Review Topics

Exact vs. finite-precision arithmetic and its effect on numerical computations.

Sensitivity of solution of a linear system and eigenvalue computation to round-off errors; condition number.

Matrix norms: definitions and computations for given matrices (note that you should know how matrix norms are "induced" by the corresponding vector norms).

Direct methods for systems of linear equations: Gaussian elimination, LU factorization and Cholesky factorization (computational algorithms, computational cost, advantages and disadvantages).

Inutility of determinants for numerical computations.

Pivoting and round-off errors.

Approximation of linear equations (differential, integral, delay, etc.) using system of algebraic equations.

Iterative solution of systems of equations using the Jacobi and Gauss-Seidel methods: algorithms, computational cost and sufficient conditions for convergence.

Properties and applications of special matrices: orthogonal, unitary, symmetric, positive-definite, self-adjoint, etc.

Matrix inverse vs. left inverse; orthogonal vs. left-orthogonal matrices.

QR algorithm: properties of the matrices Q and R, relationship to Gram-Schmidt orthogonalization and Householder transformation, reduced vs. full implementation.

Overdetermined systems of equations: solution in the least-squares sense and its geometric interpretation, relationship to the QR algorithm, applications to linear regression.

Characterization of the spectrum of a given matrix using Gelfand's formula and Gershgorin's theorem.

Computation of eigenvalues and eigenvectors: power method, inverse power method and acceleration via the Rayleigh quotient.