# Qualifiers: 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 42201 / 52201: Introduction to Numerical Computing I
• MATH/CS 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, 3rd ed., 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, 3rd ed., Springer
• Trefethen and Bau, Numerical Linear Algebra, SIAM