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A different class of generalizations arise from suitable generalizations of the notion of [[metric space]], e.g. by weakening the defining axioms for the notion of metric.<ref>{{cite book |first1=Pascal |last1=Hitzler | author-link1=Pascal Hitzler|first2=Anthony |last2=Seda |title=Mathematical Aspects of Logic Programming Semantics |publisher=Chapman and Hall/CRC |year=2010 |isbn=978-1-4398-2961-5 }}</ref> Some of these have applications, e.g., in the theory of programming semantics in theoretical computer science.<ref>{{cite journal |first1=Anthony K. |last1=Seda |first2=Pascal |last2=Hitzler | author-link2=Pascal Hitzler|title=Generalized Distance Functions in the Theory of Computation |journal=The Computer Journal |volume=53 |issue=4 |pages=443–464 |year=2010 |doi=10.1093/comjnl/bxm108 }}</ref>
==Example of numerical application - calculating high accuracy [[Pi|{{pi}}]]==
Banach theorem allows for example fast and accurate calculation of the {{pi}} number using the trigonometric
functions which numerically are the power [[Taylor_series|Taylor series]].
Because <math>\sin(\pi)=0</math> and the {{pi}} is the fixed point of for example the function
<math>f(x)=\sin(x)+x</math>
i.e.
<math>f(\pi)=\pi</math>
and also the function
<math>f</math> is around {{pi}} the contraction mapping from the obvious reasons because its derivative in {{pi}} vanishes therefore {{pi}} can be obtained from the infinite superposition for example for the argument value 3:
<math>\pi=f(f(f(f(3)...))))</math>
Already the triple superposition of this function at <math>3</math> gives {{pi}} with accuracy to 33 digits:
<math>f(f(f(3)))=3.141592653589793238462643383279502</math>.
==See also==
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