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Tables
One of the things loops are good for is generating and printing tabular data. For example, before computers were readily available, people had to calculate logarithms, sines and cosines, and other common mathematical functions by hand. To make that easier, there were books containing long tables where you could find the values of various functions. Creating these tables was slow and boring, and the result tended to be full of errors. When computers appeared on the scene, one of the initial reactions was, "This is great! We can use the computers to generate the tables, so there will be no errors." That turned out to be true (mostly), but shortsighted. Soon thereafter computers (and calculators) were so pervasive that the tables became obsolete. Well, almost. It turns out that for some operations, computers use tables of values to get an approximate answer, and then perform computations to improve the approximation. In some cases, there have been errors in the underlying tables, most famously in the table the original Intel Pentium used to perform floatingpoint division. Although a "log table" is not as useful as it once was, it still makes a good example of iteration. The following program prints a sequence of values in the left column and their logarithms in the right column: double x = 1.0;while (x < 10.0) { System.out.println (x + " " + Math.log(x)); x = x + 1.0; } The output of this program is 1.0 0.02.0 0.6931471805599453 3.0 1.0986122886681098 4.0 1.3862943611198906 5.0 1.6094379124341003 6.0 1.791759469228055 7.0 1.9459101490553132 8.0 2.0794415416798357 9.0 2.1972245773362196 Looking at these values, can you tell what base the log function uses by default? Since powers of two are so important in computer science, we often want to find logarithms with respect to base 2. To find that, we have to use the following formula:
Changing the print statement to System.out.println (x + " " + Math.log(x) / Math.log(2.0));yields 1.0 0.02.0 1.0 3.0 1.5849625007211563 4.0 2.0 5.0 2.321928094887362 6.0 2.584962500721156 7.0 2.807354922057604 8.0 3.0 9.0 3.1699250014423126 We can see that 1, 2, 4 and 8 are powers of two, because their logarithms base 2 are round numbers. If we wanted to find the logarithms of other powers of two, we could modify the program like this: double x = 1.0;while (x < 100.0) { System.out.println (x + " " + Math.log(x) / Math.log(2.0)); x = x * 2.0; } Now instead of adding something to x each time through the loop, which yields an arithmetic sequence, we multiply x by something, yielding a geometric sequence. The result is: 1.0 0.02.0 1.0 4.0 2.0 8.0 3.0 16.0 4.0 32.0 5.0 64.0 6.0 Log tables may not be useful any more, but for computer scientists, knowing the powers of two is! Some time when you have an idle moment, you should memorize the powers of two up to 65536 (that's 2^{16}).


Home Iteration Tables 