in an analogous manner. We used our adaptive integration techniques to estimate â« i ( x)dx . Algorithms Associative Arrays Aloysius G. OâRithm An associative array is a data structure, together with some access methods, that allows us as programmers to place and access data items by name rather than explicitly by position. Such arrays are implemented as a primitive type in some other programming languages (in K and in PostScript, for instance, where they are called dictionaries, and in Lisp, where they are called association lists); in some languages (Java, C, C++) they are available through libraries (and are frequently called hash tables, after a particular method of implementation). Clearly, nonAPL programmers find them useful. Both APL2 and J obtained answers for a tolerance of 1e_6 but failed when a tolerance of 1e_7 was requested, APL2 with a workspace full error and J with a limit error. Using 1e_6, the answers agreed and both used 873 subintervals. APL2 took 0.31 seconds while J took 0.29 seconds. Interestingly, Mathematica, gave an error message that the desired tolerance had not been met and then gave an answer only correct to 3 decimal places. While the differences in timing may
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