Nash - Scientific Computing with PCs (523165), страница 56
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Dobb’s Journal•For best effect, near/far perspective needs to be shown by making "closer" points larger or a differentshape, a non-trivial programming task for developers, especially given the comment above.•Grid lines are helpful to the viewer in assessing the level represented by a point for one or more ofthe three variables plotted, but such lines clutter the graph.
Users should have control of rotations andwhich (if any) grid lines to include.•So users can discern which points are "special", the graph points need to be identified by using alabelling letter or symbol as we have done in Figure 19.5.3. Unfortunately, the identifiers may makethe points so large on the display that other features are hidden. Again we have to balance betweenincluding wanted information in the graph and maintaining a clear and uncluttered image. Analternative to labelled points is the possibility of pointing an arrow cursor at a given point to haveit identified. Developing such attractive capabilities requires a lot of programming effort.•Color can be very useful in adding extra information to a display.
We have also already mentionedin Section 19.4 (in describing EXECUSTAT) how point brushing and spinning are useful in interactivedata analysis.•It may be helpful if data are pre-processed before display to focus on certain features present in thedata that we want to emphasize. Our use of the ratio-to-best transformations, or even the logarithmictransformation of the data, may be regarded in this light. Other possibilities include clustering (withcluster membership indicated) or robust regression to select and label points that may be outliers.A different type of pattern — that of similarity of cases — requires us to attempt to graph many variablesat once for a number of different cases. Several tools have been developed that try to squeeze many moredimensions out of a flat, two-dimensional page than are immediately available.
In the present examplewe could, for example, consider each row of the table that makes up Figure 19.2.1 as an observationvector.Figure 19.7.2Point cloud spinning:Left hand graph — unrotatedRight hand graph — rotated(We would probably leave out the observation labelled F because it lacks information on the threevariables time, function/gradient count, and iteration count for the program CG.) We now wish to plotsome display for each problem that uses the information for all variables. Note that we nearly always needto scale the data so that actual plotted line lengths are similar for different variables. Thus ratios-to-bestor ratios-to-maxima are commonly used, and logarithmic scaling may be necessary.
Some possibilitiesinclude:19: GRAPHICAL TOOLS FOR DATA ANALYSIS171•Star plots — for each case, we plot each variable as a radial distance "spoke" from a center. We jointhe ends of the centers for effect. If we do not join the ends of the spokes, the plot is sometimesreferred to as a glyph plot (Gnanadesikan, 1977, p.63)•Castles — each observation yields a small bar graph representing the values of all the variables. These"castles" are arrayed as in the star or glyph plot so we have every case displayed.•Chernoff’s faces — the values of each variable for a given case are represented by human facialfeatures such as roundness of face, form of smile, size of ears, roundness and slant of eyes, etc.
It ispossible to represent over a dozen variables at once in this way. Because it is important for us torecognize facial features of our own species, we can quickly spot similarities and differences, thoughassignment of variables to features can alter perceptions. (Tufte, 1983, p.142)It is relatively easy to develop the bar charts for castles, and some packages, for example Stata, have starcharts. Software for Chernoff’s faces is rarer; in our personal library we have only experimental code inBASIC to draw crude faces using regular text symbols as characters.We illustrate the use of star charts in Figure 19.7.3, which displays the execution times for the threeprogram codes simultaneously on all problems.
Note the similarity of the following sets of problems:problems A, B, C, J, and Qproblems D, E, I, K, L, M, O, R, S, and Tproblems G, H, N, and PNote that this does not "prove" the problems are similar. It simply allows us to pick out similarities anddifferences for further study.Outliers are rendered more detectable if we have a model for the phenomena that underlie the data weare analyzing. For the present data we could, for example, compute approximate models of executiontimes for the three optimization codes using, say, a cubic polynomial. Then the residuals — the deviationsfrom the model — could be computed and plotted for each method, possibly as a star chart.Figure 19.7.3Star plots to compare optimization test problemsNash S G/Nocedal ratio-to-best execution times.Previous Home Next172Copyright © 1984, 1994 J C & M M NashNash Information Services Inc., 1975 Bel Air Drive, Ottawa, ON K2C 0X1 CanadaSCIENTIFIC COMPUTING WITH PCsCopy for:Dr.
Dobb’s JournalChapter 20Your Own Strategy for Scientific Computingon PCs20.120.220.320.420.5Identifying needs and wantsAcquiring the right computing environmentsTools and methodsAdd-ons, extras and flexibilityPersonal choice and satisfactionThis last chapter is a restatement of our basic theme that PCs should suit the individual who uses them.As far as possible, it is the PC that must adapt to the user, not the user to the PC. We review the elementsof developing the hardware and software mix that will allow you to fulfil your scientific computing needs.In doing this, we consider your problems and the strategy that will be used to solve them.
By strategywe mean the overall approach, including the methods and type of software to attempt a solution. If thereis a choice of computing environment, we will also want to select the most appropriate one for theparticular need. We will, of course, continue to look at the suitability of PCs for the job.Tactics to overcome the detailed obstacles that arise during the evolution of a solution to a problem willonly be mentioned in passing.
Developing and improving such tactics requires experimentation of the typeused in the Cholesky timing study in Chapter 16, where the importance of details in computationalsolutions is made abundantly clear. While tactical ideas for use in improving the features or efficiency ofa solution may be suggested by many sources — books, magazines, conversations with colleagues — webelieve such ideas must be tested before being accepted as applicable to a specific problem, solutionmethod, or computing system.20.1 Identifying Needs and WantsSince becoming known as "PC experts" by virtue of our writings on personal computing, we have oftenbeen asked questions such as:What type of computer (or software) should I buy?orIs (name of brand) any good?We do not answer these questions directly because the person asking has not generally given us anyinformation why they want a computer or a piece of software.
Computers and software are expensive junkunless there are well-defined reasons for their acquisition. Usually — and this is a delight of PCs and theirsoftware — one later discovers several very interesting uses not envisaged at the time of purchase.Nevertheless, acquisitions should be justified by well-specified needs and wants.We discussed the capabilities of PCs in research applications in Chapters 2 and 3. A list of headings fromthese chapters could serve as a preliminary checklist for the types of applications for which one may wanta PC. We suggest that this list be augmented with some detail to reflect the size and complexity of20: YOUR OWN STRATEGY173problems anticipated, the special graphical or text processing features required, and any unusualinterfacing needs.
Without these details, one may purchase good equipment that nevertheless is unsuitablefor solving our problems.If you require software of a certain type to do your scientific work, then you must ensure that theequipment you purchase is can run this software along with any operating system or utility software tosupport it, such as memory managers or graphics drivers.Our first piece of closing advice is then:Identify what you want to do and how you want to do it before acquiring hardware andsoftware.We also would point out that while this book concerns the management of scientific computing when onehas to do it oneself, we do not recommend this option if there are staff to carry out all the annoying workof backing up files, managing installations of software, maintaining and cleaning equipment etc.
Ourmessage is mainly for those without the helpers.Choose the easiest means to get the job done — if there are competent helpers, let them help!Assuming we are forced to be self-sufficient, we should be aware of the computing and data-processingtasks that our job entails. Listing these and selecting those that place the greatest demands on ourcomputing resources, we can establish some guidelines for the computer we will need, be it PC orotherwise.For some tasks, the methods will be well-known to us. For others, we must work through formulationsto potential reasonable methods as in Chapters 13 to 17. The word "reasonable" is important here.
We maysolve a linear least squares problem as a general unconstrained function minimization as mentioned inSection 14.2. However, this is only useful in the rare instance that a function minimization algorithm isavailable while a method for least squares solutions is not. Clearly, if we do not know many possibleformulations, our choice of methods is correspondingly limited.
In the situation where the problem hasa large computational or memory requirement, being able to choose a method that can be used at allwithin the computing resources available may depend on our knowledge of the possible formulations ofour problem.We want to recognize quickly whether a particular PC or type of PC can be used to attempt a solution.In making this decision we need to know:•The total amount of input data required by the problem;•The minimum set of data that must be held in the main memory during solution to define theproblem;•The precision of each of these data elements, so that the total memory requirement for the problemdata can be determined;•The likely amount of working storage needed, along with a rough estimate of the size of the programneeded;•Amount of memory needed, that is, the sum of program storage, operating system and languageprocessor storage, and working storage;•The amount of output generated by a solution to the problem;•Typical solution times on other PCs, leading to a rough estimate of run time on the target PC;•Computational precision required, as opposed to storage precision.With the above information we can make an intelligent guess at the possibility of tackling the problemon our machine.
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