One-dimensional I test and direction vector I test with array references by induction variable

In this paper, theoretical aspects to demonstrate the accuracy of the Interval Test (the I test and the direction vector I test) to be applied for resolving the problem stated above is presented. Also, it is proved from the proposed theoretical aspects that under a specific direction vector 1 ( , , ) d θ = = = ... there are integer-valued solutions for one-dimensional arrays with subscripts formed by induction variable and under other specific direction vectors there are no integer-valued solutions. Experiments with benchmarks, cited from Parallel loop, Vector loop and TRFD (Perfect benchmark), reveal that our framework can properly enhance the precision of data dependence analysis for one-dimensional arrays with subscripts mentioned above.