Math 191, Scientific Computation I
Syllabus

1. General Information

   Math 191 is designed to be a comprehensive introduction to Numerical Analysis and Scientific Computing. Topics in the syllabus are fairly standard for first year graduate students and constitute the basic building blocks for numerical methods across the sciences. Throughout this course, attention will be paid not only to the numerical methods but also to developing in students the ability to critically analyze the output from numerical experiments. Topics which are covered implicitly and will be emphasized during the course include: High level programming techniques, makefiles, debuggers, mixing C/C++/Fortran, using tools such as matlab for prototyping and data analysis, using computational routines from outside sources (netlib, fftw, LaPack, LinPack, etc).

2. Intended audience

   The level of material is intended for beginning graduate students from different disciplines.

3. Text

   Class Lecture Notes Online.

4. Recommended Reading

   Kendall Atkinson, ``An Introduction to Numerical Analysis", 2nd edition. (Highly Recommended)
   Research papers.

5. Course Outline

   1. Preliminaries
      Mathematical preliminaries, computer representation of numbers, source of errors and their propagation, and stability.

   2. Root-finding
      Bisection, Newton's Method, Secant Method. Newton and Quasi-Newton algorithms in multi-dimensions.

   3. Interpolation Theory
      Polynomial Interpolation, Lagrange Interpolation Formula, Newton Divided Difference, Hermite Interpolation. Piecewise Polynomial Interpolation, which include Lagrange piecewise polynomial functions and spline functions.

   4. Approximation Theory
      Least Square Approximation, Orthogonal Polynomials. Discrete Fourier Transform, and the FFT.

   5. Integration and Differentiation
      Newton-Cotes Integration Formulas, Gauss Quadratures, Asymptotic Error Formulas, Adaptive Integration, Monte-Carlo integration and Numerical Differentiation.

   6. Basic numerical linear algebra
      Gauss Elimination, how to solve Ax=b using LAPACK, introduction to stability and condition numbers.

   7. Numerical ODE
      Euler's and Backward Euler's Methods, Multi-step Methods, Runge-Kutta Methods, predictor Corrector Methods, deferred corrections, Stability Regions and Stiff Problems. Methods for stiff systems of ODE's.

6. Programming

   Some programming experience is expected. You may choose any of the following programming languages: Matlab, Fortran, C, or C++. Basic and Pascal are allowed but not recommended.
   The use of computer algebra systems is also encouraged. You may use Mathematica or Maple in your theoretical homework to generate numerical algorithms, to derive convergence rates, to simplify proofs, and to compute errors.

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