# Numerical Analysis - Breadth and Depth Exams

• Understanding of concepts pertaining to the following topics is required for the breadth examination.
• In-depth understanding of these concepts and the ability to present formal arguments is required for the depth examination.
• Preparation: Class CS537 and additional readings from textbooks such as:
1. [BOOK1] S. D. Conte and C. de Boor, Elementary Numerical Analysis, An Algorithmic Approach, McGraw-Hill Book Co., New York, 1980.
2. [BOOK2] W. Cheney and D. Kincaid, Numerical Mathematics and Computing, 4th Ed., Brooks/Cole, New York 1999.
• Topics:
• Floating-Point Arithmetic [BOOK1 Ch1, BOOK2 Ch1]
• number representation and errors,
• error estimation
• condition number and stability,
• Systems of Linear Equations [BOOK1 Ch4, BOOK2 Ch6]
• basic methods, LU and QR factorizations,
• systems with matrices of special forms,
• condition number,
• error and residuals of approximations,
• backward error analysis and iterative improvement
• Polynomial Interpolation [BOOK1 Ch2, BOOK2 Ch4]
• Lagrange and Hermite interpolation,
• Newton form and divided differences,
• piece-wise polynomial interpolation and splines,
• interpolation errors
• Function Approximation [BOOK1 Ch6, BOOK2 Ch10]
• uniform polynomial approximation,
• data fitting,
• orthogonal polynomials and least-squares approximation
• Numerical Differentiation [BOOK1 Ch7, BOOK2 Ch4]
• Numerical Integration [BOOK1 Ch7, BOOK2 Ch5]
• basic rules,
• Gaussian rules,
• composite rules,
• adaptive rules,
• singular integrals
• Nonlinear Equations [BOOK1 Ch3, BOOK2 Ch 3]
• bisection and secant methods,
• Newton method,
• polynomial equations with real roots