## Scientific Computing Syllabus

**For the Oral Qualifying Exam**

For the oral qualifying exam in Scientific Computing the candidate is to prepare a syllabus by selecting topics from the list below. The total amount of material on the syllabus should be roughly equal to that covered in a standard one semester graduate course. Once you have made your selections discuss them with the professor who will examine you.

##### Hardware/Programming Issues

- Machine numbers, floating point arithmetic, accumulation of rounding errors, memory hierarchy, arrays in C and FORTRAN, C++ scope, C++ classes, organization of loops for computational efficiency.

##### Computational Linear Algebra

- Basic linear algebra, solution of linear equations: direct and iterative methods, convergence, matrix factorizations (LU, LL^T, QR, SVD), linear equations and least squares, eigenvalues and eigenvectors.

##### Iterative Methods for Nonlinear Equations

- Fixed point theorems, Convergence proofs, linear iteration methods, Newton and secant methods for scalar equations, techniques for enhancing global convergence, Newton and quasi-Newton methods for nonlinear systems.

##### Approximation Theory and Interpolation

- Interpolating polynomials, Lagrange and Newton interpolation, divided differences, piecewise polynomial approximation, least squares polynomial approximation, orthogonal decompositions: Fourier series/transforms and orthogonal polynomials.

##### Differentiation and Integration

- Divided differences, Richardson extrapolation, midpoint and trapezoidal rules, the Euler-Maclaurin formula, Gaussian quadrature, singular integrals.

##### Initial Value Problems for Ordinary Differential Equations

- Finite difference methods: order of accuracy, stability analysis, convergence results, Euler's explicit and implicit methods, local truncation errors/rounding errors/accumulated errors, higher order methods: Adams Bashforth and Adams Moulton methods, Runge-Kutta methods, backward differentiation formulas, stiffness.

##### Boundary Value Problems for Ordinary Differential Equations

- Shooting methods, finite difference methods, finite element methods, eigenvalue problems.

## References

- K.E. Atkinson,
*An Introduction to Numerical Analysis*, 2nd ed. (Wiley, 1989) - Isaacson and Keller,
*Analysis of Numerical Methods*(Dover, 1994) - Kincaid, Cheney and Cheney,
*Numerical Analysis: Mathematics of Scientific Computing* - Stoer and Bulirsch,
*Introduction to Numerical Analysis* - J. Trangenstein,
*Scientific Computing*