By Utpal Banerjee
This ebook is on dependence thoughts and common equipment for dependence checking out. right here, dependence skill info dependence and the assessments are compile-time checks. We felt the time used to be ripe to create an exceptional conception of the topic, to supply the examine neighborhood with a uniform conceptual framework during which issues healthy jointly properly. How profitable we have now been in assembly those pursuits, after all, continues to be visible. we don't try and comprise all of the minute info which are identified, nor will we care for smart methods that each one strong programmers would wish to exploit. We do attempt to persuade the reader that there's a mathematical foundation along with theories of bounds of linear features and linear diophantine equations, that degrees and path vectors are ideas that come up really natu rally, that diverse dependence assessments are rather targeted circumstances of a few basic checks, and so forth. a few mathematical adulthood is required for a superb comprehend ing of the ebook: typically calculus and linear algebra. we've got cov ered diophantine equations particularly completely and given a descrip of a few matrix thought rules that aren't very widely recognized. tion A reader acquainted with linear programming might fast recog nize numerous strategies. we've got discovered greatly from the works of M. Wolfe, and okay. Kennedy and R. Allen. Wolfe's Ph. D. thesis on the college of Illinois and Kennedy & Allen's paper on vectorization of Fortran courses are nonetheless very priceless assets in this subject.
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Additional info for Dependence Analysis for Supercomputing
Im) identical to the location represented by y in iteration (j1. h, .... , je, jm+t. jm+2,"" jn), each pair of corresponding subscripts of x, y in the respective iterations must be equal. This means there must be an integer solution to the system of equations (1). Since h, jl are values of the index variable 11, they must satisfy the limits of 11. and similarly for the other variables. This explains the conditions (2). Finally, conditions (3) simply say that the iteration pair has the given direction vector s.
If i < j, the instance S(i, is) of S will be executed before the instance T(j, h) of T, irrespectively of the lexical positions of the two statements. In the case i = j, however, S(i, is) will be executed before T(j, h) iff S precedes T lexically in the program. In ord~r that the relation i < j may hold, we must have i
I m) is executed before the instance TOt. j2, .... , je, jm+1. jm+2, ... , jn) in the serial execution of the program. For the 'if part, trace the steps backwards (see the comments at the beginning of the section). • If only dependence at a given level is to be tested, the conditions are simpler: Corollary. In the notation of the above theorem, (x, y) causes a dependence of T on S at a level u, iff there is an integer solution to the system of DIM equations (1 ~r ~ DIM), (1) in N = (n + e - u + 1) variables, subject to the conditions: and Pk ~ik ~qk, Pk ~jk ~ qk, (1 ~k~m); (u ~ k ~ e and m + 1 ~ k ~ n); (2) 39 DEPENDENCE if 1 ~u ~e.
Dependence Analysis for Supercomputing by Utpal Banerjee