However, the 49 regions of the contiguous U. Lab Contiguous United States Figure 3. Lab Population Data Figure 3. Lab Dynamic Updates Latitude is measured north-south of the equator but longitude is measured east-west of the arbitrary prime meridian through Royal Observatory, Greenwich in London, England. We are treating longitude as x and latitude as y when in reality the Earth is almost spherical so true surface locations are more complicated to calculate.
Line 48 of Code Listing 3. This is important because as the breakpoints change we may need to re-color: cnvs. Thus, by picking a new random angle after each movement our result will approximate what would have happened if we had modeled the actual collisions. Figures 3. Lab Standard Deviation An increasing average distance tells us the particles are moving away from the center but it does not tell us they are moving away from each other; how do we know they are not all making the exact same movements together, or that a tight circle is not maintained with an expanding radius?
Answer: calculate the standard deviation. Lab Model Parameters The vertical line in Figure 3. What happens to the vertical and horizontal lines over time?
To avoid this truncation issue we might also initialize the particles along a circle of radius instead of clumped together. As the bin width decreases i. You might use arrow keys to interactively change the bin settings Fix one particle in place at the center. Thus, this problem is said to be coupled. Two parallel options are possible.
Alternatively, this application lends itself to spatial decomposition. Of course if a particle leaves its region then a node will have to communicate this as well.
Note the very slow convergence. Lab Plot Values Since the sign alternates and the denominator grows we can bound our error without even knowing! Table 3. Decimal Hexadecimal 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 Binary Decimal Hexadecimal 8 8 9 9 10 A B 11 12 C D 13 14 E F 15 Binary Floating-Point Representation Computers do arithmetic in binary base 2 because it is easier to build the machines that way. However, there are special cases when the exponent bits are all the same.
Thus, the smallest not be stored. Fine-Grain Parallelism Figure 3. An alternative is to pair the terms up so that each pair-sum can be calculated simultaneously. Since log2 25, , 90 3 Visualization Table 3.
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Regardless, to solve problems of practical interest it will certainly be necessary that our code runs fast for large cases, and as habits govern so much of our behavior we will do best by writing code for small cases with potential large cases already in mind. Small problems appear only in the lower-left corner. With every order of magnitude increase in problem size the relative difference in runtime grows by two orders of magnitude. As one example, the Declaration of Independence contains just under words while Federalist 78 has just over To determine how many times each unique word appears in these two texts we might use either a linear or quadratic algorithm.
R chip the linear code ran in 0. Note that as the problem size doubled the runtime of the linear code also doubled but the quadratic code quadrupled. A fair objection: 4 seconds is not a long time to wait for a code to run regardless of the algorithm. Since understanding requires accurate results and otherwise what is the point?
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Lab Population Cloud We want to build a word cloud that quickly relays information about the main topics of a work in a visual manner. Code Listing 4. This will separate the data at newlines only and not at any other whitespace characters. Your assignment is to use the state names and populations to display full names, at random, where font size is determined by relative population size. Algorithm 4.
Lab Uncommon Words Files of common words can be found on the Internet, or you can build your own. Then, a word cloud like the one shown in Figure 4. Navy, photo credit Dr. Brian Mason. Databases Accumulation of data continues to grow at a faster-and-faster rate, so much so that researchers in astronomy are now facing issues related to the mere transfer of data from observational projects located in Chile, see Figure 4.
However, these problems will be solved because the promise of new learning from massive amounts of data is so attractive. For instance, language translation software that uses computational statistics has become a real alternative to formal linguistics, not in the sense that theory could be abandoned but, as we have seen regarding experiment and computer simulation, theory can be augmented by data.
Lab Comparison Comparing the word clouds in Figures 4. Implement a feature that cycles through all 85 papers showing one cloud on the screen at a time and somehow indicating background color, footnote who wrote each article. Note that frequencies of multiple similar words could be combined, too. In addition, it is okay if the words in your cloud overlap.
Image courtesy of the G. A clay tablet over years old depicts their version of Algorithm 4. Even if our initial guess is absurd the value of x stops changing after only seven steps. A vertical line shows how an input produces an output while a horizontal line shows how we then use that ouptut as the next input in order to build a sequence of better-and-better approximations. Figure 4. Our previous approximation of 2 converged to 1.
Pretty close. Say we were manufacturing a car and needed this calculation; could an industrial saw cut metal with enough precision for the difference to matter? Afterwards, could we even measure the cut metal this precisely? And how does the sqrt function calculate its result anyway? Evidence has suggested that this ability is possible only because an exceedingly large number of very small bristles use intermolecular attraction in a massively parallel way.
Image courtesy of Tim Vickers. But now how can we solve a cubic equation for V? Note how an input of 0. Lab Chaotic Figure 4. Lab Logistic Map Figure 4. This is known as the logistic map and we might think of our vertical scale as the population of deer, for instance, changing in response to r, which could represent an environmental condition of some kind. These periodic regions might represent cyclic changes in our deer population, from either seasonal or migratory effects perhaps, but they are stable nevertheless. The bifurcation points themselves would appear even sharper if more pre-drawing loops were used.
At right we see It is easier to change xp at each step and interpolate from there to r than the other way around. PhotoImage img a single Tk image object is drawn Otherwise we wait until steps to stop the loop.