Speaker: Shizhong Zhao

Title: Sources of errors in floating point arithmetic

Abstract: For any given arithmetic expressions, there are often
miscalculations in the floating-point arithmetic due to representation
errors. This calculation mode of a fixed number of digits has the
following drawbacks: for a given arithmetic expression, (1) sometimes,
there are contaminated digits in the results, regardless of how many
significant digits there are, which means that no correct answer could
ever be obtained. (2) The calculation results may not be consistent
when the number of significant digits varies. (3) For the same number
of significant digits, the calculation results from two equivalent
expressions may be different. (4) In particular, rules such as the
distributive and associative laws no longer hold. (5) Reductions in
the number of operations may not necessarily result in a more accurate
value. (6) Similarly, an improvement in the accuracy of the
intermediate operation processes cannot ensure a more accurate
calculation result. (7) "Catastrophic cancellation" may occur. With
regards to the first drawback, the concept of "Error Number" and its
calculation method, which can be used to calculate the number of
contaminated digits, are proposed in this study.