Introduction to Numerical Analysis (Applied MS)

Preparatory Course: Math 4513

1.

Binary fractions, computer floating point format, machine arithmetic, rounding procedures, roundoff error, conditioning and stability of algorithms.

2.

Lagrange interpolation, divided difference and difference tables, Hermite interpolation, errors of interpolation, interpolation at equally spaced and at Chebyshev points.

3.

Solution of a single nonlinear equation by various methods: Bisection, Newton iteration, secant iteration, simplified Newton method. Contraction mapping theorem on R and related error estimates, orders of methods, higher order methods.

4.

Solving systems of linear equations by Gaussian elimination with pivoting, and by the mathematically equivalent PLU-factorization. Norms, condition numbers of matrices, backward error analysis (Theorem of v. Neumann-Goldstein-Wilkinson). QR-factorization and applications to orthogonalization and solution of linear systems.

5.

Solving systems of nonlinear equations by Newton or simplified Newton iteration. Damped Newton Method. Contraction mapping theorem on $Rn$ and related error estimates.

6.

Numerical differentiation. Numerical integration: Newton-Cotes formulas, composite trapezoidal and Simpson rules, Gaussian rules, general interpolatory quadrature formulas. Euler-Maclaurin summation formula and the Romberg table. Errors: Upper bounds and asymptotics, examples of super-convergence: Gaussian formulas and trapezoidal rule for periodic integrands.

7.

Initial value problems for systems of ordinary differential equations. Runge-Kutta, Adams-Moulton, and backward differentiation methods, local truncation errors, global discretization error, order and stability of methods. Boundary value problems and finite difference methods.

8.

Eigenvalue problems for matrices. Power method, QR-algorithm.

The study of numerical analysis should include experience with and careful consideration of the effects of running algorithms on actual computing machines for a full understanding of subject matter.

REFERENCES: Kendall E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., Wiley, 1989; R.L. Burden & J.D. Faires, Numerical Analysis, 5th ed., PWS-Kent, 1993; Sam Conte & Carl de Boor, Elementary Numerical Analysis, 3rd ed., McGraw Hill, 1980; Peter Henrici, Elements of Numerical Analysis, Wiley, 1964; David Kincaid & Ward Cheney, Numerical Analysis: The Mathematics of Scientific Computing, 2nd ed., Brooks/Cole, 1996; J. Stoer & R. Bulirsch, Introduction to Numerical Analysis, 2nd Ed., Springer Verlag, 1993.