• 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: `Linear Algebra' by Friedberg, Insel and Spence, chapters 1 through 7; `Linear Algebra and its Applications' by Strang, chapters 1 to 7.2 plus appendices A and B; `Applied Linear Algebra' by Noble and Daniel, chapters 1 through-9. Students should be familiar with applications to difference equations, differential equations, quadratic forms, and minimum principles. Suggested supplementary texts are: `Finite Dimensional Vector Spaces' by Halmos, and `Linear Algebra' by Hoffman and Kunze.

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 7 of `Numerical Analysis for Applied Science' by Allen and Isaacson or chapters 2 through 8 of `Analysis of Numerical Methods' by Isaacson and Keller. 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. The following topics deserve special attention:

• application of Taylor's Theorem in one or more dimensions
• application of Richardson extrapolation
• finite difference approximation of derivatives
• analysis of fixed-points of iterative mappings

 Detailed Outline, Linear Algebra

1. Solution of linear equations. LU Decomposition.
2. Vector spaces, dual spaces, quotient spaces.
3. Linear transformations and their matrices. Isomorphisms.
4. Rank and invertibility. The Fredholm alternative.
5. Properties of determinants.
6. Diagonalization. The Cayley-Hamilton theorem.
7. Inner product spaces, Gram-Schmidt orthogonalization, least squares, adjoints, and Schur triangular form.
8. The spectral theorem for normal operators.
9. Jordan canonical form and generalized eigenvectors.
10. Norm and condition number. Applications of linear algebra to differential equations, difference equations, diagonalization of quadratic forms, maxima and minima, Rayleigh's principle.  

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.
5. Numerical analysis of eigenvalues and eigenvectors: power and inverse power methods, QR method.
6. Numerical integration: Newton-Cotes, Romberg quadrature, Gauss quadrature.
7. Numerical methods for ODES: single step methods, multistep methods, consistency, stability, error analysis, stiffness, shooting techniques.

Copies of past qualifying exams are available here.