Glossary
- Numerical analysis::
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A general term for this subject. Roughly speaking, it replaces manipulating real number by manipulating finite decimals.
- Rounding::
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In this chapter, the concept of regular rounding is replaced by the concept of measuring functions.
- Measuring with a meterstick::
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It is a geometric way of conducting rounding real numbers.
- Vector::
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Vector, not sequence, is used to name a linearly ordered set.
Definition of the Subject
Abstract: Numerical analysis, an ancient subject, lacks mathematical theory behind its practices. In this chapter, two basic concepts of approximation are formalized. The first is the approximation defined by a mechanical procedure called measuring with a meterstick, such as when we use a meterstick to measure the diameter of a test tube. The mathematics behind this concept of approximation is called granular topology, which contains the usual topology as a sub-topology. The second concept of approximation is hidden in the approximate arithmetic....
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References
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Lin TY (2015) A paradox in rounding errors approximate computing for big data. SMC 2015:2567–2573
Lin TY (2019) Approximate computing in numerical analysis variable interval computing – extended abstract. SMC 2019:2013–2018
Lin TY, Liu Q (1994) Rough approximate operators-axiomatic. In: Ziarko W (ed) Rough set theory, rough sets, fuzzy sets and knowledge discovery. Springer-Verlag, pp 256–260. (Co-author: Qing Liu)
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Measurement and Significant Figures https://chem.libretexts.org/Bookshelves/Introductory_Chemistry/Map%3A_Fundamentals_ofGeneral_Organic_and_Biological_Chemistry_(McMurry_et_al.)/1%3A_Matter_and_Measurements/1.09%3A_Measurement_and_Significant_Figures
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Some actual quotes from the website:
1) “In a measurement, all of the certain digits (those read directly from the scale) and one uncertain digit (one number made by estimating between marks on the scale) are significant”
2) “Remember that calculators do not understand significant figures. You are the one who must apply the rules of significant figures to a result from your calculator”
3) “Assigning significant figures and using the rules to round the result of computations gives us a systematic way of indicating the degree of trustworthiness of the result”
4) “The foundation of significant figures is preserving the degree of uncertainty of the instruments”
Stanat D, McAllister D (1977) Discrete mathematics in computer science. Prentice-Hall
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Lin, T.Y. (2023). A Theory of Approximate Arithmetic for Numerical Analysis: A Mathematical Foundation. In: Meyers, R.A. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_773-2
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DOI: https://doi.org/10.1007/978-3-642-27737-5_773-2
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Latest
A Theory of Approximate Arithmetic for Numerical Analysis: A Mathematical Foundation- Published:
- 16 February 2023
DOI: https://doi.org/10.1007/978-3-642-27737-5_773-2
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Original
A Theory of Approximate Arithmetic for Numerical Analysis: A Mathematical Foundation- Published:
- 04 February 2023
DOI: https://doi.org/10.1007/978-3-642-27737-5_773-1