NUMERICAL ANALYSIS

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/CS 4/52201-2, 6/72251-2. 
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.