User:Egm6341.s11/Lecture plan

 Under construction throughout the semester. Egm6321.f10 21:49, 31 October 2010 (UTC)

= All versions =

Spring 2011, Spring 2010

EML 5526, Spring 2011, Lecture plan

= Recorded lectures, TA user page =

Recorded lectures in E-Learning at UF

= Lecture notes, report table =

Additional presentations (video, wiki, static html) made in class may not be recorded on these transparencies.

A mtg number n followed by a version letter in parentheses ([a-z]) indicates that this pdf file had been updated with a 2nd version "b", 3rd version "c", etc. The second letter "x" after a version letter designates an extra lecture; this letter became "m" after the extra lecture had become a make-up lecture. If you had downloaded this pdf file before, you want to clear the cache of your browser so to get the new version.

djvu: (install the viewers DjView4 or evince) [[media:nm1.s11.mtg1.djvu|Mtg 1 (b)]], [[media:nm1.s11.mtg2.djvu|Mtg 2]], [[media:nm1.s11.mtg3.djvu|Mtg 3 (e)]], [[media:nm1.s11.mtg4.djvu|Mtg 4 (b)]], [[media:nm1.s11.mtg5.djvu|Mtg 5 (d)]], [[media:nm1.s11.mtg6.djvu|Mtg 6 (b,m)]], [[media:nm1.s11.mtg7.djvu|Mtg 7 (d,m)]], [[media:nm1.s11.mtg8.djvu|Mtg 8 (d)]], [[media:nm1.s11.mtg9.djvu|Mtg 9 (e)]], [[media:nm1.s11.mtg10.djvu|Mtg 10 (b,m)]], [[media:nm1.s11.mtg11.djvu|Mtg 11 (c)]], [[media:nm1.s11.mtg12.djvu|Mtg 12]], [[media:nm1.s11.mtg13.djvu|Mtg 13 (b)]], [[media:nm1.s11.mtg14.djvu|Mtg 14 (b)]], [[media:nm1.s11.mtg15.djvu|Mtg 15 (b)]], [[media:nm1.s11.mtg16.djvu|Mtg 16 (b,m)]], [[media:nm1.s11.mtg17.djvu|Mtg 17 (c)]], [[media:nm1.s11.mtg18.djvu|Mtg 18 (c)]], [[media:nm1.s11.mtg19.djvu|Mtg 19 (c)]], [[media:nm1.s11.mtg20.djvu|Mtg 20 (b)]], [[media:nm1.s11.mtg21.djvu|Mtg 21 (c)]], Mtgs 22+23: Exam 1 (thus the Mtg no. from 22 to 26 should be incremented by 2, i.e., [[media:nm1.s11.mtg22.djvu|Mtg 22]] should be numbered Mtg 24, ..., [[media:nm1.s11.mtg26.djvu|Mtg 26]] should be numbered Mtg 28) [[media:nm1.s11.mtg22.djvu|Mtg 22 (b,m)]], [[media:nm1.s11.mtg23.djvu|Mtg 23 (a,m)]], [[media:nm1.s11.mtg24.djvu|Mtg 24 (d)]], [[media:nm1.s11.mtg25.djvu|Mtg 25 (c)]], [[media:nm1.s11.mtg26.djvu|Mtg 26 (b)]], (Mtg numbers are readjusted from Mtg 29 on) [[media:nm1.s11.mtg29.djvu|Mtg 29 (b,m)]] [[media:nm1.s11.mtg29.old.djvu|{old}]], [[media:nm1.s11.mtg30.djvu|Mtg 30 (3,m)]], [[media:nm1.s11.mtg31.djvu|Mtg 31 (b)]], [[media:nm1.s11.mtg32.djvu|Mtg 32 (b,x)]], [[media:nm1.s11.mtg33.djvu|Mtg 33 (b)]], [[media:nm1.s11.mtg34.djvu|Mtg 34 (b)]], [[media:nm1.s11.mtg35.djvu|Mtg 35]], [[media:nm1.s11.mtg36.djvu|Mtg 36 (b,x)]], [[media:nm1.s11.mtg37.djvu|Mtg 37 (b)]], [[media:nm1.s11.mtg38.djvu|Mtg 38]], [[media:nm1.s11.mtg39.djvu|Mtg 39]], [[media:nm1.s11.mtg40.djvu|Mtg 40 (c)]] ([[media:egm6341.s10.mtg38.djvu|S10 Mtg 38]], [[media:egm6341.s10.mtg39.djvu|S10 Mtg 39]], [[media:egm6341.s10.mtg40.djvu|S10 Mtg 40]], [[media:egm6341.s10.mtg41.djvu|S10 Mtg 41]]), [[media:nm1.s11.mtg41.djvu|Mtg 41 (c)]] ([[media:egm6341.s10.mtg41.djvu|S10 Mtg 41]]), [[media:nm1.s11.mtg42.djvu|Mtg 42 (b)]] ([[media:egm6341.s10.mtg38.djvu|S10 Mtg 38]], [[media:egm6341.s10.mtg39.djvu|S10 Mtg 39]], [[media:egm6341.s10.mtg40.djvu|S10 Mtg 40]], [[media:egm6341.s10.mtg41.djvu|S10 Mtg 41]], [[media:egm6341.s10.mtg42.djvu|S10 Mtg 42]]), Mtgs 43+44: Exam 2

Report table

Course billboard

Inspiring video on teamwork and learning: A hole in the wall: How children learn without a teacher

Mourning the Death of Handwriting, By Claire Suddath. Time Magazine, Monday, Aug. 03, 2009.

Op-Art: The Write Stuff, by Inga Dubay and Barbara Getty, NY Times, 8 Sep 2009.

= References =

$\displaystyle \color{red}{\spadesuit}$NEW:Symbolic computation
=== $\displaystyle \color{red}{\spadesuit}$NEW:Fast Clenshaw-Curtis quadrature, Matlab central ===

by Greg von Winckel, 12 Feb 2005 (Updated 16 Feb 2005). This extremely fast and efficient algorithm uses MATLAB's ifft routine to compute the Clenshaw-Curtis nodes and weights in linear time. The routine appears optimal for 2^N+1 points. Running on an average laptop, this routine computed N=2^20+1 (1048577 points) in about 4.5 seconds. Great for integrating highly oscillatory functions.

Books
$$\displaystyle \clubsuit$$ K. Atkinson, An Introduction to Numerical Analysis, Wiley, 1989. UF library QA297 .A84 1989 Google Amazon

$$\displaystyle \clubsuit$$ Suli & Mayers, An introduction to numerical analysis, Cambridge, 2003. UF library E-book Google Amazon

$$\displaystyle \color{red}{\clubsuit}$$  NEW: 

NIST Digital Library of Mathematical Functions, companion of the NIST Handbook of Mathematical Functions, ed. by F.W.J. Olver, et al., Cambridge U. Press, 2010. "Together these works represent a successor to the highly successful Handbook of Mathematical Functions (M. Abramowitz and I. Stegun, Eds.; see below), which was published by the National Bureau of Standards in 1964." NA Digest, Vol.10, No.19, May 2010.

$$\displaystyle \clubsuit$$ M. Abramowitz & I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, 1972. Read online, Download Wikipedia

$$\displaystyle \clubsuit$$ L.N. Trefethen, Spectral Methods in Matlab, SIAM, Philadelphia, 2000. google matlab codes, e.g., clencurt;

$$\displaystyle \color{red}{\clubsuit}$$  NEW: 

see also Fast Clenshaw-Curtis quadrature by von Winckel, 2005.

$$\displaystyle \clubsuit$$ L.N. Trefethen, Approximation Theory and Approximation Practice: A 21st Century Treatment in the Form of 32 Executable Chebfun M-Files, 2010 draft of first 8 chapters, pdf.

$$\displaystyle \clubsuit$$ J.D. Murray, Mathematical biology I: An introduction, 3rd edition, Springer, 2002. UF library QH323.5.M88 2002 google amazon

$$\displaystyle \clubsuit$$ R.H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, AIAA, 1999. google amazon. Battin was a speaker in the symposium Apollo: Reflections and Lessons (40th anniversary of the first moon landing), MIT, 11 Jun 2009.

$$\displaystyle \clubsuit$$ D. Zwillinger, Handbook of Differential Equations, Third Edition, Academic Press, 1998. ISBN-10: 0127843965. ISBN-13: 978-0127843964. UF library QA371.Z88 1989, 2 copies, one for in-library use. Google books Amazon.com

Web references
$$\displaystyle \spadesuit$$ Algorithm (wikipedia)

$$\displaystyle \spadesuit$$ Related MIT OpenCourseWare courses

$$\displaystyle \spadesuit$$ Wolfram|Alpha is ... "learning resource available to your students at no cost that works as a computational knowledge engine. Wolfram|Alpha is not a search engine like Google or Yahoo!, because unlike a traditional search engine, Wolfram|Alpha has the capability to instantly compute the answer to previously unasked questions instead of scouring the web and returning links to pages that already exist. The results are displayed in an easy-to-read, understandable format that can be used as a primary source for educational and academic purposes."

$$\displaystyle \color{red}{\spadesuit}$$  NEW: 

Stephen Wolfram: Computing the theory of everything, video on TED.com

$$\displaystyle \spadesuit$$ EqWorld, The World of Mathematical Equations. It is a good idea to verify the sources, as the site is not responsible for accuracy and correctness; see Rights and obligations of contributors and website administration.
 * 2nd-order linear ODEs
 * 2nd-order non-linear ODEs
 * ODE education

$$\displaystyle \spadesuit$$ ODE (Wikipedia): Be careful; always verify the sources.

$$\displaystyle \color{red}{\spadesuit}$$  NEW: 

Sitmo: Resources for financial engineers, latex equation editor, Clenshaw-Curtis quadrature, etc.

Papers
$$\displaystyle \clubsuit$$ L.N. Trefethen and Z. Battles, The Chebfun project, Oxford.
 * "Vision statement": Trefethen, Computing numerically with functions instead of numbers (Math. in Comp. Sci., 2007)
 * One-page summaries: Trefethen, Chebfun and Chebop (IMAJNA Newsletter, 2008), Approximation theory and practice  (Oxford Maths. Inst. Newsletter, 2009)
 * Survey of chebfun and chebop: Platte and Trefethen, Chebfun: a new kind of numerical computing (ECMI Proceedings, 2009)
 * Original chebfun paper: Battles and Trefethen, An extension of Matlab to continuous functions and operators (SIAM J. Sci. Comp., 2004)
 * Introduction of Chebops: Driscoll, Bornemann and Trefethen, The chebop system for automatic solution of differential equations (BIT Numer. Math., 2008)

$$\displaystyle \clubsuit$$ P. Kessler, Trapezoidal rule error, Math 128a, Berkeley, 2006. These notes were referred to in a paper by S.G. Johnson, 2008.

$$\displaystyle \clubsuit$$ S. G. Johnson, Notes on the convergence of trapezoidal-rule quadrature, online MIT course notes (2008). This document mentioned Kessler's notes in a footnote.

$$\displaystyle \clubsuit$$ Lloyd N. Trefethen, Is Gauss quadrature better than Clenshaw-Curtis?, SIAM Review 50 (1), 67-87 (2008).

$$\displaystyle \clubsuit$$ B.A. Cipra, The Best of the 20th Century: Editors Name Top 10 Algorithms, SIAM News, May 16, 2000. A look at ten algorithms named some of the "best" of the computer age. The Fast Fourier Transform (FFT) is one of them.

$$\displaystyle \clubsuit$$ D.N. Rockmore, The FFT — an algorithm the whole family can use, Computing in Science & Engineering, January/February 2000, Volume 2, Number 1, pp. 60--6. Contained a historical account of the development of the FFT.

$$\displaystyle \clubsuit$$ Sri Welaratna, 30 years of FFT Analyzers, Sound and Vibration (January 1997, 30th anniversary issue). A historical review of hardware FFT devices. See Fast Fourier transform (wikipedia).

$$\displaystyle \clubsuit$$ B.A. Conway, The choices available to space trajectory optimizers, seminar given at UF on Tue, 23 Feb 2010. By listening to this talk, I noticed an application of the Simpson's rule; I thought about teaching this application and searched the literature as shown below. Prof. Conway later graciously shared his talk slides with me (I did the literature search below, and found the paper by Hargraves & Paris 1987 and other papers before I received his talk slides, which I also shared with the students in the class). Another seminar on the use of the Hermite-Simpson algorithm was given at UF on 30 Mar 2010.

$$\displaystyle \clubsuit$$ Gauss, Legendre, ... pseudospectral methods
 * Web of Science literature search: Topic=(gauss legendre pseudospectral method*) Timespan=All Years. Databases=SCI-EXPANDED, SSCI, A&HCI.
 * Hermite-Legendre-Gauss-Lobatto Direct Transcription in Trajectory Optimization. Author(s): Williams P. Source: JOURNAL OF GUIDANCE CONTROL AND DYNAMICS  Volume: 32   Issue: 4   Pages: 1392-1395   Published: JUL-AUG 2009
 * Direct trajectory optimization using nonlinear programming and collocation, HARGRAVES, C. R., ; PARIS, S. W., Boeing Aerospace Co., Seattle, WA, Journal of Guidance, Control, and Dynamics 1987 0731-5090 vol.10 no.4 (338-342)
 * Direct optimization using collocation based on high-order Gauss-Lobatto quadrature rules, Herman, A.L., Conway, B.A., Journal of Guidance, Control, and Dynamics 1996 0731-5090 vol.19 no.3 (592-599).
 * Computational optimal control of the terminal bunt manoeuvre - Part 1: minimum altitude case, S. Subchan, R. Zbikowski, Volume 28 Issue 5, Pages 311 - 395 (September/October 2007). Bunt (maneuver): "2. (Engineering / Aeronautics) to cause (an aircraft) to fly in part of an inverted loop or (of an aircraft) to fly in such a loop".
 * Bryson, Denham, A steepest-ascent method for solving optimum programming problems, Journal of Applied Mechanics, Jun 1962, p.247-257.
 * Direct Trajectory Optimization by a Chebyshev Pseudospectral Method Fariba Fahroo, ; I. Michael Ross, Journal of Guidance, Control, and Dynamics 2002 0731-5090 vol.25 no.1 (160-166)
 * Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type Author(s): Parand K, Shahini M, Dehghan M Source: JOURNAL OF COMPUTATIONAL PHYSICS  Volume: 228   Issue: 23   Pages: 8830-8840   Published: DEC 10 2009
 * Leaky mode analysis of optical waveguides by legendre and laguerre pseudospectral method Author(s): Huang CC Source: MICROWAVE AND OPTICAL TECHNOLOGY LETTERS  Volume: 50   Issue: 10   Pages: 2507-2509   Published: OCT 2008
 * Pseudospectral methods for infinite-horizon optimal control problems Author(s): Fahroo F, Ross IM Source: JOURNAL OF GUIDANCE CONTROL AND DYNAMICS  Volume: 31   Issue: 4   Pages: 927-936   Published: JUL-AUG 2008

= News =

$\displaystyle \color{red}{\clubsuit}$ NEW: Is openness in the digital space killing creativity?, Digital Planet, BBC, 15 Mar 2011.

$$\displaystyle \color{red}{\spadesuit}$$  NEW: 

Bing, Now With More Wolfram, NY Times, November 11, 2009. ... Microsoft today announced that it will begin integrating statistics and other data from computational engine WolframAlpha into its search engine Bing.

$$\displaystyle \color{red}{\spadesuit}$$  NEW: 

Ask.com to Return to Old Service, NY Times, November 9, 2010. ... Other search engines have taken on Google with little success. Cuil, one of the upstarts, closed in September after two years. But new challengers include Blekko and WolframAlpha.

$$\displaystyle \color{red}{\spadesuit}$$  NEW: 

Fun With WolframAlpha, NY Times, June 5, 2009.

= Motivation: Simulations =

$$\displaystyle \color{red}{\spadesuit}$$  NEW: </b>

Humans to Asteroids: Watch Out!, NY Times, By RUSSELL SCHWEICKART Published: October 25, 2010. Deflect That Asteroid, NY Times, November 2, 2010

An asteroid breakup 160 Myr ago as the probable source of the K/T impactor, William F. Bottke, David Vokrouhlický & David Nesvorný, Nature 449, 48-53 (6 September 2007).

Asteroids: Spun in the sun, William F. Bottke, Nature 446, 382-383 (22 March 2007).

Asteroids: How to make a flying saucer, William F. Bottke, Nature 454, 173-174 (10 July 2008).

Asteroid, NASA. Pictures of asteroids Ida, Eros, and Chicxulub impact (Yucatan peninsula), extinction of dinausaurs.

Large asteroid impacting the Earth, simulation

Hubble Space Telescope Captures Rare Jupiter Collision, 07.24.09

Near Earth Objects Program, JPL.

Solar system collision: Set Target Earth (land only), Projectile Rock, Projectile diameter 10 km, Projectile velocity 60 km / sec; click Kaboom; see photo of Chicxulub impact.

= Numerical integration of functions: Preliminaries =

Atkinson, 1989, p.249

Norms
Atkinson, 1989, p.10

Integral mean value theorem
Atkinson, 1989, p.4

Mean value theorem (wikipedia)

Simpson's rule
On the history of the Newton-Raphson method:

Computation, comparison

 * {| style="width:100%" border="0"

$$  \displaystyle I = \int\limits_0^x \frac{e^t - 1}{t} d t = Ei(x) - ln(x) - \gamma
 * style="width:95%" |
 * style="width:95%" |

$$  (1)
 * <p style="text-align:right">
 * }

The exact result of the above integral is given in Abramowitz & Stegun, p.230, with the Euler's constant $$\gamma\,\;$$ given in Abramowitz & Stegun, p.255.

A more accurate Euler's constant can be found at Wolfram|Alpha.

See also Euler–Mascheroni constant (wikipedia) and Euler–Mascheroni constant (wolfram).

Since the exponential integral is defined as


 * {| style="width:100%" border="0"

$$  \displaystyle Ei(x) := ln(x) + \gamma + \sum_{n=1}^{\infty} \frac{x^n}{n n!} $$  (2)
 * style="width:95%" |
 * style="width:95%" |
 * <p style="text-align:right">
 * }

the integral in Eq.(1) is nothing but the series in the 3rd term in Eq.(2), which can also be obtained by integrating the Taylor series.

Newton-Cotes formula
On Thu, 11 Feb 2010, I listened to the following interesting NPR program at the University of Illinois at Urbana-Champaign; of course Newton plays an important role in the current course (and in many other areas):

Composite rule
= Lagrange interpolation error: Theorem =

Atkinson, 1989, p.134

Derivative mean-value theorem
Atkinson, 1989, p.4

Mean value theorem (wikipedia)

Rolle's theorem
Rolle's theorem (wikipedia)

Application: Cauchy distribution
Lorentzian distribution of spectral line width (Lecture 08), in Astronomy course on Nebulae, K. Wood, Univ. of St Andrews, Scotland, UK.

Portrait of Gauss on the Deutsch mark: Notice the normal distribution curve next to Gauss' portrait. Gauss "developed the Gaussian Bell Curve, to account for the statistical variation in individual observations of stellar locations." So even though Gauss was not the first to have come across the normal distribution, he may have invented it independently of de Moivre, and applied it to astronomy.

See also the quotation on Pythagoras' theorem (and the corresponding NPR program in which there was a mention of Stigler's law of misnomy).

Composite rules
= Higher-order error analysis of trapezoidal rule =

Euler-MacLaurin series
Euler–Maclaurin formula

googled for "matlab bernoulli number"


 * Matlab function for Bernoulli numbers (matlab central)


 * MATLAB routines for computation of Special Functions CETA (Center for Electromagnetic Theory and Applications), MIT.

Corrected trapezoidal rules $$CT_k (n)$$
Another notation $$CT_k (n) \equiv CT^{(2i+1)}_n (f)$$ with (2i+1)th derivative and (n+1) integration points.

Elliptic integral of second kind


Ellipse (wikipedia)



Orbital mechanics
The 18th-century battle over lunar motion, Siegfried Bodenmann, Physics Today, Jan 2010. In a dispute with more than just scientific import, Alexis Clairaut, Leonhard Euler, and Jean le Rond d’Alembert each employed their own strategies to establish that they were the first to understand a puzzling feature of the Moon’s orbit.

The Sky Is Falling; The Threat of Near Earth Objects (audio) (From the US National Academies) The US spends approximately $4 million each year searching for near-Earth objects to detect those that may collide with Earth. What is the true threat that we are facing and what can we do about it?

The Phuto Files (video): An excellent, funny, lively, and highly informative NOVA program on the history of the discovery and the current status of the (dwarf) planet Pluto.

The case for Pluto (talk video), Alan Boyle, MSNBC, Distinctive voices @ the Beckman Center, National Academies, 2010. While watching this presentation, it may be useful to take a look at the excellent article Pluto (wikipedia) now and then to identify the names of different people mentioned in the talk; the main story of the talk is also recorded in this Wikipedia article.

Clenshaw-Curtis quadrature
The Clenshaw-Curtis quadrature is used in spectral methods for efficient integration, and the spectral methods are among the important numerical methods of the 20th century:

L.N. Trefethen, Approximation Theory and Approximation Practice: A 21st Century Treatment in the Form of 32 Executable Chebfun M-Files, 2010 draft of first 8 chapters, pdf.

Discrete cosine transform
Fourier transform
 * Sine and cosine transforms
 * Discrete cosine transform
 * Computation, Fast Fourier transform

S.A. Khayam, The Discrete Cosine Transform (DCT): Theory and Application (pdf), 2003: Application of DCT to image/video compression.

Chebyshev polynomial expansion
= Application: Optimal control of trajectory =

Supersonic interceptor at minimum time
Bryson, Denham, A steepest-ascent method for solving optimum programming problems, Journal of Applied Mechanics, Jun 1962, p.247-257.

Minimum altitude aircraft bunting maneuver
Computational optimal control of the terminal bunt manoeuvre - Part 1: minimum altitude case, S. Subchan, R. Zbikowski, Volume 28 Issue 5, Pages 311 - 395 (September/October 2007). Bunt (maneuver): "2. (Engineering / Aeronautics) to cause (an aircraft) to fly in part of an inverted loop or (of an aircraft) to fly in such a loop".

Application: Population dynamics
J.D. Murray, Mathematical biology I: An introduction, 3rd edition, Springer, 2002. UF library QH323.5.M88 2002 google amazon

Butterfly effects
= Orthogonal polynomials =

Orthogonality
<b> Lecture plan of other courses </b>