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

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

Two courses are recommended for students preparing for Part C:

 

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.