Preliminary Exam in Applied Mathematics - Part C. Linear Algebra and Numerical Methods

Two courses are recommended for students preparing for Part C:

• Math 631: Linear Algebra
• Math 614: Numerical Methods I

Material Covered

The linear algebra part of the exam focuses on material that can be found in each of the following three texts: 'Matrix Theory' by Franklin, 'Applied Numerical Linear Algebra' by Demmel, and 'Numerical Linear Algebra and Applications' by Datta. Students should be familiar with solving linear systems, matrix decompositions, applications to differential equations, variational principles, condition number and effect on system solutions, pivoting, and numerical methods such as eigenvalue estimation and iterative methods for solving systems of equations.

The numerical methods part of the exam focuses on analyzing the behavior of numerical algorithms and estimating the resources needed to implement them. The material covered can be found in chapters 1 through 8 of `An Introduction to Numerical Analysis' by K. Atkinson. Questions on conditions for convergence, stability, rate of convergence and operation counts are to be expected. Students are expected to be able to apply basic methods of calculus, ordinary differential equations and linear algebra to the analysis of numerical algorithms.

Detailed Outline, Linear Algebra

1. Theory of Linear Equations
2. Determinants
3. Vector Spaces, Subspaces , Span, Basis, Dimension, Rank
4. Elementary Row Operations, LU Decomposition
5. Least-squares Solutions.
6. Eigenvalues, Eigenvectors, and Canonical Forms
7. The Cayley Hamilton Theorem
8. Unitary Matrices
9. The Gram-Schmidt Process
10. Hermitian Matrices, Positive Definiteness,
11. Unitary Triangularization
12. Normal Matrices
13. Eigenvalue-Eigenvector Decomposition in ODEs
14. Variational Principles and Perturbation Theory
15. The Rayleigh Principle
16. The Courant Minimax Theorem
17. The Inclusion Principle
18. Criteria for Positive Definiteness
19. Hadamard’s Inequality
20. Weyl’s Inequality
21. Gershgorin’s Theorem
22. Vector and Matrix Norms – Condition Number
23. Perturbation Theorems and Conditioning
24. Gaussian Elimination Techniques with Pivoting
25. Singular Value Decomposition
26. Eigenvalue Computation
27. Iterative methods (Jacobi, Gauss-Seidel, SOR)

Detailed Outline, Numerical Methods

Algorithms considered on the exam are generally drawn from the following:

1. Direct linear solvers: algorithms, partial pivoting, versions for banded and symmetric matrices, operation counts, error analysis.
2. Indirect linear solvers: matrix splitting, Gauss-Seidel, SOR, Jacobi, conjugate gradient.
3. Nonlinear solvers: stability of fixed points, bisection and secant methods, Newton's method in one or more variables, rate of convergence.
4. Interpolation and approximation: polynomial interpolation, splines, least squares, minimax approximation.
5. Numerical integration: Newton-Cotes,Gauss quadrature, Richardson extrapolation.
6. Numerical methods for ODEs: single step methods, multistep methods, consistency, stability, error analysis, stiffness, absolute stability.

Copies of past qualifying exams are available here.