Preliminaries: The student is expected to be familiar with those topics normally covered in a one-year, senior-level course in numerical methods, including computer arithmetic, solving linear systems of equations (by direct methods), polynomial interpolation, numerical quadrature methods, linear least-squares data fitting, solving non-linear equations, and basic numerical methods for ODE initial-value problems.

Error Analysis: Floating-point arithmetic, roundoff-error analysis, mathematical conditioning. Interpolation: Lagrange formula, Neville's algorithm, Newton formula and divided differences, error in polynomial interpolation, Hermite interpolation, trigonometric interpolation, discrete Fourier analysis, fast Fourier transform, interpolation by spline functions.

Integration: Newton-Cotes formulas, Peano kernel theorem, Euler-Maclaurin summation formula, asymptotic expansions, extrapolation and Romberg integration, Gaussian quadrature, orthogonal polynomials.

Systems of Linear Equations: Gaussian elimination, LU-decomposition, Cholesky decomposition, backwards error analysis, matrix and vector norms and condition numbers.

Linear Least-Squares: Orthogonalization, Gram-Schmidt, Householder and Givens transformations, QR-factorization, condition of linear least-squares problems, pseudoinverse.

Eigenproblems: Matrix normal forms (Jordan, Schur), similarity reduction to tri-diagonal or Hessenberg forms, power method, inverse iteration, Rayleigh quotients, LR-method, QR-method, singular value decomposition.

Suggested Courses: MATH 42201 / 52201: Introduction to Numerical Computing I
MATH 42202 / 52202: Introduction to Numerical Computing II
MATH 62251 / 72251: Numerical Analysis I
MATH 62252 / 72252: Numerical Analysis II

Suggested References: 
Conte and de Boor, Elementary Numerical Analysis: an Algorithmic Approach, McGraw-Hill. 
Dahlquist and Bjorck, Numerical Methods , Prentice-Hall. 
Golub and Van Loan, Matrix Computations, Johns Hopkins. 
Kahaner, Moler, and Nash, Numerical Methods and Software, Prentice-Hall. 
Stewart, Introduction to Matrix Computations, Academic Press. 
Stoer and Bulirsch, Introduction to Numerical Analysis, Springer.
Golub and Van Loan, Matrix Computations, 3rd ed., Johns Hopkins.
Stoer and Bulirsch, Introduction to Numerical Analysis, 3rd ed., Springer.
Trefethen and Bau, Numerical Linear Algebra, SIAM.