ࡱ> QSPd#` v5bjbj5G5G *HW-W-v-VVVVVVVj. . . . ,Z jW  $ Xl>9W$Xh[2WV 2WVV GWV V rTVVh v g. fL]W0WZ[f[h[Vh42W2WvWjjj. jjj. jjjVVVVVV  NUMERICAL ANALYSIS SOFTWARE Due May 3 20 points Please download this file from my web site and use Word to insert your answers to each question following the question. Purpose: To introduce you to sources of mathematical software and to sources of reference information that can be obtained. The internet has a wealth of high quality software for scientific computing. Much of it is free. Useful sites that contain this software are netlib ( HYPERLINK "http://www.netlib.org" www.netlib.org) and the national institute of standards ( HYPERLINK "http://math.nist.gov" http://math.nist.gov) as well as sites related to mathemematical programming languages (for example  HYPERLINK "http://www.mathworks.com" www.mathworks.com for Matlab software). GAMS, which is at the national institute of standards site, is a tool for locating mathematical software that could be useful to help solve scientific computing problems. MathSciNet is a tool for locating background references for particular numerical methods or for any mathematical topic. MathSciNet includes the facility to search through mathematics reviews, which has reviews of nearly all articles written in journals that relate to mathematics. This includes many computer science journals and engineering journals. The steps that you need to follow are:  1. Use internet to get into the GAMS facility at address  HYPERLINK "http://math.nist.gov" http://math.nist.gov . Scroll in the GAMS window until you see "Project summary" in color. Click on this with the mouse and read through the project summary. Below write a short paragraph stating some of the key points in the summary. 2. Now we can try to search GAMS for software relating to some of the methods were currently studying. GAMS is organized by problem type but we want to search GAMS by the type of numerical method. This is easiest to do using the search facilities in the "text in module abstract". Return to the GAMS main page. Scroll up or down on the GAMS page until you find "text in module abstract". Click on this "text in module abstract". In the "search for" window type "predictor* corrector*". The stars here are wild cards that will make the search more flexible. For example with the stars we can locate "predictor-corrector" as well as "predictor corrector". Now click on the search box in the search window. Below list the number of modules and packages located. 3. Pick one of the packages found, click on it and then read the package abstract. In the space below list the package name and a brief description of what the package does. 4. Use the back arrow key to return to the page of packages and modules for predictor corrector methods. Click on one of the modules and read the abstract of the module. Below list the module name and briefly describe the module. 5. Note that some of the packages are proprietary and some are not. Modules in packages that are not proprietary can usually be downloaded through GAMS by clicking on a "fullsource" box. Find a module that is not in a proprietary package and "fullsource" down load it. Probably you will be able to look at the code on-line but sometimes a window will pop up asking where you would like to save the file. You would need to compile the program (many of the programs are in Fortran) to run it. You don't need to try to compile in this assignment. Below list the name of the package and module that you downloaded and as well the number of lines of code in the module. To get the number of lines of code in the module you may need to download the file and open it in Word (use tools/ word count) or Textpad (use view/ line numbers) etc. Extra credit: Choose a module and recreate equivalent code, on your own, from scratch (just kidding). 6. Now use the back arrow key in Netscape to return to the "search in module abstract" page. Search for "runge* kutta*". List below the number of modules and packages located. 7. In the list you will find methods with names Runge-Kutta-(some other name). For example some of the methods will be the Runge-Kutta-Fehlberg method that we discussed in class. List three other methods of the form Runge-Kutta-something.  8. Pick one of the three methods listed above or pick Runge-Kutta-Fehlberg. Suppose that you needed to read about the method in order to discover the ideas underlying the method. You can look up related journal articles by searching another data base MathSciNet. To use MathSciNet when you are on campus go to  HYPERLINK "http://www.ams.org/mathscinet" www.ams.org/mathscinet . From off campus you will need to go through the SJSU library at  HYPERLINK "http://www.sjlibrary.org/gateways/academic/" http://www.sjlibrary.org/gateways/academic/ . Select databases and Mathscinet. You will need to enter the username "spartans" and the current semester's proxy server password . Once at mathscinet click on "Start Full Search". Scroll down to the "anywhere" and type in the third name followed by a * in the label runge-kutta-(some name) for the method that you chose. For example if you chose fehlberg you would type in "fehlberg*". Now click on start search. You should get a list of references that have this name somewhere in the abstract of the reference. Below list the total number of articles found. One of the challenges in searching a data base is narrowing the number of hits to a set that is of interest to you. This is an art which we won't pursue here.. You can also search MathSciNet by author or title. For example one could search for the publications of the infamous mathematician T. Kaczynski (aka the Unibomber). I am not assigning this. 9. You can get a summary of the article by clicking on the review number which is the first field listed in the reference. For example it might be a number like "96d:65206". Pick one article and click on the review number. Insert the review or summarize the review in your own words below. 10. You can view online some of the articles that are reviewed in MathSciNet. These articles will have a box indicating To original article or To journal. Often to access these you need to have a paid subscription to the journal (however if you go through the SJSU library at  HYPERLINK http://www.library.sjsu.edu www.library.sjsu.edu you can get electronic access to many of the librarys journals). Some journals are at a marvelous site JSTOR ( HYPERLINK http://www.jstor.org www.jstor.org). This site is publicly accessible and all the articles except for the last five year --for more than 100 journals. To see how to get to JSTOR from MathSciNet look for the article Differentable interpolants for high-order Runge-Kutta methods by J. H. Verner. You might want to do a MathSciNet search for verner* to find this article. When you find the article click on To Original Article which will transfer you to JSTOR. Read through the abstract that will appear at the top of the article. Below state what the author says about the possibility of improving the lower bound mentioned in the abstract. 11. Now pick some numerical method that we have discussed in class or that you have heard about from another source. Search GAMS to see how many modules it locates for this method. Also search MathSciNet for related references. If you don't get any hits in your searches you may need to try a variation of the name you use in the search, add wild cards or choose another method. You will probably not find many or any GAMS modules for the simplest methods such as the Trapezoid method since there are better methods. Below list the method that you choose, the number of modules GAMS found and the number of references MathSciNet found. Describe any troubles of surprises that you encountered in your search. 12. Another good source of background material is Google scholar. For the same numerical method that you used in question 11 go to Google scholar  HYPERLINK "http://scholar.google.com/" http://scholar.google.com/ and search for the numerical method. Note that if you use quotes (which force an exact match) you can narrow the number of hits somewhat. What term did you search for and how many (from the first line of the search results) results are shown? 13. Of course other excellent sources of quality software in numerical analysis are mathematical programming languages. Matlab, Maple and Mathematica are the best known such language. However there are free alternatives such as Octave ( HYPERLINK "http://www.octave.org/" http://www.octave.org/ ), which is compatible to some extent with Matlab, and Maxima ( HYPERLINK "http://maxima.sourceforge.net/" http://maxima.sourceforge.net/ ) which is a computer algebra system similar to Maple and Mathematica. To see an impressive Matlab demo of a solution to an ordinary differential equation, open up Matlab and type lorenz. The Lorenz equations are a simple system of differential equations, for example  EMBED Equation.3 ,  EMBED Equation.3  and  EMBED Equation.3 , that has a path that is bounded but not periodic nor convergent nor predictable (it is a "strange attractor"). Run the lorenz program and cut and paste the graph here. Note that I could not get Matlab's "edit / copy figure" to work for this picture. However you can make the figure full screen, press the "print screen" key to put it in the clipboard and then paste the result into an application such as word. Extra credit: Keep running the lorenz example by pressing start over again until you can predict in advance where the solution will end up ------- just kidding (again). 14. Another interesting demonstration is odedemo (in Matlab 5) or odeexamples (in Matlab 7). Try the demo. For one (you choose) of the example ODE's cut and paste a plot below. With odedemo a plot with the phase2d or phase3d selection is often more interesting. With odeexamples the plot style is predetermined. 15. The Matlab 5 odedemo is actually much more interesting because one can choose the numerical method (ode45, ode23, ode113, ode15s or ode23s) to use. For this problem you will need Matlab 5 not Matlab 7. Run odedemo with ode45, orbitode, plot = phase2d and refine = 4. Now run ode15s with orbitode, plot = phase2d and refine = 4 Which code runs more quickly. 16. Repeat problem 14 (that is compare the time of ode45 and ode15s) with the problem the buiode example, output = plot and refine = 1. Which code runs more quickly 17. Documentation for Matlab's ode routines (ode45, ode23, ode113, ode15s,ode23s, etc.) is at  HYPERLINK "http://www.mathworks.com/access/helpdesk/help/pdf_doc/otherdocs/ode_suite.pdf" http://www.mathworks.com/access/helpdesk/help/pdf_doc/otherdocs/ode_suite.pdf . Scan through this paper and briefly describe the methods used in any two of the above routines. Specify the routine and a brief one sentence description of the method for each routine. 18. Scan through the above document again. Which method does the article recommend as a user's first choice and what does the article recommend as a second choice if the first method has difficulties? 19. The Transactions on Mathematical Software provides another summary of sources of mathematical software at  HYPERLINK "http://math.nist.gov/toms/Resources.html" http://math.nist.gov/toms/Resources.html. Browse around this site. 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