The IEEE 754 Floating Point Standard, established in 1985, has been the predominant method for representing real numbers in computing systems. However, this standard encounters challenges such as precision errors and limitations in numerical range. This thesis introduces and examines the Symmetric Level Index (SLI) Arithmetic as a potential alternative, offering an extended numerical range compared to the IEEE 754 standard. Our research delves into the application of an existing hybrid framework that combines SLI and Floating-Point (FP) arithmetic. This hybrid framework is tested through various computational experiments, including the Conjugate Gradient Method(CG) for solving linear systems, the evaluation of the Softmax Function, and Automatic Differentiation (AD) computations. The analysis demonstrates SLI’s capability to handle large-scale data inputs, mitigating numerical instabilities such as overflow and underflow, commonly encountered classical IEEE 754 Floating Point standard. Notably, in the Softmax Function evaluation, the Hybrid SLI-FP System exhibits robustness against overflow and underflow issues and maintains stability across a wide range of input values. This study also explores Hybrid SLI-FP System’s potential in enhancing parallel computing tasks. It highlights systems’s ability to efficiently manage parallel algorithm bottlenecks, such as parallel reduction operations, which is crucial in parallel scalability. Furthermore, the thesis explores the effective application of the Hybrid SLI-FP System in AD and CG methods. The AD evaluation highlights system’s capability in computing accurate derivatives across a wide range of values, especially advantageous in the denormalized (subnormal) range of FP numbers. Similarly, in the CG method, the Hybrid SLI-FP System demonstrates enhanced accuracy and computational efficiency for large values, underscoring its suitability for large-scale scientific computations.
Citation Formats
S. EŞME, “APPLICATIONS OF SYMMETRIC LEVEL INDEX ARITHMETIC,” M.S. - Master of Science, Middle East Technical University, 2024.