MATH 524: NUMERICAL ANALYSIS II, Spring 2008
Syllabus
Description:
This course forms a one semester graduate level introduction to
numerical analysis. This is a followup course of MATH 523.
The focus is on the analysis of classical
numerical algorithms and techniques, for the solution of variety of
problems in mathematical analysis and algebra. In short, this course
will provide an introduction to the basics of the mathematical theory
behind scientific and engineering computing. The students who take
this course should have a very good and stable knowledge of single
and multi-variable calculus, linear algebra and be familiar with basic
facts from functional, real and complex analysis and the theory of
partial differential equations.
Brief topic list:
Approximation and interpolation -- linear theory;
Numerical Linear Algebra;
Numerical solution of nonlinear equations and optimization;
Numerical Solution of Ordinary Differential Equations;
Numerical Solution of Partial Differential Equations;
Tentative schedule: (subject to minor changes)
- Approximation and Interpolation:
- Best Approximation; Jackson theorem;
- Kolmogorov Characterization Theorem; Chebyshev polynomials and
best polynomial approximation in $L_\infty(a,b)$ and $L_2(a,b)$;
- Lagrange Interpolation; Lebesgue constants, divergence and
convergence of the interpolation;
- Piecewise polynomial approximation and interpolation; Splines;
- The Fast Fourier Transform.
- Numerical Quadrature
- The Peano Kernel Theorem
- Asymptotic error expansions; Richardson Extrapolation; Romberg
Integration
- Gaussian Quadrature;
- Hermite interpolation and Gaussian Quadrature;
- Numerical quadrature on simplicies and cubes.
- Adaptive quadrature. * (optional)
- Direct Methods of Numerical Linear Algebra
- Gaussian elimination and LU decomposition;
- Conditioning and round-off errors.
- Eigenvalue problems; Lanczos and Arnoldi methods;
- Classical Iterative Methods (Richardson, Jacobi, Gauss-Seidel
methods); M matrices and positive matrices; Convergence of the
classical iterative methods.
- Numerical solution of nonlinear systems and optimization
- One-point iteration; Newton's method; Broyden's
method. Convergence of Newton's method.
- Line search methods;
- Conjugate gradients; Chebyshev polynomials and convergence of
Conjugate Gradient method.
- Numerical Solution of Ordinary Differential Equations
- Convergence of Euler's Method;
- Linear multi-step methods and stability;
- One step methods (Runge Kutta methods);
- Consistency, stability and convergence. Lax Equivalence Theorem.
- Numerical Solution of Partial Differential Equations
- Boundary value problems for 2nd order elliptic PDEs; The five-point discretization of the Laplacian.
- Difference methods for the heat equation;
- Variational formulations and finite element methods; Piecewise polynomial approximation in two and three dimensions. Solvability and error estimates.
- Difference methods for hyperbolic equations;
Text:
Notes will be distributed in class.
Some possible reference books:
- Introduction to Numerical Analysis, 3rd ed., by J. Stoer and
R. Bulirsch;
- Matrix Computations, 3rd ed., by G.H. Golub and
C.F. van Loan.
- Analysis of Numerical Methods, by Eugene Isaacson and Herbert Bishop
Keller; Dover Publications 1994.
Academic Integrity Policy:
All Penn State Policies regarding ethics and honorable behavior apply
to this course.
Modified Jan 10, 2008,
by Wen Shen.