Eingabe IE[br][table][tr][td]Zeilenoperationen-Matrix P [i]Product([math]\overleftarrow{IP}[/math])[/i][br][br][i]{n,m,a}[/i] Zeile n += Zeile m * (a)[br][i]{n,n,a}[/i] Zeile n = Zeile n * (a)[br][i]{n,m,0}[/i] tausch Zeile n >< Zeile m[br]{n,m,a} ==> e[sub]n,m[/sub] = a [/td][td][/td][td]Spaltenoperationen-Matrix Q [i]Product(IQ)[br][br]{-n,m,a}[/i]  Spalte n += Spalte m * (a)[br][i]{-n,n,a}[/i]  Spalte n = Spalte n * (a)[br][i]{-n,m,0}[/i] tausch Spalte n >< Spalte m[br]{-n,m,a} ==> e[sub]m,n[/sub] = a[/td][/tr][/table]P: {z , s , a}, Q: {-s , z , a} ==> Elementarmatrix (e[sub]z,s[/sub]) = id(e[sub]i,i[/sub]=1) mit e[sub]z,s[/sub]=a, z (zeilennummer), s (Spaltennummer) [br][br](6) Abfolge der Elementarmatrizen, Stellung Slider [color=#ff0000]p[/color] (rot) [br][math]\small {\begin{array}{rrrrrrrrrr}\textcolor{blue}{\textbf{1}}&-1&2&3&4&2&3&4&3&-3 \\ \textcolor{blue}{\textbf{2}}&4&1&1&1&3&2&2&4&4 \\ \textcolor{blue}{\textbf{0}}&0&-0.25&-0.25&-0.5&0&-0.2&-0.4&0&0 \\ \end{array}}\textcolor{red}{\overleftarrow{P}[{4, 3, -0.14}]\overrightarrow{Q}}{\begin{array}{rr}1&1 \\ 1&1 \\ 1&1 \\ \end{array}}[/math][br][br](8) Zuordnung Abfolge in P[sub]zeilen[/sub] A Q[sub]spalten[/sub] Operationen:[br][color=#ff0000]![/color] [color=#0000ff][sub]reverse[/sub]IP[/color] links [math]\stackrel{nach}{\leftarrow}[/math] rechts IQ links [math]\stackrel{nach}{\rightarrow}[/math] rechts [color=#ff0000]![/color][br][math] \left\{ \left(\begin{array}{rrrrrrrrr}4&3&4&3&2&4&3&2&\textcolor{blue}{\textbf{1}}\\3&4&2&2&3&1&1&1&\textcolor{blue}{\textbf{2}}\\\frac{-1}{7}&0&\frac{-2}{5}&\frac{-1}{5}&0&\frac{-1}{2}&\frac{-1}{4}&\frac{-1}{4}&\textcolor{blue}{\textbf{0}}\\\end{array}\right), I{A}I, \left(\begin{array}{rr}1&3\\4&4\\0&0\\\end{array}\right) \right\} [/math][br][br](10) Ergebnis P A Q wie in Abfolge eingestellt[br][math]PAQ \, := \, \left(\begin{array}{rrrr}4&1&1&3\\0&\frac{15}{4}&\frac{11}{4}&\frac{5}{4}\\0&0&\frac{7}{5}&-1\\0&0&0&\frac{1}{7}\\\end{array}\right)[/math][br][br](11) Letzte mit Slider p ausgewählte Operation P[sub]p-1[/sub] A Q[sub]p-1[/sub] [sub](Ausblenden, Eingabe abschließen mit ; )[/sub][br][math] \left\{ \left(\begin{array}{rrrr}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&\frac{-1}{7}&1\\\end{array}\right), \left(\begin{array}{rrrr}4&1&1&3\\0&\frac{15}{4}&\frac{11}{4}&\frac{5}{4}\\0&0&\frac{7}{5}&-1\\0&0&\frac{1}{5}&0\\\end{array}\right), \left(\begin{array}{rrrr}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\\\end{array}\right) \right\} [/math][br][br](12...) Zusammenstellung der LR-Zerlegung[br][br][math]\text{\large\,LR/LU Zerlegung: P^{Zeilen-Permutationen}_{*}, Q^{Spalten-Permutaionen}_{*}[/math][br][br][math]\large\text{P_{*} A Q_{*} = L R \textcolor{blue}{\to} L^{-1} P_{*} A Q_{*} = R \textcolor{blue}{\uparrow}\\PAQ = R ∧ IP = L^{-1} {P_{*}} \textcolor{blue}{\to} IP \overline{P_{*}} = L^{-1} \textcolor{blue}{\to} (IP \overline{P_{*}})^{-1} = L} [/math][br][br][i]P[sub]*[/sub] :=Reverse(Sequence(If( Element(Ip, jj,3)==0,Element(IP, jj ),{0} ), jj,1,Length(IP) )\{{0}})[br]Q[sub]*[/sub] :=IQ[/i] [br][br][i]L:=(Product(Reverse(IP)) Product(Reverse(P_{*})))⁻¹[/i][br][math]L \, := \, \left(\begin{array}{rrrr}1&0&0&0\\\frac{1}{4}&1&0&0\\\frac{1}{2}&\frac{2}{5}&1&0\\\frac{1}{4}&\frac{1}{5}&\frac{1}{7}&1\\\end{array}\right)[/math][br][br][math]\large {P_{*}\cdot A\cdot Q_{*} = L\cdot R}[/math]

A:={{1, 1, 1,1}, {1,1, 3, 4}, {3, 4, 2,1},{3,2,1,2}};[br]IE :={{1,2,0},{-1,4,0},{2,1,-1/4},{3,1,-1/4},{4,1,-1/2},{2,3,0},{3,2,-4/15 3/4},{4,2,-4/15 3/2 },{3,4,0},{-3,4,0},{4,3,-5/7 1/5},{1,1,1},{1,1,1}};[br][br]Zeilen/Spalten-Pivotsuche[br]A:={{4, -5, 3, 6}, {5, -6, 6, 5}, {-4, 2, 9, 3}, {2, -3, 0, 1}}[br]IE :={{1,3,0},{-1,3,0},{2,1,-6/9},{3,1,-3/9},{-2,3,0},{3,2,-3/23 16/3},{4,2,-3/23 2},{-3,4,0},{4,3,-23/67 5/23},{1,1,1}}; [br][br]A:={{4, 2, 0, -2}, {2, -3, -2, 1}, {0, -2, 3, 4}, {-2, -2, 2, 5}}[br]IE :={{1,4,0},{-1,4,0},{2,1,-1/5},{3,1,-4/5},{4,1,2/5},{2,4,0},{-2,4,0},{3,2,-5/16 8/5},{4,2,-5/16 12/5},{3,4,0},{-3,4,0},{4,3,-2/7}};[br][br]ohne Pivotsuche (ausnutzen der Matrixstruktur: erzeuge PAQ==L durch Spalten-Operationen)[br]A:={{1, -5, 25, -125, 625}, {1, -3, 9, -27, 81},{1, 0, 0, 0, 0}, {1, 2, 4, 8, 16}, {1, 5, 25, 125, 625}}[br]IE :={{1,3,0},{-5,2,27},{-4,2,-9},{-3,2,3},{-5,3,-49},{-4,3,8},{-5,4,6},{1,1,1},{1,1,1}};[br]P[sub]*[/sub] A = L R [br][math]\small{\left(\begin{array}{rrrrr}0&0&1&0&0\\0&1&0&0&0\\1&0&0&0&0\\0&0&0&1&0\\0&0&0&0&1\\ \end{array}\right)_{IP(1)} \left(\begin{array}{rrrrr}1&-5&25&-125&625\\1&-3&9&-27&81\\1&0&0&0&0\\1&2&4&8&16\\1&5&25&125&625\\ \end{array}\right)=\left(\begin{array}{rrrrr}1&0&0&0&0\\1&-3&0&0&0\\1&-5&10&0&0\\1&2&10&70&0\\1&5&40&400&1200\\ \end{array}\right)_{PAQ} \left(\begin{array}{rrrrr}1&0&0&0&0\\0&1&-3&9&-27\\0&0&1&-8&49\\0&0&0&1&-6\\0&0&0&0&1\\ \end{array}\right)_{(Product(IQ))⁻¹} }[/math][br][br]