# On the Arithmetic of Errors

Markov, Svetoslav; Hayes, Nathan

Serdica Journal of Computing (2010)

- Volume: 4, Issue: 4, page 447-462
- ISSN: 1312-6555

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topMarkov, Svetoslav, and Hayes, Nathan. "On the Arithmetic of Errors." Serdica Journal of Computing 4.4 (2010): 447-462. <http://eudml.org/doc/11399>.

@article{Markov2010,

abstract = {An approximate number is an ordered pair consisting of a (real)
number and an error bound, briefly error, which is a (real) non-negative
number. To compute with approximate numbers the arithmetic operations
on errors should be well-known. To model computations with errors one
should suitably define and study arithmetic operations and order relations
over the set of non-negative numbers. In this work we discuss the algebraic
properties of non-negative numbers starting from familiar properties of real
numbers. We focus on certain operations of errors which seem not to have
been sufficiently studied algebraically. In this work we restrict ourselves to
arithmetic operations for errors related to addition and multiplication by
scalars. We pay special attention to subtractability-like properties of errors
and the induced “distance-like” operation. This operation is implicitly used
under different names in several contemporary fields of applied mathematics
(inner subtraction and inner addition in interval analysis, generalized
Hukuhara difference in fuzzy set theory, etc.) Here we present some new
results related to algebraic properties of this operation.* The first author was partially supported by the Bulgarian NSF Project DO 02-359/2008
and NATO project ICS.EAP.CLG 983334.},

author = {Markov, Svetoslav, Hayes, Nathan},

journal = {Serdica Journal of Computing},

keywords = {Computer Arithmetic; Error Analysis; Interval Arithmetic; Approximate Numbers; Algebra of Errors; Quasilinear Spaces; inner subtraction; inner addition; interval analysis; generalized Hukuhara difference; fuzzy set theory; computer arithmetic; error analysis; interval arithmetic; approximate numbers; algebra of errors; quasilinear spaces},

language = {eng},

number = {4},

pages = {447-462},

publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},

title = {On the Arithmetic of Errors},

url = {http://eudml.org/doc/11399},

volume = {4},

year = {2010},

}

TY - JOUR

AU - Markov, Svetoslav

AU - Hayes, Nathan

TI - On the Arithmetic of Errors

JO - Serdica Journal of Computing

PY - 2010

PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences

VL - 4

IS - 4

SP - 447

EP - 462

AB - An approximate number is an ordered pair consisting of a (real)
number and an error bound, briefly error, which is a (real) non-negative
number. To compute with approximate numbers the arithmetic operations
on errors should be well-known. To model computations with errors one
should suitably define and study arithmetic operations and order relations
over the set of non-negative numbers. In this work we discuss the algebraic
properties of non-negative numbers starting from familiar properties of real
numbers. We focus on certain operations of errors which seem not to have
been sufficiently studied algebraically. In this work we restrict ourselves to
arithmetic operations for errors related to addition and multiplication by
scalars. We pay special attention to subtractability-like properties of errors
and the induced “distance-like” operation. This operation is implicitly used
under different names in several contemporary fields of applied mathematics
(inner subtraction and inner addition in interval analysis, generalized
Hukuhara difference in fuzzy set theory, etc.) Here we present some new
results related to algebraic properties of this operation.* The first author was partially supported by the Bulgarian NSF Project DO 02-359/2008
and NATO project ICS.EAP.CLG 983334.

LA - eng

KW - Computer Arithmetic; Error Analysis; Interval Arithmetic; Approximate Numbers; Algebra of Errors; Quasilinear Spaces; inner subtraction; inner addition; interval analysis; generalized Hukuhara difference; fuzzy set theory; computer arithmetic; error analysis; interval arithmetic; approximate numbers; algebra of errors; quasilinear spaces

UR - http://eudml.org/doc/11399

ER -

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