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LUDecomposition (LUD)

LU decomposition for ordinary matrices.

fricas
)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))
fricas
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}}
\
{{a_{12}}\over{a_{11}}}&{{{{a_{11}}\ {a_{22}}}-{{a_{12}}\ {a_{21}}}}\over{a_{11}}}&{{{{a_{11}}\ {a_{32}}}-{{a_{12}}\ {a_{3
1}}}}\over{a_{11}}}
\
{{a_{13}}\over{a_{11}}}&{{{{a_{11}}\ {a_{23}}}-{{a_{13}}\ {a_{21}}}}\over{{{a_{11}}\ {a_{22}}}-{{a_{12}}\ {a_{21}}}}}&{{{{\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}}}}\over{{{a_{11}}\ {a_{22}}}-{{a_{12}}\ {a_{21}}}}}
(2)
Type: Matrix(Fraction(Polynomial(Integer)))
fricas
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[{{\left(
\begin{array}{@{}l}
\displaystyle
{{\left({{a_{11}}\ {a_{22}}}-{{a_{12}}\ {a_{21}}}\right)}\ {a_{33}}}+ 
\
\
\displaystyle
{{\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}}}
(4)
Type: List(Fraction(Polynomial(Integer)))
fricas
diagProduct(B.LU)=determinant A
fricas
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)))
fricas
%::Boolean

\label{eq6} \mbox{\rm true} (6)
Type: Boolean




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