@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr: a skos:ConceptScheme .
psr:-W7VR5LK6-1
  skos:prefLabel "linear span"@en, "sous-espace vectoriel engendré"@fr ;
  a skos:Concept ;
  skos:narrower psr:-NGJB8TFQ-R .

psr:-NGJB8TFQ-R
  skos:prefLabel "sous-espace de Krylov"@fr, "Krylov subspace"@en ;
  skos:inScheme psr: ;
  a skos:Concept ;
  skos:definition """In linear algebra, the order-<i>r</i> <b>Krylov subspace</b> generated by an <i>n</i>-by-<i>n</i> matrix <i>A</i> and a vector <i>b</i> of dimension <i>n</i> is the linear subspace spanned by the images of <i>b</i> under the first <i>r</i> powers of <i>A</i> (starting from <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^{0}=I}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>A</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>=</mo>
<br/>        <mi>I</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle A^{0}=I}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b37b528693b2abe92a15abb9aa17895cd9dd27" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.068ex; height:2.676ex;" alt="A^{0}=I"></span>), that is,
<br/>
<br/><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 {\\\\mathcal {K}}_{r}(A,b)=\\\\operatorname {span} \\\\,\\\\{b,Ab,A^{2}b,\\\\ldots ,A^{r-1}b\\\\}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi class="MJX-tex-caligraphic" mathvariant="script">K</mi>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>r</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>A</mi>
<br/>        <mo>,</mo>
<br/>        <mi>b</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mi>span</mi>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mo fence="false" stretchy="false">{</mo>
<br/>        <mi>b</mi>
<br/>        <mo>,</mo>
<br/>        <mi>A</mi>
<br/>        <mi>b</mi>
<br/>        <mo>,</mo>
<br/>        <msup>
<br/>          <mi>A</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mi>b</mi>
<br/>        <mo>,</mo>
<br/>        <mo>…<!-- … --></mo>
<br/>        <mo>,</mo>
<br/>        <msup>
<br/>          <mi>A</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>r</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mi>b</mi>
<br/>        <mo fence="false" stretchy="false">}</mo>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\mathcal {K}}_{r}(A,b)=\\\\operatorname {span} \\\\,\\\\{b,Ab,A^{2}b,\\\\ldots ,A^{r-1}b\\\\}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e581dbf19b48f88030faf8bb4e5dc75318f5c04e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.044ex; height:3.176ex;" alt="{\\\\displaystyle {\\\\mathcal {K}}_{r}(A,b)=\\\\operatorname {span} \\\\,\\\\{b,Ab,A^{2}b,\\\\ldots ,A^{r-1}b\\\\}.}"> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Krylov_subspace">https://en.wikipedia.org/wiki/Krylov_subspace</a>)"""@en, """En algèbre linéaire, le <b>sous-espace de Krylov d'ordre</b> <i>r</i> associé à une matrice <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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>A</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle A}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A"></span> de taille <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}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>n</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle n}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span> et un vecteur <i>b</i> de dimension <i>n</i> est le sous-espace vectoriel linéaire engendré par les vecteurs images de <i>b</i> par les <i>r</i> premières puissances de <i>A</i> (à partir de <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^{0}=I}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msup>
<br/>          <mi>A</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo>=</mo>
<br/>        <mi>I</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle A^{0}=I}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9b37b528693b2abe92a15abb9aa17895cd9dd27" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.068ex; height:2.676ex;" alt="{\\\\displaystyle A^{0}=I}"></span> ), c'est-à-dire 
<br/>
<br/><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 {\\\\mathcal {K}}_{r}(A,b)=\\\\operatorname {span} \\\\,\\\\{b,Ab,A^{2}b,\\\\ldots ,A^{r-1}b\\\\}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi class="MJX-tex-caligraphic" mathvariant="script">K</mi>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>r</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>A</mi>
<br/>        <mo>,</mo>
<br/>        <mi>b</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mi>span</mi>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mo fence="false" stretchy="false">{</mo>
<br/>        <mi>b</mi>
<br/>        <mo>,</mo>
<br/>        <mi>A</mi>
<br/>        <mi>b</mi>
<br/>        <mo>,</mo>
<br/>        <msup>
<br/>          <mi>A</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>2</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mi>b</mi>
<br/>        <mo>,</mo>
<br/>        <mo>…<!-- … --></mo>
<br/>        <mo>,</mo>
<br/>        <msup>
<br/>          <mi>A</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>r</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mi>b</mi>
<br/>        <mo fence="false" stretchy="false">}</mo>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\mathcal {K}}_{r}(A,b)=\\\\operatorname {span} \\\\,\\\\{b,Ab,A^{2}b,\\\\ldots ,A^{r-1}b\\\\}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e581dbf19b48f88030faf8bb4e5dc75318f5c04e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.044ex; height:3.176ex;" alt="{\\\\displaystyle {\\\\mathcal {K}}_{r}(A,b)=\\\\operatorname {span} \\\\,\\\\{b,Ab,A^{2}b,\\\\ldots ,A^{r-1}b\\\\}.}"> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Sous-espace_de_Krylov">https://fr.wikipedia.org/wiki/Sous-espace_de_Krylov</a>)"""@fr ;
  skos:broader psr:-KL7BX9Z3-T, psr:-W7VR5LK6-1 ;
  dc:created "2023-08-04"^^xsd:date ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Sous-espace_de_Krylov>, <https://en.wikipedia.org/wiki/Krylov_subspace> ;
  dc:modified "2023-08-04"^^xsd:date .

psr:-KL7BX9Z3-T
  skos:prefLabel "opérateur"@fr, "operator"@en ;
  a skos:Concept ;
  skos:narrower psr:-NGJB8TFQ-R .