Applied Numerical Methods
Fall '98, Math 151A, UCLA
Lecture: MS 5117 MWF 1-1:50pm
Instructor: Professor Eitan
Tadmor office hours. -- MS 7945 Tu: 2-3pm
Fr: 2:30-3:30pm
Discussion: MS 5117 Tu 1-1:50pm
Teaching assistant: J. Tanner
office hours. -- MS 6142 Tr: 10-12, 1-2pm
Course description [Postscript
file]
151a: Applied Numerical Methods. Fall '98
Undergraduate course in Applied Mathematics
Prof. Eitan Tadmor
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Solution of Nonlinear Systems
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The bisection method
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Newton's Method
... local vs. global convergence, multiple roots
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Extensions of Newton's method
... divided differences: the secant method, Steffensen's method...,
... Systems of equations
... high-order extensions
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Fixed point iterations
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Accelerations
... Aitken's process, Steffenssen's method
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*Polynomial Equations
... Local methods - Newton, Bairstow,...
... Global methods - Bernoulli, Graeffe, Laguerre, the square root method,...
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Approximation Theory
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General overview
... On the choice of norm: L2
vs. L¥
... Weierstrass' density theorem, Bernstein polynomials,
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Interpolation I. Lagrange interpolant
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Interpolation II. Newton interpolant
... Divided differences. Error estimates.
... Equidistant points. Synthetic calculus
... Forward backward and centered formulae.
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Interpolation III. Error estimates.
... Runge effect, region of analyticity
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*Addt'l topics
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Interpolation IV. Interpolation with derivatives
... Hermite interpolation; Splines
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Trigonometric interpolation
... FFT, truncation + aliasing, error estimates,
... elliptic solvers, fast summations ( - discrete convolution),...
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Least Squares Approximations I. A general overview
... Gramm mass matrix, ill-conditioning of monomials in L2
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Least Squares Approximations II. Fourier expansion
... Bessel, Parseval, ...
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Least Squares Approximations III. Orthogonal polynomials
... Examples: Legendre, Chebyshev,...
... Sturm's sequence; more examples: Jacobi, Hermite,..
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*Trigonometric polynomials
... Complex & real representation; Chebyshev interpolation,..
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MinMax Approximation
... Smoothness & Error Estimates
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Numerical Differentiation
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The method of undetermined coefficients
... The differentiated algebraic interpolant;
... *The differentiated spline interpolant
... Error estimates
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Equidistant Differentiation
... The synthetic approach - operational calculus
... Richardson's extrapolation
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*Addt'l topics
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Trigonometric differentiation
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Spline differentiation
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Numerical Integration - Quadratures
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Equidistant points
... Newton-Cotes formulae
... Composite Simpson's rule
... Romberg & adaptive integration
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Gauss Quadratures
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Solution of Linear Systems - Direct Methods
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Introduction
... Cramer's rule costs N!; diagonal, triangular and unitary
systems
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Gauss elimination
... The basic algorithm
... pivoting, multipliers, update, back substitution
... Operations count
... storage & pivoting for special matrices:
... symmetric, positive definite, banded, Hessenberg,...
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LU decomposition
... Elementary matrices
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Backward error analysis
... norms, a priori estimates, ill-conditioning,
... condition number, backward error estimates
... Pivoting: partial & full pivoting
... amplification factors: diagonally dominant, symmetric pos. definite,
... banded & Hessenberg matrices
... Equilibration: counterexamples,
... equilibration & pivoting - scaling, iterative refinement
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Direct LU decompositions
Crout, Doolittle & Cholesky, tridiagonal systems
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*Addt'l topics
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Other direct methods
... Cayley-Hamilton, Conjugate Gradient, block decomposition,
... Sherman-Woodbury formula,QR decomposition
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Special Fast Solvers
... FFT, Circulant matrices, Toeplitz,...
References
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BF
Bunder & Faires, NUMERICAL ANALYSIS, PWS Co. Boston, 1993
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At
K. Atkinson, An Introduction to NUMERICAL ANALYSIS, Wiley,
1987.
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CdB S. Conte & C. deBoor, ELEMENTARY NUMERICAL
ANALYSIS, McGraw-Hill, User friendly; Shows how 'it' works; Proofs,
exercises and notes
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DB G. Dahlquist & A. Bjorck, NUMERICAL METHODS,
Prentice-Hall, User friendly; Shows how 'it' works; Exercises
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IK
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E. Isaacson & H. Keller, ANALYSIS of NUMERICAL METHODS,
Wiley, The 'First'; Proofs; out-dated in certain aspects; Encrypted
message in Preface
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RR A. Ralston & P. Rabinowitz, FIRST COURSE
in NUMERICAL ANALYSIS, 2nd ed., McGraw-hill, Detailed; Scholarly
written; Comprehensive; Proofs exercises and notes
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SB J. Stoer & R. Bulrisch, INTRODUCTION
TO NUMERICAL ANALYSIS, Springer-Verlag, detailed account on approximation,
linear solvers & eigensolvers, ODE solvers,..
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W
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B. Wendroff, THEORETICAL NUMERICAL ANALYSIS, Academic Press,
1966, Only the 'Proofs'; elegant presentation
Eitan Tadmor
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