LR Zerlegung R³ - LU decomposition linear equations
- Author:
- hawe
- Topic:
- Linear Equations, Matrices
(11) Pivot = 0 - ohne Pivotsuche
= 1 - mit Spalten-Pivotsuche
= 2 - mit Zeilen/Spalten-Pivotsuche
Pi - Zeilen-Tausch-Matrizen ∈ Pivoti,
Li - Matrizen für Gauß-Eliminationsschritt Spalte i ∈ Gaussi,
Qi - Spalten-Tausch-Matrizen ∈ Pivoti
L1P1AQ1
L1 (1.Spalte unter 1.Diagonal-Element Nullen erzeugen,
P1,Q1 Zeilen-Spalten-Tausch für Pivot 1. Diagonal-Element [i=1]
ADD n=4
(20) L2P2L1P1AQ1Q2:=Product(Gauss_{2})
(21) Pivot_{3}:={T(3, If(Pivot==2,PivotZ(L2P2L1P1AQ1Q2, 3),Pivot==1,SPvt(L2P2L1P1AQ1Q2,3),3) ),L2P2L1P1AQ1Q2,T(3,If(Pivot==2,PivotS(L2P2L1P1AQ1Q2, 3),3) )}
(22) P3L2P2L1P1AQ1Q2Q3:= Product(Pivot_{3})
(23) Gauss_{3}:={Product(LE(P3L2P2L1P1AQ1Q2Q3, 3)), P3L2P2L1P1AQ1Q2Q3}
R=(L3 P3 L2 P2 L1 P2 P3) (P3 P2 P1) A (Q1 Q2 Q3)
(25) R:=Product(Gauss_{3})
(26) L:=( Element(Gauss_{3}, 1) Element(Pivot_{3},1) Element(Gauss_{2}, 1) Element(Pivot_{2},1) Element(Gauss_{1}, 1) Element(Pivot_{2},1) Element(Pivot_{3},1) )^(-1)
(27) P:=Element(Pivot_{3},1) Element(Pivot_{2},1)*Element(Pivot_{1},1)
(28) Q:=Element(Pivot_{1}, 3)*Element(Pivot_{2}, 3)*Element(Pivot_{3}, 3)
(29) L R = P A Q .........
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LR-Zerlegung3x3MitTotalPivotSuche