Titlebook: Dependence Analysis for Supercomputing; Utpal Banerjee Book 1988 Springer Science+Business Media New York 1988 Calc.Mathematica.algebra.bo

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Basic Concepts,his area; for more detail, consult any standard books on set theory and graph theory. We would recommend [Halmos 1965] for relations and [Knuth 1974] for graphs. Order relations on vectors are studied in Section 2.2; they are useful in understanding the ordering of loop iterations which plays a very

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Dependence,loops. Two statements S and T are chosen arbitrarily in the program. They define a number of nests: the common nest . consisting of loops containing both S and T, the nest . . of loops containing S but not T, the nest . . of loops containing T but not S, .(S) = (. .) consisting of all loops containi

SLAY

Bounds of Linear Functions,, one must decide if there is an integer solution to a system of linear diophantine equations satisfying a system of linear inequalities (constraints). If the system of equations (without any further constraints) has no integer solution, then there is no dependence. When integer solutions (to the sy

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CLAP

Dependence Tests,e that problem. Through some of the informal examples on dependence tests, the reader has already been given a glimpse of what to expect in this chapter. The problem is always to solve a system of linear diophantine equations subject to a set of linear constraints. The equations and constraints may

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Linear Diophantine Equations,details on the gcd and Euclid’s algorithm, see the excellent treatment in [Knuth 1980, Section 4.5.2]. The matrix methods used in sections 5.4, 5.5 are taken from [Kertzner 1981]; we have changed his notation and given detailed proofs and algorithms.

Infantry

Dependence Tests,tion, and then test certain necessary conditions for the existence of a solution to the system or to each individual equation, subject to the constraints. As we saw in Chapter 5, there are a number of undetermined integer parameters in the general solution to a system of equations. If the number of

GORGE

https://doi.org/10.1007/978-3-322-95312-4details on the gcd and Euclid’s algorithm, see the excellent treatment in [Knuth 1980, Section 4.5.2]. The matrix methods used in sections 5.4, 5.5 are taken from [Kertzner 1981]; we have changed his notation and given detailed proofs and algorithms.

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Evolutionstheoretischer Ansatz der Arbeit,tion, and then test certain necessary conditions for the existence of a solution to the system or to each individual equation, subject to the constraints. As we saw in Chapter 5, there are a number of undetermined integer parameters in the general solution to a system of equations. If the number of