An error assessment of matrix multiplications on posit matrices

Abstract: The posit is a floating-point number format as proposed by John Gustafson to improve upon the accuracy of the IEEE-754 format, which is the standard today. The goal of this paper was to look specifically at matrix multiplication and examine how the posit format compared to the IEEE-754. The quire which is part of the posit standard was not included in this paper due to limitations. We used the library softPosit in Python to construct arrays of matrices referred to as matrix chains. These matrices were filled with numbers in one format and bit size at a time. These chains were then multiplied together with normal matrix multiplication, and then we compared the error of the two formats. An IEEE- 754 matrix chain with more bits than the ones being compared was used as the reference to compare the accuracy between the IEEE-754 and the posit, as it pertains to matrix multiplication. The result was that the posit format could yield more accurate matrix multiplications, especially for matrix multiplications with few matrices of low dimension. When dimensions and number of matrices increased, however, the posit matrix produced an error that was greater than that of the IEEE-754 matrix. The conclusion was that posits, if used sensibly, can be a more accurate format for matrix multiplication but it is important to consider the properties inherent to the posit when dealing with matrix multiplication of matrices inhabited by posits.