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
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INTERPOLATION AND APPROXIMATION
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Polynomial approximation theory
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Weierstrass Theorem
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Bernstein polynomials
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Lagrange polynomial interpolation
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Error analysis
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Divided differences
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Piecewise polynomial interpolation
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piecewise Lagrange and Hermite interpolation
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cubic spline interpolation
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error analysis
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Least squares approximation
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normal equations
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orthogonal polynomials
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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
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QUADRATURE
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Peano kernel theorem
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Euler-Maclaurin Expansion
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Extrapolation and Romberg integration
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Adaptive quadrature
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Gaussian quadrature
{SB} Chapter 3, {A Sections 5.3, 5.5, 5.6
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EIGENVALUE PROBLEMS
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Similarity transformations
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Rayleigh quotients
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Singular value decomposition
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Householder transformations
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Power and Inverse Power methods
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QR algorithm
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Shift strategies
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Convergence theory
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Eigenvalue Perturbation theory
{SB} Chapter 6, {S} Chapters 6 and 7
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ITERATIVE METHODS FOR LINEAR SYSTEMS
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The classical iterations
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Jacobi method
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Gauss-Seidel method
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SOR
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Convergence theory
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The conjugate gradient method
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Preconditioning
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convergence properties
{SB} Sections 8.1-8.4, 8.7{GvL} Chapter 10
References
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A. K. Atkinson, An Introduction to Numerical Analysis, Wiley,
1978. QA297.A84
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GvL. G. Golub and C. van Loan,
Matrix computations, John
Hopkins University Press, 1983. QA188.G65
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P. M. J. D. Powell, Approximation Theory and Methods, Cambridge
University Press, 1981 QA221.P65
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S. G. W. Stewart,
Introduction to Matrix Computations, Academic
Press, 1973. QA188.S7
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SB. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis,
Springer, 1980. QA297.S8213
The CMSC/MAPL 667 NUMERICAL ANALYSIS EXAM
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NONLINEAR SYSTEMS OF EQUATIONS
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Newton's method
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Broyden's method
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Rates of convergence
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Minimization
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Newton's method
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Secant methods
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Rates of convergence
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Nonlinear least squares
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Gauss - Newton methods
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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
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ORDINARY DIFFERENTIAL EQUATIONS
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Multistep methods
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Error estimation and stepsize control
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Stiffness
(SB), sections 7.2.6 through 7.2.15 (SG), chapters 3,4,5,6
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BVP'S FOR DIFFERENTIAL EQUATIONS
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Difference methods for ODE's and PDE's
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Variational methods for ODE's and PDE's
(BVP), sections 4, 3, 5.2 (J), chapters 2, 3, 4, 5
References:
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(BPV) I. Babuska, M. Prager, and E. Vitasek, Numerical Processes in
Differential Equations, Wiley, 1966
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(DB) G. Dahlquist and A. Bjork, Numerical Methods, Prentice
Hall, 1974, QA297.D3313
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(DS) J.E. Dennis Jr. and R. B. Schnabel, Numerical Methods for Unconstrained
Optic and Nonlinear Equations, Prentice Hall, 1983, QA402.5.D44
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(J) C. Johnson, Numerical Solution of Partial Differential Equations
by the Finite Element Method, Cambridge University Press, 1987. TA347.F5J62
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(K) R. B. Kellogg, Notes on the solution of nonlinear systems deposited
in the reserve section of the Engineering Library.
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(OR) J. M. Ortega and W.C. Rheinboldt, Iterative solution of Nonlinear
Equations in Several Variables, Academic Press, 1970
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(SB) J. Stoer and R. Bulirsch, Introduction to Numerical Analysis,
Springer, 1980, QA297.S8213
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(SG) L. F. Shapine and M. K. Gordon,
Computer Solution of Ordinary
Differential Equations, W.H. Freeman, 1975