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: