On theDimensionalEstimate ofRounding- Errorsofatypical Computing Process
Raheel Noor · 2007
and jquamarghotmail.com Abstract-The phenomenon ofroundoff-error propagation isa wellknownproblemincomputations involving floating point arithmetic. Prominent worksinthefield oferroranalysis include (1)theerror-analysis basedondifferential error-propagation modelforcomputer algebra system (CAS), (2)theidentification andreformulation ofinstability inacodegenerated byCAS,(3) estimating theboundson errorsinsymbolic andnumerical environments. Themainconcern inthese attempts istocontrol error-propagation byusing numerically stable code. Beside these attempts, onlyfewefforts aremadetowardsthetheoretical understanding oftheunderlying process oferrorpropagation. In thispaper, we attempt toshowthattheroundoff-errors may propagate asa random-fractal process. We applyconcepts of nonlinear time-series analysis onaseries constituting successive roundoff-errors generated during thecomputation ofHenon-map solutions. We estimate thecorrelation dimension, whichisa measureofthefractal dimension, oftheseries as5.5± 0.05. This lowvalue ofcorrelation dimension showsthattheerror series can bemodeled byalowdimensional dynamical system. IndexTerms-roundoff erroranalysis, nonlinear time-series analysis, phase-space reconstruction, delaycoordinate embedding, correlation dimension