Exam Overview
- Exam Focus: Numerical Analysis
- Exam Format: 4-Hour Written Exam
- View Past Numerical Analysis Exams »
Syllabus Topics
This exam will cover the following topics:
- General Numerical Methods
- Numerical Linear Algebra
- Numerical Methods for Ordinary Differential Equations
- Numerical Methods for Partial Differential Equations
General Numerical Methods
Principles of Numerical Mathematics
- Well-posedness and condition number of a problem
- Stability and convergence of numerical methods
- Machine representation of numbers
Rootfinding for Nonlinear Equations
- The bisection, the secant and Newton's methods
- Fixed-point iterations
- Solution of nonlinear systems of equations
Polynomial Interpolation
- Lagrange polynomials (and their Newton form)
- Hermite interpolation
- Approximation by splines
Numerical Differentiation and Integration
- Finite-difference approximations of derivatives
- Midpoint, trapezoidal, Simpson, Newton-Cote quadratures
- Richardson extrapolation
Orthogonal Polynomials in Approximation Theory
- Approximation of functions by Fourier series
- Gaussian integration and interpolation
- Fourier trigonometric polynomials
Numerical Linear Algebra
Fundamentals
- Orthogonal vectors and matrices
- Vector and matrix norms
- The singular value decomposition (SVD)
- Conditioning and condition number
Least Squares Problem
- Normal equations
- QR factorization
Solutions of Linear Systems of Equations
- Direct methods - LU factorization; Cholesky factorization
- Iterative methods - Jacobi, Gauss-Seidel, SOR, Conjugate Gradient
Eigenvalue Problem
- Power method
- QR method for symmetric matrices
Numerical Methods for Boundary Value Problems
- Boundary value problems for ODEs
- Boundary value problems for elliptic PDEs
Numerical Methods for Ordinary Differential Equations
Numerical Methods for Initial Value Problems
- One-step methods
- Linear multistep methods
- Runge-Kutta methods
- Consistency, stability and convergence
Numerical Methods for Partial Differential Equations
Finite-Difference Methods
- Accuracy and derivation of spatial discretizations
- Explicit and implicit schemes for parabolic equations
- Consistency, stability and convergence, Lax equivalence theorem
- Von Neumann stability, amplification factor
- CFL condition for hyperbolic equations
- Upwind schemes for hyperbolic equations
- Leapfrog, Lax-Friedrichs and Lax-Wendroff schemes
- Crank-Nicolson scheme for the heat equation
- Discrete approximation of boundary conditions
Finite Element Methods: Derivation and Basic Properties
Finite Volume Methods: Derivation and Basic Properties
Splitting Methods
- Dimensional splitting, ADI methods
- Operator splitting methods for convection-diffusion equations
References
- Numerical Analysis, 6th edition, by Richard L. Burden and J. Douglas Faires
- An Introduction to Numerical Analysis, 2nd edition, by Kendall E. Atkinson
- Numerical Mathematics, by Alfio Quarteroni, Riccardo Sacco and Fausto Saleri
- Numerical Linear Algebra, by Lloyd N. Trefhen and David Bau
- Matrix Computations, by Gene H. Golub and Charles F. Van Loan
- Finite Difference Schemes and Partial Differential Equations, by John C. Strikwerda
- Finite Difference Methods for Ordinary and Partial Differential Equations, by Randall J. LeVeque
- Numerical Methods for Evolutionary Differential Equations, by Uri M. Ascher