Auslöschung (numerische Mathematik)

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role="main"> <a id="top"></a> <div id="siteNotice"></div> <div class="mw-indicators"> </div> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Auslöschung (numerische Mathematik)</span></h1> <div id="bodyContent" class="vector-body"> <div id="siteSub" class="noprint">aus Wikipedia, der freien Enzyklopädie</div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="contentSub2"></div> <div id="jump-to-nav"></div> <a class="mw-jump-link" href="#mw-head">Zur Navigation springen</a> <a class="mw-jump-link" href="#searchInput">Zur Suche springen</a> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Unter <b>Auslöschung</b> (<a href="/wiki/Englische_Sprache" title="Englische Sprache">engl.</a> <i><span lang="en">cancellation</span></i>) versteht man in der <a href="/wiki/Numerik" class="mw-redirect" title="Numerik">Numerik</a> den Verlust an Genauigkeit bei der <a href="/wiki/Subtraktion" title="Subtraktion">Subtraktion</a> fast gleich großer <a href="/wiki/Gleitkommazahl" title="Gleitkommazahl">Gleitkommazahlen</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Beispiele"><span class="tocnumber">1</span> <span class="toctext">Beispiele</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Zahlenbeispiel"><span class="tocnumber">1.1</span> <span class="toctext">Zahlenbeispiel</span></a></li> <li class="toclevel-2 tocsection-3"><a href="#Beispiel:_Algorithmus_des_Archimedes_zur_Kreiszahlberechnung"><span class="tocnumber">1.2</span> <span class="toctext">Beispiel: Algorithmus des Archimedes zur Kreiszahlberechnung</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-4"><a href="#Faustregel"><span class="tocnumber">2</span> <span class="toctext">Faustregel</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#Differentialrechnung"><span class="tocnumber">3</span> <span class="toctext">Differentialrechnung</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#Einzelnachweise"><span class="tocnumber">4</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ausl%C3%B6schung_(numerische_Mathematik)&veaction=edit&section=1" title="Abschnitt bearbeiten: Beispiele" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ausl%C3%B6schung_(numerische_Mathematik)&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Beispiele"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Zahlenbeispiel">Zahlenbeispiel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ausl%C3%B6schung_(numerische_Mathematik)&veaction=edit&section=2" title="Abschnitt bearbeiten: Zahlenbeispiel" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ausl%C3%B6schung_(numerische_Mathematik)&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Zahlenbeispiel"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Wir subtrahieren die Zahlen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=2{,}345678}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>2,345</mn> <mn>678</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=2{,}345678}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61a90ea0cb9108a3b7b7927c87eb313e519d6f41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.112ex; height:2.509ex;" alt="{\displaystyle a=2{,}345678}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=2{,}346789}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>2,346</mn> <mn>789</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=2{,}346789}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db4d716c4ea58b017c292d7ced655b1c4dc02845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.88ex; height:2.509ex;" alt="{\displaystyle b=2{,}346789}"></span> voneinander und erhalten als Ergebnis </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b-a=0{,}001111}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo>=</mo> <mn>0,001</mn> <mn>111</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b-a=0{,}001111}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b68048accf29b5853ad4413bbfcf1df3c9af5182" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.95ex; height:2.509ex;" alt="{\displaystyle b-a=0{,}001111}"></span>.</dd></dl> <p>Stammen nun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> bereits aus vorherigen Berechnungen, so werden die niedrigwertigen Stellen durch <a href="/wiki/Rundungsfehler" title="Rundungsfehler">Rundungsfehler</a> beeinflusst sein. Stimmen nun aber die höherwertigen Stellen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> überein, so löschen sich die gültigen Stellen zu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> aus, und die Differenz ergibt sich ausschließlich aus Rundungsfehlern. </p><p>Angenommen, bei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> seien die ersten drei Ziffern korrekt, und alle niedrigwertigeren Ziffern durch Rundungsfehler verfälscht. Verkürzen wir die Zahlen auf ihre korrekten Ziffern, so ergibt sich </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{,}34-2{,}34=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>34</mn> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>34</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{,}34-2{,}34=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acf48133ff143a27cfe2cf970cc3e154e03cf990" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.37ex; height:2.509ex;" alt="{\displaystyle 2{,}34-2{,}34=0}"></span>,</dd></dl> <p>während sich im Ergebnis der ersten, vermeintlich genauen Berechnung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b-a=0{,}001111}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo>=</mo> <mn>0,001</mn> <mn>111</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b-a=0{,}001111}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b68048accf29b5853ad4413bbfcf1df3c9af5182" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.95ex; height:2.509ex;" alt="{\displaystyle b-a=0{,}001111}"></span> keine einzige korrekte Ziffer mehr findet. </p><p>Angenommen, in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> seien die ersten vier Ziffern noch korrekt, so ergibt sich </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{,}346-2{,}345=0{,}001}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2,346</mn> <mo>−</mo> <mn>2,345</mn> <mo>=</mo> <mn>0,001</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{,}346-2{,}345=0{,}001}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/483af6ee8615c5d4fbf93a2b4a34338b3b47b753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.829ex; height:2.509ex;" alt="{\displaystyle 2{,}346-2{,}345=0{,}001}"></span>,</dd></dl> <p>wohingegen wir uns oben mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b-a=0{,}001111}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo>=</mo> <mn>0,001</mn> <mn>111</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b-a=0{,}001111}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b68048accf29b5853ad4413bbfcf1df3c9af5182" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.95ex; height:2.509ex;" alt="{\displaystyle b-a=0{,}001111}"></span> einen absoluten Fehler von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0{,}001111-0{,}001000=0{,}000111}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0,001</mn> <mn>111</mn> <mo>−</mo> <mn>0,001</mn> <mn>000</mn> <mo>=</mo> <mn>0,000</mn> <mn>111</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0{,}001111-0{,}001000=0{,}000111}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/178656f72bb5c199bdf572274f6aa706e6e6d83e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.291ex; height:2.509ex;" alt="{\displaystyle 0{,}001111-0{,}001000=0{,}000111}"></span> und damit einen relativen Fehler von ungefähr 10 % eingehandelt haben. </p> <div class="mw-heading mw-heading3"><h3 id="Beispiel:_Algorithmus_des_Archimedes_zur_Kreiszahlberechnung">Beispiel: Algorithmus des Archimedes zur Kreiszahlberechnung</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ausl%C3%B6schung_(numerische_Mathematik)&veaction=edit&section=3" title="Abschnitt bearbeiten: Beispiel: Algorithmus des Archimedes zur Kreiszahlberechnung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ausl%C3%B6schung_(numerische_Mathematik)&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Beispiel: Algorithmus des Archimedes zur Kreiszahlberechnung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:PiArchimedes.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/PiArchimedes.png/220px-PiArchimedes.png" decoding="async" width="220" height="311" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/PiArchimedes.png/330px-PiArchimedes.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/PiArchimedes.png/440px-PiArchimedes.png 2x" data-file-width="595" data-file-height="841" /></a><figcaption>Berechnung von pi nach Archimedes</figcaption></figure> <p><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> von Syrakus bewies, dass sich der Umfang eines Kreises zu seinem Durchmesser genauso verhält, wie die Fläche des Kreises zum Quadrat des Radius. Er nannte dieses (heute als <a href="/wiki/Kreiszahl" title="Kreiszahl">Kreiszahl</a> bezeichnete) Verhältnis noch nicht π, gab aber eine Anleitung, wie man sich mit Hilfe von ein- und umschriebenen Vielecken dem Verhältnis bis zu einer beliebig hohen Genauigkeit nähern kann, vermutlich eines der ältesten numerischen Verfahren der Geschichte. Und er führte die Berechnung bis zum 96-Eck mit dem folgenden Resultat durch: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3{,}1408450\dots =3+{\frac {10}{71}}<\pi <3+{\frac {10}{70}}=3{,}1428571\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3,140</mn> <mn>8450</mn> <mo>⋯</mo> <mo>=</mo> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>71</mn> </mfrac> </mrow> <mo><</mo> <mi>π</mi> <mo><</mo> <mn>3</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>70</mn> </mfrac> </mrow> <mo>=</mo> <mn>3,142</mn> <mn>8571</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3{,}1408450\dots =3+{\frac {10}{71}}<\pi <3+{\frac {10}{70}}=3{,}1428571\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de3d2dbb6e2b367079ecc1fc344156bfc60e515" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:54.167ex; height:5.343ex;" alt="{\displaystyle 3{,}1408450\dots =3+{\frac {10}{71}}<\pi <3+{\frac {10}{70}}=3{,}1428571\dots }"></span></dd></dl> <p>Wie man dem Zahlenbeispiel entnehmen kann, hatte Archimedes keine Chance, beim 96-Eck die Auslöschung überhaupt nur wahrzunehmen. </p><p>In heutiger Sprache beginnt man mit direkt berechenbaren Seitenlängen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}=AB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}=AB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ae05d761f80210e10c23ff1adec53906972339b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.914ex; height:2.509ex;" alt="{\displaystyle s_{n}=AB}"></span> von in einem Einheitskreis (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle MA=MB=MC=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>A</mi> <mo>=</mo> <mi>M</mi> <mi>B</mi> <mo>=</mo> <mi>M</mi> <mi>C</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle MA=MB=MC=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/976aee9297e602b823a7fdf7ef463ddc3a8ad232" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:23.058ex; height:2.176ex;" alt="{\displaystyle MA=MB=MC=1}"></span>) einbeschriebenen Vielecken, z. B. dem Zweieck <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{2}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{2}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0815b8ab997ab24c088428f016f1248cb534af5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.406ex; height:2.509ex;" alt="{\displaystyle s_{2}=2}"></span>, dem Dreieck <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{3}={\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{3}={\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87b8ac105aae752a9da0af5e02edd89f5714921f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.341ex; height:2.843ex;" alt="{\displaystyle s_{3}={\sqrt {3}}}"></span>, dem Viereck <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{4}={\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{4}={\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42da958156c570dea9d7972dc27cdd89271a1956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.341ex; height:3.009ex;" alt="{\displaystyle s_{4}={\sqrt {2}}}"></span> oder dem Sechseck <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{6}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{6}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4b802b61fd3c038dd5fb9d464b0b73fc4d4df97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.406ex; height:2.509ex;" alt="{\displaystyle s_{6}=1}"></span>. </p><p>Dann ist für Vielecke mit doppelter Eckenzahl deren Seitenlänge <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{2n}=AC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>A</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{2n}=AC}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fb1cf49bbdb87585d89607303e6af5de9670ff6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.739ex; height:2.509ex;" alt="{\displaystyle s_{2n}=AC}"></span> mit der Hilfsstrecke <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{n}=MS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>M</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{n}=MS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/858a23a8d0a3a970a3717c04d198ce1c7c869998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.46ex; height:2.676ex;" alt="{\displaystyle \rho _{n}=MS}"></span> und zweimaliger Anwendung des Satzes von Pythagoras (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AM^{2}=MS^{2}+AS^{2},1=\rho _{n}^{2}+s_{n}^{2}/4,\rho _{n}={\sqrt {1-s_{n}^{2}/4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>M</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>A</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mn>1</mn> <mo>=</mo> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>,</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AM^{2}=MS^{2}+AS^{2},1=\rho _{n}^{2}+s_{n}^{2}/4,\rho _{n}={\sqrt {1-s_{n}^{2}/4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d367ee54c2f0271350ae2f1f7703bb9826b3d521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:53.275ex; height:4.843ex;" alt="{\displaystyle AM^{2}=MS^{2}+AS^{2},1=\rho _{n}^{2}+s_{n}^{2}/4,\rho _{n}={\sqrt {1-s_{n}^{2}/4}}}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AC^{2}=AS^{2}+SC^{2},s_{2n}^{2}=s_{n}^{2}/4+(1-\rho _{n})^{2}=s_{n}^{2}/4+1-2\rho _{n}+\rho _{n}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>A</mi> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>S</mi> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AC^{2}=AS^{2}+SC^{2},s_{2n}^{2}=s_{n}^{2}/4+(1-\rho _{n})^{2}=s_{n}^{2}/4+1-2\rho _{n}+\rho _{n}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f8a53327132b85b79794f622aee2edebf71a13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:66.648ex; height:3.343ex;" alt="{\displaystyle AC^{2}=AS^{2}+SC^{2},s_{2n}^{2}=s_{n}^{2}/4+(1-\rho _{n})^{2}=s_{n}^{2}/4+1-2\rho _{n}+\rho _{n}^{2}}"></span>) leicht herleitbar: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{2n}={\sqrt {2-2{\sqrt {1-{\tfrac {s_{n}^{2}}{4}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>4</mn> </mfrac> </mstyle> </mrow> </msqrt> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{2n}={\sqrt {2-2{\sqrt {1-{\tfrac {s_{n}^{2}}{4}}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9977cf8dbd61e63263feaa0b09c43e21223c8b16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.617ex; height:7.676ex;" alt="{\displaystyle s_{2n}={\sqrt {2-2{\sqrt {1-{\tfrac {s_{n}^{2}}{4}}}}}}}"></span></dd></dl> <p>Mit den vier <a href="/wiki/Grundrechenart" title="Grundrechenart">Grundrechenarten</a> und dem <a href="/wiki/Wurzelziehen" class="mw-redirect" title="Wurzelziehen">Wurzelziehen</a> kann man also beginnend mit dem Zweieck die Seitenlänge und den Umfang eines einbeschriebenen Vielecks und damit indirekt eine Näherung für <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> berechnen. In der Praxis ist das Ergebnis jedoch enttäuschend. Die folgende Tabelle zeigt beginnend mit n=2 den Abstand <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\rho _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\rho _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5ff06d88943faa95727eb5496b8059838c748e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.423ex; height:2.676ex;" alt="{\displaystyle 1-\rho _{n}}"></span> der Seitenmitte S zum Kreisrand, die Seitenlängen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d671890050b21484dde3087d000700c97fc3b03c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.309ex; height:2.009ex;" alt="{\displaystyle s_{n}}"></span> des eingeschriebenen und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}=A'B'=s_{n}\cdot 1/\rho _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <msup> <mi>B</mi> <mo>′</mo> </msup> <mo>=</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⋅</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}=A'B'=s_{n}\cdot 1/\rho _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/711505cd9651a8d64fd6d76bfc911942af6a6ec3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.45ex; height:3.009ex;" alt="{\displaystyle S_{n}=A'B'=s_{n}\cdot 1/\rho _{n}}"></span> des umschriebenen n-Ecks und deren Flächen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}=ns_{n}\rho _{n}/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}=ns_{n}\rho _{n}/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/186676aa63b3d9d2b83c0e22ce2da276fd35c472" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.996ex; height:2.843ex;" alt="{\displaystyle a_{n}=ns_{n}\rho _{n}/2}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}=nS_{n}/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}=nS_{n}/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b0d889dbd917b424a20d5711c7ae64dcc90d541" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.423ex; height:2.843ex;" alt="{\displaystyle A_{n}=nS_{n}/2}"></span>, die beim Einheitskreis gegen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> konvergieren sollten. Die Rechnung wurde in <a href="/wiki/C_(Programmiersprache)" title="C (Programmiersprache)">C</a> mit <a href="/wiki/Doppelte_Genauigkeit" title="Doppelte Genauigkeit">doppelter Genauigkeit</a> nach <a href="/wiki/IEEE_754" title="IEEE 754">IEEE 754</a> und somit ca. 15 Dezimalstellen durchgeführt. Die Zahlenwerte sind aber auch mit jedem <a href="/wiki/Taschenrechner" title="Taschenrechner">Taschenrechner</a>, der Quadratwurzeln beherrscht, nachvollziehbar: </p> <pre><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></b> <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\rho _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\rho _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5ff06d88943faa95727eb5496b8059838c748e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.423ex; height:2.676ex;" alt="{\displaystyle 1-\rho _{n}}"></span></b> <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d671890050b21484dde3087d000700c97fc3b03c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.309ex; height:2.009ex;" alt="{\displaystyle s_{n}}"></span></b> <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.643ex; height:2.509ex;" alt="{\displaystyle S_{n}}"></span></b> <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span></b> <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730f6906700685b6d52f3958b1c2ae659d2d97d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}"></span></b> 2 1.000e+00 2.00e+00 Inf 0.00000000000000 Inf 4 2.929e-01 1.41e+00 2.00e+00 2.00000000000000 4.00000000000000 8 7.612e-02 7.65e-01 8.28e-01 2.82842712474619 3.31370849898476 16 1.921e-02 3.90e-01 3.98e-01 3.06146745892072 3.18259787807453 32 4.815e-03 1.96e-01 1.97e-01 3.12144515225805 3.15172490742926 64 1.205e-03 9.81e-02 9.83e-02 3.13654849054593 3.14411838524589 128 3.012e-04 4.91e-02 4.91e-02 3.14033115695474 3.14222362994244 256 7.530e-05 2.45e-02 2.45e-02 3.14127725093262 3.14175036916881 512 1.882e-05 1.23e-02 1.23e-02 3.14151380114509 3.14163208070397 1024 4.706e-06 6.14e-03 6.14e-03 3.14157294036989 3.14160251025961 2048 1.177e-06 3.07e-03 3.07e-03 3.14158772527060 3.14159511774302 4096 2.941e-07 1.53e-03 1.53e-03 3.14159142155216 3.14159326967027 8192 7.353e-08 7.67e-04 7.67e-04 3.14159234553025 3.14159280755978 1.638e+04 1.838e-08 3.83e-04 3.83e-04 3.14159257570956 3.14159269121694 3.277e+04 4.596e-09 1.92e-04 1.92e-04 3.14159264036917 3.14159266924601 6.554e+04 1.149e-09 9.59e-05 9.59e-05 3.14159264171161 3.14159264893082 1.311e+05 2.872e-10 4.79e-05 4.79e-05 3.14159260647332 3.14159260827812 2.621e+05 7.181e-11 2.40e-05 2.40e-05 3.14159291071407 3.14159291116527 5.243e+05 1.795e-11 1.20e-05 1.20e-05 3.14159169662728 3.14159169674009 1.049e+06 4.488e-12 5.99e-06 5.99e-06 3.14159655369072 3.14159655371892 2.097e+06 1.122e-12 3.00e-06 3.00e-06 3.14159655370129 3.14159655370834 4.194e+06 2.804e-13 1.50e-06 1.50e-06 3.14151884046467 3.14151884046643 8.389e+06 7.017e-14 7.49e-07 7.49e-07 3.14120796828205 3.14120796828249 1.678e+07 1.754e-14 3.75e-07 3.75e-07 3.14245127249408 3.14245127249419 3.355e+07 4.441e-15 1.87e-07 1.87e-07 3.14245127249412 3.14245127249415 6.711e+07 1.110e-15 9.42e-08 9.42e-08 3.16227766016838 3.16227766016838 1.342e+08 2.220e-16 4.71e-08 4.71e-08 3.16227766016838 3.16227766016838 2.684e+08 0.000e+00 2.11e-08 2.11e-08 2.82842712474619 2.82842712474619 5.369e+08 0.000e+00 0.00e+00 0.00e+00 0.00000000000000 0.00000000000000 </pre> <p>Man erkennt deutlich am Anfang die Konvergenz gegen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>. Nach Erreichen etwa der halben Stellenzahl beim 32768-Eck macht sich jedoch die Auslöschung bei der Subtraktion der fast gleich großen Zahlen 2 und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{\sqrt {1-s_{n}^{2}/4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\sqrt {1-s_{n}^{2}/4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eacb8572b6b028b6b7700e27861c47880f19f928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:12.123ex; height:4.843ex;" alt="{\displaystyle 2{\sqrt {1-s_{n}^{2}/4}}}"></span> bemerkbar. Das Ergebnis wird jetzt wieder ungenauer und am Ende falsch (2 − 2.000…000xxx = 0). </p><p>In vielen Fällen, so auch hier, kann man die Auslöschung vermeiden, einfach indem man die betroffenen Subtraktionen vermeidet. Hier gelingt das mit einer Umformung der Formel in eine äquivalente Form ohne Subtraktion unter Anwendung von </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}-b^{2}=(a+b)(a-b)\quad \Rightarrow \quad a-b={\frac {a^{2}-b^{2}}{a+b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}-b^{2}=(a+b)(a-b)\quad \Rightarrow \quad a-b={\frac {a^{2}-b^{2}}{a+b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a554926a94b74bea390c6fed2d4efc431ca921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:48.467ex; height:6.009ex;" alt="{\displaystyle a^{2}-b^{2}=(a+b)(a-b)\quad \Rightarrow \quad a-b={\frac {a^{2}-b^{2}}{a+b}}}"></span></dd></dl> <p>mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=2,b=2{\sqrt {1-{\tfrac {s_{n}^{2}}{4}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>b</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>4</mn> </mfrac> </mstyle> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=2,b=2{\sqrt {1-{\tfrac {s_{n}^{2}}{4}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/345799c1d59b4a2cbb7e7a8bf6e45718289a6992" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.682ex; height:6.176ex;" alt="{\displaystyle a=2,b=2{\sqrt {1-{\tfrac {s_{n}^{2}}{4}}}}}"></span> </p><p>Es ergibt sich: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{2n}={\sqrt {2-2{\sqrt {1-{\frac {s_{n}^{2}}{4}}}}}}={\sqrt {\frac {4-4(1-{\frac {s_{n}^{2}}{4}})}{2+2{\sqrt {1-{\frac {s_{n}^{2}}{4}}}}}}}={\sqrt {2{\frac {1-1+{\frac {s_{n}^{2}}{4}}}{1+{\sqrt {1-{\frac {s_{n}^{2}}{4}}}}}}}}={\sqrt {{\frac {1}{2}}s_{n}^{2}{\frac {1}{1+{\sqrt {1-{\frac {s_{n}^{2}}{4}}}}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>4</mn> </mfrac> </mrow> </msqrt> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>4</mn> <mo>−</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>4</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>4</mn> </mfrac> </mrow> </msqrt> </mrow> </mrow> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>4</mn> </mfrac> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>4</mn> </mfrac> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mn>4</mn> </mfrac> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{2n}={\sqrt {2-2{\sqrt {1-{\frac {s_{n}^{2}}{4}}}}}}={\sqrt {\frac {4-4(1-{\frac {s_{n}^{2}}{4}})}{2+2{\sqrt {1-{\frac {s_{n}^{2}}{4}}}}}}}={\sqrt {2{\frac {1-1+{\frac {s_{n}^{2}}{4}}}{1+{\sqrt {1-{\frac {s_{n}^{2}}{4}}}}}}}}={\sqrt {{\frac {1}{2}}s_{n}^{2}{\frac {1}{1+{\sqrt {1-{\frac {s_{n}^{2}}{4}}}}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/074148652d3ee18532321ae657b3c9154f3bae94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:87.822ex; height:10.676ex;" alt="{\displaystyle s_{2n}={\sqrt {2-2{\sqrt {1-{\frac {s_{n}^{2}}{4}}}}}}={\sqrt {\frac {4-4(1-{\frac {s_{n}^{2}}{4}})}{2+2{\sqrt {1-{\frac {s_{n}^{2}}{4}}}}}}}={\sqrt {2{\frac {1-1+{\frac {s_{n}^{2}}{4}}}{1+{\sqrt {1-{\frac {s_{n}^{2}}{4}}}}}}}}={\sqrt {{\frac {1}{2}}s_{n}^{2}{\frac {1}{1+{\sqrt {1-{\frac {s_{n}^{2}}{4}}}}}}}}}"></span></dd></dl> <p>Natürlich ist es ein glücklicher Zufall, dass sich im Zähler die Subtraktion „weghebt“. Jetzt verläuft die Rechnung wie erwünscht: </p> <pre><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span></b> <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-\rho _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\rho _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5ff06d88943faa95727eb5496b8059838c748e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.423ex; height:2.676ex;" alt="{\displaystyle 1-\rho _{n}}"></span></b> <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d671890050b21484dde3087d000700c97fc3b03c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.309ex; height:2.009ex;" alt="{\displaystyle s_{n}}"></span></b> <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.643ex; height:2.509ex;" alt="{\displaystyle S_{n}}"></span></b> <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span></b> <b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730f6906700685b6d52f3958b1c2ae659d2d97d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}"></span></b> 2.000e+00 1.000e+00 2.00e+00 Inf 0.00000000000000 Inf 4.000e+00 2.929e-01 1.41e+00 2.00e+00 2.00000000000000 4.00000000000000 8.000e+00 7.612e-02 7.65e-01 8.28e-01 2.82842712474619 3.31370849898476 1.600e+01 1.921e-02 3.90e-01 3.98e-01 3.06146745892072 3.18259787807453 3.200e+01 4.815e-03 1.96e-01 1.97e-01 3.12144515225805 3.15172490742926 6.400e+01 1.205e-03 9.81e-02 9.83e-02 3.13654849054594 3.14411838524590 1.280e+02 3.012e-04 4.91e-02 4.91e-02 3.14033115695475 3.14222362994246 2.560e+02 7.530e-05 2.45e-02 2.45e-02 3.14127725093277 3.14175036916897 5.120e+02 1.882e-05 1.23e-02 1.23e-02 3.14151380114430 3.14163208070318 1.024e+03 4.706e-06 6.14e-03 6.14e-03 3.14157294036709 3.14160251025681 2.048e+03 1.177e-06 3.07e-03 3.07e-03 3.14158772527716 3.14159511774959 4.096e+03 2.941e-07 1.53e-03 1.53e-03 3.14159142151120 3.14159326962931 8.192e+03 7.353e-08 7.67e-04 7.67e-04 3.14159234557012 3.14159280759964 1.638e+04 1.838e-08 3.83e-04 3.83e-04 3.14159257658487 3.14159269209225 3.277e+04 4.596e-09 1.92e-04 1.92e-04 3.14159263433856 3.14159266321541 6.554e+04 1.149e-09 9.59e-05 9.59e-05 3.14159264877699 3.14159265599620 1.311e+05 2.872e-10 4.79e-05 4.79e-05 3.14159265238659 3.14159265419140 2.621e+05 7.181e-11 2.40e-05 2.40e-05 3.14159265328899 3.14159265374019 5.243e+05 1.795e-11 1.20e-05 1.20e-05 3.14159265351459 3.14159265362739 1.049e+06 4.488e-12 5.99e-06 5.99e-06 3.14159265357099 3.14159265359919 2.097e+06 1.122e-12 3.00e-06 3.00e-06 3.14159265358509 3.14159265359214 4.194e+06 2.804e-13 1.50e-06 1.50e-06 3.14159265358862 3.14159265359038 8.389e+06 7.017e-14 7.49e-07 7.49e-07 3.14159265358950 3.14159265358994 1.678e+07 1.754e-14 3.75e-07 3.75e-07 3.14159265358972 3.14159265358983 3.355e+07 4.441e-15 1.87e-07 1.87e-07 3.14159265358978 3.14159265358980 6.711e+07 1.110e-15 9.36e-08 9.36e-08 3.14159265358979 3.14159265358980 1.342e+08 2.220e-16 4.68e-08 4.68e-08 3.14159265358979 3.14159265358979 2.684e+08 0.000e+00 2.34e-08 2.34e-08 3.14159265358979 3.14159265358979 </pre> <p>Schon bei dem 268435456-Eck erreicht man die volle Genauigkeit von knapp 16 Dezimalstellen. Das <a href="/wiki/Abbruchbedingung" title="Abbruchbedingung">Abbruchsignal</a> gibt die 0 in der zweiten Spalte. </p> <div class="mw-heading mw-heading2"><h2 id="Faustregel">Faustregel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ausl%C3%B6schung_(numerische_Mathematik)&veaction=edit&section=4" title="Abschnitt bearbeiten: Faustregel" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ausl%C3%B6schung_(numerische_Mathematik)&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Faustregel"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Subtrahiert man zwei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>-stellige, fast gleich große Zahlen, die in den ersten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> Stellen übereinstimmen, so gehen im Ergebnis von den eigentlich möglichen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> Stellen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> verloren. Es sind also nur noch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p-k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>−</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p-k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3175ee0304d3a109340f43d086e857533ee785c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.31ex; height:2.509ex;" alt="{\displaystyle p-k}"></span> Stellen ungleich Null. Die Information, dass die ersten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> Stellen sich zu Null aufgehoben haben, geht dabei verloren. Die Genauigkeit des Ergebnisses vermindert sich um diese <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> Stellen. </p><p>Unterscheiden sich die Zahlen in den letzten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p-k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>−</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p-k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3175ee0304d3a109340f43d086e857533ee785c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.31ex; height:2.509ex;" alt="{\displaystyle p-k}"></span> Stellen lediglich um Rundungsfehler, dann hat das Ergebnis keine Aussagekraft. Es sollte als solches nicht in weitere Berechnungen einfließen. </p> <div class="mw-heading mw-heading2"><h2 id="Differentialrechnung">Differentialrechnung</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ausl%C3%B6schung_(numerische_Mathematik)&veaction=edit&section=5" title="Abschnitt bearbeiten: Differentialrechnung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ausl%C3%B6schung_(numerische_Mathematik)&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Differentialrechnung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bei der <a href="/wiki/Numerische_Differentiation" title="Numerische Differentiation">numerischen Berechnung von Ableitungen</a> durch <a href="/wiki/Differenzenquotient" title="Differenzenquotient">Differenzenquotienten</a> wie zum Beispiel </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x)\approx {\frac {f(x+h)-f(x)}{h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'(x)\approx {\frac {f(x+h)-f(x)}{h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b2e10702ce6a85055c04f9633878bff9ae2037" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.934ex; height:5.843ex;" alt="{\displaystyle f'(x)\approx {\frac {f(x+h)-f(x)}{h}}}"></span></dd></dl> <p>tritt bei zu kleinem <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> Auslöschung auf, da die Funktionswerte dann nahezu gleich sind. </p> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ausl%C3%B6schung_(numerische_Mathematik)&veaction=edit&section=6" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ausl%C3%B6schung_(numerische_Mathematik)&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Wolfgang Dahmen, Arnold Reusken: <i>Numerik für Ingenieure und Naturwissenschaftler.</i> 2. Auflage. 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Auslöschung (numerische Mathematik) – Wikipedia