical Analysis - Breadth and Depth Exams

  • Understanding of concepts pertaining to the following topics is required for the breadth examination.
  • In-depth understanding of these concepts and the ability to present formal arguments is required for the depth examination.
  • Preparation: Class CS537 and additional readings from textbooks such as:
    1. [BOOK1] S. D. Conte and C. de Boor, Elementary Numerical Analysis, An Algorithmic Approach, McGraw-Hill Book Co., New York, 1980.
    2. [BOOK2] W. Cheney and D. Kincaid, Numerical Mathematics and Computing, 4th Ed., Brooks/Cole, New York 1999.
  • Topics:
    • Floating-Point Arithmetic [BOOK1 Ch1, BOOK2 Ch1]
      • number representation and errors,
      • error estimation
      • condition number and stability,
    • Systems of Linear Equations [BOOK1 Ch4, BOOK2 Ch6]
      • basic methods, LU and QR factorizations,
      • systems with matrices of special forms,
      • condition number,
      • error and residuals of approximations,
      • backward error analysis and iterative improvement
    • Polynomial Interpolation [BOOK1 Ch2, BOOK2 Ch4]
      • Lagrange and Hermite interpolation,
      • Newton form and divided differences,
      • piece-wise polynomial interpolation and splines,
      • interpolation errors
    • Function Approximation [BOOK1 Ch6, BOOK2 Ch10]
      • uniform polynomial approximation,
      • data fitting,
      • orthogonal polynomials and least-squares approximation
    • Numerical Differentiation [BOOK1 Ch7, BOOK2 Ch4]
    • Numerical Integration [BOOK1 Ch7, BOOK2 Ch5]
      • basic rules,
      • Gaussian rules,
      • composite rules,
      • adaptive rules,
      • singular integrals
    • Nonlinear Equations [BOOK1 Ch3, BOOK2 Ch 3]
      • bisection and secant methods,
      • Newton method,
      • polynomial equations with real roots