MATHEMATICS 642:573:01 -- SYLLABUS
Fall 2000
Inst: B. Walsh --- Hill Center 728 (Busch) --- (732) 445-3733

This syllabus is tentative, and is subject to change. Its initial condition showed the content of the course as it has been given in the past; revisions (dated at the bottom, and also on the header that shows at the top of the terminal screen) show how the course is progressing. A current version will always be available from the Section Home Page. (Prospective) students who browse this location should be aware that the current state of this syllabus is not 100% determinative . The contents of the course will be adjusted (insofar as is reasonable and in conformity with the catalogue description) to accommodate the interests of the students who enroll. [In particular, if it does not appear that sufficiently many (> 4) students will continue into the Math 642:574 course (Spring 2001) to assure that the continuation course will run, then major adjustments may be necessary.]

The preliminary version of the syllabus contained no selections of textbook or MATLAB problems.   As the courses progresses, the syllabus will contain problem assignments; these will be adjusted in the course of the semester. Students are responsible for the content of all assigned problems; this content may occur in examination questions.

Links and references to Additional notes will appear on the Section Home Page.  Some of these notes may contain (additional, non-textbook) homework problems; these will be indicated in the syllabus.

In the syllabus, references to "A" are to the course textbook ("Atkinson"); references of the form "Nk" are references to the k-th entry in the list of notes on the course web page.



 
 
- Date Sections Subjects Textbook Exercises Notes & References
- - - - - -
1 Sep 6 1.1, 1.2 Review of elementary analysis.  Computer representation of numbers: the joys of floating-point representation. A Ch. 1 p. 43 ff.: 1(a,b), 4, 8(b), 32 for base 2. General analysis: A §1.1. Vector/matrix facts: A §§7.2-7.3 and N2 §5. Floating-point numbers: A §1.2, N1 §2.
2 Sep 11 1.3, 1.4, 1.5, 1.6; 3.1 Sources of error, and its propagation; stability.  Introduction to polynomial interpolation. A Ch. 3 p. 185 ff.: 8, 11, 20, 28, 29. General error: A §1.3. In machine operations: N1 §2.0. In analytical formulæ: A §1.4. Loss of digits: N1 §2.1. Build-up in summation loops: A §1.5, N1 §§2.3-2.4. (In)stability: A §1.6.  Review of the algebra of polynomials: N3 §1.
3 Sep 13 3.1, 3.2, 3.3 Lagrange & Newton forms of the interpolating polynomial. Neville's recursion; recursive computation of divided differences. The error term in polynomial interpolation. Evaluation of polynomials in Newton form. A Ch. 3 p. 185 ff.: 2, 12, 26, N3 §§2-4. A Ch. 3 §§1-3.
4 Sep 18 [continued] Tabular differences and rudiments of the classical "difference calculus."  Repeated nodes.  The Hermite-Gennochi form of the error term. [continued] 21, 24, 30. N3 §§5-7. A Ch. 3 §§3-6. N5 may make useful reading throughout A's Ch. 3.
5 Sep 20 2.1, 2.2, 2.3, 2.5, 2.10. Fixed-point methods; theory of contractive mappings in R^n. Rootfinding: the bisection, Newton[-Raphson] and secant methods.  Newton's method in dimensions > 1. A Ch. 2 p. 117 ff.: 11, 12, 18, 21, 22, 23, 28, 36, 54. A Ch. 2 §§5, 1, 2, 3, 10. N6 §§1, 3, 5.
6 Sep 25 2.4 Error analysis in rootfinding.  Müller's method. - A Ch. 2 §4. N6 § 4.
7 Sep 27 2.6 Rates of convergence. Aitken extrapolation. A Ch 2. p. 123: 30. Try using Maple or MATLAB to do the symbolic calculations. N6 § 2 & § 6. A Ch. 2. § 6.
8 Oct 2 4.1, 4.2, 4.3, 4.6 Introduction to the notions of uniform and (weighted) L^2 approximation.  Bernštein polynomials and the Bohman-Korovkin theorem.  Theorems of de la Vallée-Poussin and Chebyshev on approximation by interpolation; Jackson's theorem on speed of approximation. - -
9 Oct 4 4.4, 4.5 Orthogonal polynomials and L^2 approximation.  L^2 convergence of interpolants of a continuous function.  Trigonometric (= periodic) versions of the same ideas. - (notes)
10 Oct 9 7.1, 7.2, 7.3 Review of linear algebra; properties of the ell-p norms on R^n and C^n. - -
11 Oct 11 [continued], 7.4 Eigenvalues, spectral radius, inversion and convergence of iterates. - -
12 Oct 16 3.7 Piecewise polynomial interpolation; cubic splines. - -
13 Oct 18 [continued] - - -
14 Oct 23 - Catch-up and review. - -
15 Oct 25 Midterm Exam  HOUR EXAMINATION - -
16 Oct 30 5.1, 5.2, 5.3 Introduction to approximate integration; Peano-kernel error formulas.  (Briefly: general Newton-Cotes formulas.) - -
17 Nov 1 5.4 The Euler-Maclaurin sum formula; asymptotic error formulas; Richardson-Romberg extrapolation. - -
18 Nov 6 [continued]; 5.3, 5.6 Further topics.  (Weighted) Gaussian quadrature - -
19 Nov 8 5.7 Numerical (discretized) differentiation. - -
20 Nov 13 6.1, 6.2 Introduction to numerical methods for ODE initial-value problems.  Euler's method. - -
21 Nov 15 [continued] Additional considerations.  Systems of ODEs (and higher order equations). - -
22 Nov 20 6.10 Higher-order Taylor methods.  Methods of Runge-Kutta type. - -
23 Nov 27 - Adaptive error-estimation and adaptive solution methods. - (notes)
24 Nov 29 6.3, 6.4, 6.5 Introduction to multistep methods.  The trapezoidal and midpoint methods. - -
25 Dec 4 - Underlying linear algebra of multistep methods: convergence and stability. - (notes)
26 Dec 6 6.6 A low-order predictor-corrector algorithm. - -
27 Dec 11 6.8 Convergence and stability for multistep (and predictor-corrector) methods. - -
28 Dec 13 - Catch-up and review. - -
- - - - - -
29 Dec ?? Final Exam  All covered material. Time and location to be announced in class. -

Notes:

  last revised 1623 EDT 9/27/2000