Syllabus for Ph. D. Preliminary Examination
in Symbolic and Numerical Computing
within Computational Science Cluster
of Mathematical Computer Science

1. MCS 571 Numerical Methods for Partial Differential Equations.
2. MCS 589 Advanced Topics in Computer Science (Database Computing).
3. Symbolic Computation using one of
   1. MCS 561 Algebraic Symbolic Computation.
   2. MCS 563 Analytic Symbolic Computation.

Students are required to answer 5 questions, but your five best answers determine your score.
A nearly complete answer carries more weight than several partial answers adding to the same numerical score.
Syllabi for each of the areas:
1. Numerical Methods for PDEs and Related Numerical Analysis
   • Normal Preparation: MCS 571 Numerical Methods for Partial Differential Equations.
   • Other Background: MCS 471 Numerical Analysis, MCS 472 Advanced Numerical Analysis.
   • Topics:
     1. Floating point arithmetic, chopping, rounding, error propagation, operation counts.
     2. Matrix Iteration Methods. Methods of Jacobi, Gauss-Siedel.
     3. Numerical solution of Parabolic PDE. Finite Differences, Finite Element Methods, Explicit Methods, Convergence, Stability, Fourier Stability, Crank-Nicolson Implicit, Alternating Directions Implicit, Successive-Over-Relaxation, Eigenvalue Decompositions, Higher Level Schemes, Artificial Boundaries, Irregular Regions.
     4. Numerical solution of Elliptic PDE. Finite Differences, Explicit Methods, Iterative Methods (Jacobi, Gauss-Seidel), Conjugate Gradient Method, Multigrid Method, Finite Element (Galerkin) Method, Stability.
     5. Numerical solution of Hyperbolic PDE. Finite Differences, D'Alembert's exact Solution, Method of Characteristics, Courant-Friedrichs-Lewy Condition, Lax-Wendroff Method, Characteristic Meshes, Nonlinear Problems, Artificial Viscosity, Stability.
     6. Numerical Solution of Nonlinear PDE. Quasi-Linearization, Predictor- Corrector.
     7. Computer Implementation.
   • References:
     1. G. D. Smith, Numerical Soution of Partial Differential Equations, Oxford, 1985
     2. A. R. Mitchell and D. F. Griffiths, The Finite Difference Method in Partial Differential Equations, Wiley, 1980.
     3. C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis, Addison-Wesley, 1994.
     4. W. L. Briggs, A Multigrid Tutorial, SIAM, 1987.
     5. G. Strang and G. Fix, An Analysis of the Finite Element Method, Prentice-Hall, 1973.
2. Database Computing.
   • Normal preparation: MCS 589 Advanced Topics in Computer Science (Database Computing)
   • Other Background: MCS 565 Mathematical Theory of Databases.
   • Topics:
     1. Object managers and persistence: persistence, object input-output, mapping, swizzling, large objects.
     2. Data models: extended relational, functional, object model, calculus approach, algebra approach, logic approach.
     3. Queries: querying using sets and relations, SQL, querying using objects and sets, Object SQL, parallel queries.
     4. Techniques to improve performance: caching, migration, prefetching, replication, clustering, indicies, incremental queries
     5. Distributed stores and distributed queries: data centered parallelism, message passing, distributed queries.
     6. Data mining: statistical queries, correlations, cluster techniques, visualization.
   • References:
     1. R. G. G. Cattell, Object Data Management, Addison-Wesley, Reading, Massachusetts, 1991.
        Other books on object oriented databases are:
     2. Stanley B. Zdonik and David Maier, editors, Readings in Object-Oriented Database Systems, Morgan Kaufmann Publishers, San Mateo, California, 1990.
     3. Alan Dearle, Gail M. Shaw, and Stanley B. Zdonik, Implementing Persistent Object Bases, Morgan Kaufman Publishers, San Mateo, 1991.
     4. Won Kim and Frederick H. Lochovsky, editors, Object-Oriented Concepts, Databases, and Applications, ACM Press, New York, 1989.
     5. Michael Stonebraker, Readings in Database Systems, Morgan Kaufman Publishers, San Mateo, 1988.
     6. Peter M. D. Gray, Logic, Algebra, and Databases, Ellis Horwood Limited, Chichester, and John Wiley & Sons, New York, 1984.
     7. Francois Bancilhon, Claude Delobel, and Paris Kanellakis, editors, Building an Object-Oriented Database System, Morgan Kaufman Publishers, San Mateo, California, 1992.
     8. Peter M. D. Gray, Krishnarao G. Kulkarni, and Norman W. Paton, Object-Oriented Databases, Prentice Hall, New York, 1992.
     9. Won Kim, Introduction to Object-Oriented Databases, The MIT Press, Cambridge, Massachusetts, 1990.
3. Symbolic Computing.
   • Normal Preparation: One of
     
     MCS 561 Algebraic Symbolic Computation:
     MCS 563 Analytic Symbolic Computation.
   • Other Background: MCS 460 Introduction to Symbolic Computation. See also former course: MCS 560 (Math 560) Methods of Symbolic Computation.
   • Topics:
     1. Internal representation of symbolic data structures.
     2. Multi-Precision Arithmetic.
     3. Polynomial Algorithms. Arithmetic of Polynomials, Rational Functions and Power Series.
     4. Factorization Techniques.
     5. Solving Systems of Equations.
     6. Programming Computer Algebra Algorithms.
     7. Automatic Differentiation Algorithms.
     8. Integration Algorithms.
     9. Series, Perturbations, Normal Forms and Differential Equations.
     10. Combined Numeric/Symbolic Algorithms.
   • References:
     1. B. W. Char et al, Maple V Language Reference Manual, Springer-Verlag, 1991.
     2. B. W. Char et al, Maple V Library Reference Manual, Springer-Verlag, 1991.
     3. K. O. Geddes, S. R. Czapor and G. Labahn, Algorithms for Computer Algebra, Kluwer Academic Publishers, Boston, 1992.
     4. Richard H. Rand and Dieter Armbruster, Perturbation Methods, Bifurcation Theory, and Computer Algebra, Springer-Verlag, 1987.
     5. Andre Heck, Introduction to Maple, Springer-Verlag, 1993.

Web Source: http://www.math.uic.edu/~hanson/prelsncsyl.html

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