Minutes for Meeting of the Graduate Committee September 9, 1997

Issues discussed:

• Foreign language requirement specifications

  The Committee voted to add the following sentence to the description of the Ph.D. language exam in the Handbook.

  The passage will be approximately 175 words in length excluding any symbolic expressions and the exam period will be thirty minutes.

• Adding the requirements for no more than x hours of topics courses, seminar, thesis to the Plan of Study requirements in addition to or instead of the catalog.

  This was approved as an addition to the Plan of Study for each program.

  Ph.D. and Ed.D.

  The Plan of Study for a doctoral degree may contain no more than 9 hours of each of the topics courses Math 6290,6390, 6490, 6590, 6690, 6790, 6890, no more than 12 hours of Math 6010, and no more than 24 hours of Math 6000.

  M.S.

  The Plan of Study for a master's degree may contain no more than 12 hours of Math 5010. No hours of Math 5000 may be included on the Plan of Study if the creative component option is chosen. A maximum of two hours of Math 5000 may be included if the report option is chosen and six hours may be included if the thesis option is chosen. Topics and seminar hours at the 6000 level may be included but are restricted as noted in the description of the Plan of Study for doctoral degrees.

• Syllabus for the Numerical Analysis
  The Committee agreed to support the change of syllabus and to bring this to the faculty at the next departmental meeting.

  Current

  1. Floating point arithmetic
  2. Error propagation, condition numbers
  3. Lagrange interpolation, Newton form, divided differences, error formulas for polynomial interpolation
  4. Nonlinear equations; bisection, secant, Newton methods, Convergence theory, rates and orders of convergence, Aitken's 2 - acceleration method
  5. Systems of linear equation, Gaussian elimination and triangular factorization, rounding error analysis, vector and matrix norms, condition numbers, von Neumann-Goldstine-Wilkinson Theorem
  6. Numerical Integration, Newton-Cotes (trapezoidal, Simpson), Gaussian formulas, error estimation including asymptotic for composite formulas, Richardson extrapolation

  Proposed

  1. Binary fractions, computer floating point format, machine arithmetic, rounding procedures, roundoff error, conditioning and stability of algorithms.
  2. Lagrange interpolation, divided difference and difference tables, Hermite interpolation, errors of interpolation, interpolation at equally spaced and at Chebyshev points.
  3. Solution of a single nonlinear equation by various methods: Bisection, Newton iteration, secant iteration, simplified Newton method. Contraction mapping theorem on ${\bf R}$ and related error estimates, orders of methods, higher order methods.
  4. Solving systems of linear equations by Gaussian elimination with pivoting, and by the mathematically equivalent PLU-factorization. Norms, condition numbers of matrices, backward error analysis (Theorem of v. Neumann-Goldstein-Wilkinson). QR-factorization and applications to orthogonalization and solution of linear systems.
  5. Solving systems of nonlinear equations by Newton or simplified Newton iteration. Damped Newton Method. Contraction mapping theorem on ${\bf R}^n$ and related error estimates.
  6. Numerical differentiation. Numerical integration: Newton-Cotes formulas, composite trapezoidal and Simpson rules, Gaussian rules, general interpolatory quadrature formulas. Euler--Maclaurin summation formula and the Romberg table. Errors: Upper bounds and asymptotics, examples of super-convergence: Gaussian formulas and trapezoidal rule for periodic integrands.
  7. Initial value problems for systems of ordinary differential equations. Runge-Kutta, Adams-Moulton, and backward differentiation methods, local truncation errors, global discretization error, order and stability of methods. Boundary value problems and finite difference methods.
  8. Eigenvalue problems for matrices. Power method, QR-algorithm.

  Remarks

  • In the above list, items 7. and 8. should be included in any survey of Numerical Analysis, however, these topics traditionally have not been covered on the Departmental Exam.
  • The study of numerical analysis should include experience with and careful consideration of the effects of running algorithms on actual computing machines for a full understanding of subject matter.

  REFERENCES:

  1. Kendall E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., Wiley, 1989.
  2. R.L. Burden \& J.D. Faires, Numerical Analysis, 5th ed., PWS-Kent, 1993.
  3. Sam Conte \& Carl de Boor, Elementary Numerical Analysis, 3rd ed., McGraw Hill, 1980.
  4. Peter Henrici, Elements of Numerical Analysis, Wiley, 1964.
  5. David Kincaid \& Ward Cheney, Numerical Analysis: The Mathematics of Scientific Computing, 2nd ed., Brooks/Cole, 1996.
  6. J. Stoer \& R. Bulirsch, Introduction to Numerical Analysis, 2nd Ed., Springer Verlag, 1993.

Updates on Work in Progress

The projects below were divided among the committee members. In each case these will act as subcommittees of two with Alspach as the second member.

• Handbook update: web, printed and email versions.
  Ullrich
• Redesign of pamphlet.
  Witte
• Plaques for Jobe and Scroggs award.
  Mantini
• Ed.D. to Ph.D. conversion.
  Cogdell