MathAction Jet LUDecomposition

Jet LUDecomposition
last edited 12 years ago by page

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LUDecomposition (LUD)

LU decomposition for ordinary matrices.

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(1) -> )expose LUDecomposition

 LUDecomposition is now explicitly exposed in frame initial 
A:=matrix [[subscript('a,[10*i+j]) for i in 1..3] for j in 1..3]

\label{eq1}\left[ \begin{array}{ccc} {a_{11}}&{a_{21}}&{a_{31}} \ {a_{12}}&{a_{22}}&{a_{32}} \ {a_{13}}&{a_{23}}&{a_{33}} (1)

Type: Matrix(Polynomial(Integer))

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diagProduct(x) == reduce(*,[x(i,i) for i in 1..nrows(x)])

Type: Void

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B:=LUDecomp A;

Type: Record(LU: Matrix(Fraction(Polynomial(Integer))),Perm: Vector(Integer),Pivots: List(Fraction(Polynomial(Integer))))

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B.LU

\label{eq2}\left[ \begin{array}{ccc} {a_{11}}&{a_{21}}&{a_{31}} \ {\frac{a_{12}}{a_{11}}}&{\frac{{{a_{11}}\ {a_{22}}}-{{a_{12}}\ {a_{21}}}}{a_{11}}}&{\frac{{{a_{11}}\ {a_{32}}}-{{a_{12}}\ {a_{3 1}}}}{a_{11}}} \ {\frac{a_{13}}{a_{11}}}&{\frac{{{a_{11}}\ {a_{23}}}-{{a_{13}}\ {a_{21}}}}{{{a_{11}}\ {a_{22}}}-{{a_{12}}\ {a_{21}}}}}&{\frac{{{\left({{a_{11}}\ {a_{22}}}-{{a_{12}}\ {a_{21}}}\right)}\ {a_{33}}}+{{\left(-{{a_{11}}\ {a_{23}}}+{{a_{13}}\ {a_{21}}}\right)}\ {a_{32}}}+{{\left({{a_{12}}\ {a_{23}}}-{{a_{13}}\ {a_{22}}}\right)}\ {a_{31}}}}{{{a_{11}}\ {a_{22}}}-{{a_{12}}\ {a_{21}}}}} (2)

Type: Matrix(Fraction(Polynomial(Integer)))

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B.Perm

\label{eq3}\left[ 1, \: 2, \: 3 \right] (3)

Type: Vector(Integer)

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B.Pivots

\label{eq4}\begin{array}{@{}l} \displaystyle \left[{\frac{{{\left({{a_{11}}\ {a_{22}}}-{{a_{12}}\ {a_{21}}}\right)}\ {a_{33}}}+{{\left(-{{a_{11}}\ {a_{23}}}+{{a_{13}}\ {a_{21}}}\right)}\ {a_{32}}}+{{\left({{a_{12}}\ {a_{23}}}-{{a_{13}}\ {a_{22}}}\right)}\ {a_{31}}}}{{{a_{11}}\ {a_{22}}}-{{a_{12}}\ {a_{21}}}}}, \right. \ \ \displaystyle \left.\:{\frac{{{a_{11}}\ {a_{22}}}-{{a_{12}}\ {a_{21}}}}{a_{1 1}}}, \:{a_{11}}\right] (4)

Type: List(Fraction(Polynomial(Integer)))

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diagProduct(B.LU)=determinant A

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Compiling function diagProduct with type Matrix(Fraction(Polynomial(
 Integer))) -> Fraction(Polynomial(Integer))

\label{eq5}\begin{array}{@{}l} \displaystyle { \begin{array}{@{}l} \displaystyle {{\left({{a_{11}}\ {a_{22}}}-{{a_{12}}\ {a_{21}}}\right)}\ {a_{33}}}+{{\left(-{{a_{11}}\ {a_{23}}}+{{a_{13}}\ {a_{21}}}\right)}\ {a_{32}}}+ \ \ \displaystyle {{\left({{a_{12}}\ {a_{23}}}-{{a_{13}}\ {a_{22}}}\right)}\ {a_{31}}} (5)

Type: Equation(Fraction(Polynomial(Integer)))

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%::Boolean

\label{eq6} \mbox{\rm true} (6)

Type: Boolean