## NUMERICAL ANALYSIS

The numerical analysis graduate courses, MAPL 666 (Numerical Analysis I) and MAPL 667 (Numerical Analysis II), are survey courses to give students an overview of the subject and to prepare some students for more advanced coursework in the area. MAPL/CMSC 666 includes iterative methods for linear systems, piecewise interpolation, eigenvalue problems, and numerical integration. MAPL/CMSC 667 covers nonlinear systems of equations, ODE and boundary value problems. All qualifying exams in numerical analysis for MATH and MAPL graduate students will be based on material from 666 and 667. Students interested in numerical analysis but having no previous experience should start with 466 and proceed to 666-667. More experienced students should begin with 666.

### The CMSC/MAPL 666 NUMERICAL ANALYSIS EXAM

• INTERPOLATION AND APPROXIMATION
1. Polynomial approximation theory
• Weierstrass Theorem
•  Bernstein polynomials
2. Lagrange polynomial interpolation
• Error analysis
•  Divided differences
3. Piecewise polynomial interpolation
• piecewise Lagrange and Hermite interpolation
•  cubic spline interpolation
•  error analysis
4. Least squares approximation
• normal equations
•  orthogonal polynomials
• Orthogonalization and QR factorization
{SB} Sections 2.1, 2.4, 3.6, 4.7.1, 4.8.1-3, {P} Chapters 1-6, 11, 16.4, 18{S} Chapter 5
1. Peano kernel theorem
2.  Euler-Maclaurin Expansion
3.  Extrapolation and Romberg integration
{SB} Chapter 3, {A Sections 5.3, 5.5, 5.6
• EIGENVALUE PROBLEMS
1. Similarity transformations
2.  Rayleigh quotients
3.  Singular value decomposition
4.  Householder transformations
5.  Power and Inverse Power methods
6.  QR algorithm
• Shift strategies
•  Convergence theory
7. Eigenvalue Perturbation theory
{SB} Chapter 6, {S} Chapters 6 and 7
• ITERATIVE METHODS FOR LINEAR SYSTEMS
1. The classical iterations
2.  Jacobi method
3.  Gauss-Seidel method
4.  SOR
5.  Convergence theory
•  Preconditioning
•  convergence properties
{SB} Sections 8.1-8.4, 8.7{GvL} Chapter 10

#### References

• A. K. Atkinson, An Introduction to Numerical Analysis, Wiley, 1978. QA297.A84
• GvL. G. Golub and C. van Loan, Matrix computations, John Hopkins University Press, 1983. QA188.G65
• P. M. J. D. Powell, Approximation Theory and Methods, Cambridge University Press, 1981 QA221.P65
• S. G. W. Stewart, Introduction to Matrix Computations, Academic Press, 1973. QA188.S7
• SB. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer, 1980. QA297.S8213

### The CMSC/MAPL 667 NUMERICAL ANALYSIS EXAM

• NONLINEAR SYSTEMS OF EQUATIONS
1. Newton's method
2.  Broyden's method
3.  Rates of convergence
4.  Minimization
5.  Newton's method
• Secant methods
•  Rates of convergence
6. Nonlinear least squares
7.  Gauss - Newton methods
8.  Levenberg - Marquardt algorithm
(DS), Chapters 5,8 (SB), Sections 5.1, 5.2, 5.3, 5.4, 5,11 (K) (OR), Chapters 9, 10 (DS), Sections 6.1, 6.2, 6.3, Chapters 9 and 10
• ORDINARY DIFFERENTIAL EQUATIONS
1. Multistep methods
2.  Error estimation and stepsize control
3.  Stiffness
(SB), sections 7.2.6 through 7.2.15 (SG), chapters 3,4,5,6
• BVP'S FOR DIFFERENTIAL EQUATIONS
1. Difference methods for ODE's and PDE's
2.  Variational methods for ODE's and PDE's
(BVP), sections 4, 3, 5.2 (J), chapters 2, 3, 4, 5

#### References:

• (BPV) I. Babuska, M. Prager, and E. Vitasek, Numerical Processes in Differential Equations, Wiley, 1966
•  (DB) G. Dahlquist and A. Bjork, Numerical Methods, Prentice Hall, 1974, QA297.D3313
• (DS) J.E. Dennis Jr. and R. B. Schnabel, Numerical Methods for Unconstrained Optic and Nonlinear Equations, Prentice Hall, 1983, QA402.5.D44
• (J) C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987. TA347.F5J62
• (K) R. B. Kellogg, Notes on the solution of nonlinear systems deposited in the reserve section of the Engineering Library.
• (OR) J. M. Ortega and W.C. Rheinboldt, Iterative solution of Nonlinear Equations in Several Variables, Academic Press, 1970
•  (SB) J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer, 1980, QA297.S8213
•  (SG) L. F. Shapine and M. K. Gordon, Computer Solution of Ordinary Differential Equations, W.H. Freeman, 1975