MathAction SandBoxPoirier

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    - SandBoxPoirier <-- You are here.

Hall

fricas

(1) -> x := operator('x)

$\label{eq1}x$

(1)

Type: BasicOperator?

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sif(n)==1/(subscript('x,[n])-subscript('x,[n-1]))

Type: Void

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sigma(n)==1/sif(n)^2*(sif(n+1)-2*sif(n)+sif(n-1))

Type: Void

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sigma(n)

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Compiling function sif with type Variable(n) -> Fraction(Polynomial(
      Integer))

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Compiling function sif with type Polynomial(Integer) -> Fraction(
      Polynomial(Integer))

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Compiling function sigma with type Variable(n) -> Fraction(
      Polynomial(Integer))

$\label{eq2}\frac{{{x_{n}}^{3}}+{{\left(-{x_{n + 1}}-{5 \ {x_{n - 1}}}+{3 \ {x_{n - 2}}}\right)}\ {{x_{n}}^{2}}}+{{\left({{\left({4 \ {x_{n - 1}}}-{2 \ {x_{n - 2}}}\right)}\ {x_{n + 1}}}+{5 \ {{x_{n - 1}}^{2}}}-{4 \ {x_{n - 2}}\ {x_{n - 1}}}\right)}\ {x_{n}}}+{{\left(-{3 \ {{x_{n - 1}}^{2}}}+{2 \ {x_{n - 2}}\ {x_{n - 1}}}\right)}\ {x_{n + 1}}}-{{x_{n - 1}}^{3}}+{{x_{n - 2}}\ {{x_{n - 1}}^{2}}}}{{{\left({x_{n - 1}}-{x_{n - 2}}\right)}\ {x_{n}}}+{{\left(-{x_{n - 1}}+{x_{n - 2}}\right)}\ {x_{n + 1}}}}$

(2)

Type: Fraction(Polynomial(Integer))

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r(n)==1/4*(sigma(n+1)-sigma(n))

Type: Void

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numer r(n)

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Compiling function sigma with type Polynomial(Integer) -> Fraction(
      Polynomial(Integer))

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Compiling function r with type Variable(n) -> Fraction(Polynomial(
      Integer))

$\label{eq3}\begin{array}{@{}l} \displaystyle {{\left(-{x_{n + 2}}+{x_{n + 1}}+{x_{n - 1}}-{x_{n - 2}}\right)}\ {{x_{n}}^{4}}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({x_{n + 1}}+{9 \ {x_{n - 1}}}-{6 \ {x_{n - 2}}}\right)}\ {x_{n + 2}}}- \ \ \displaystyle {{x_{n + 1}}^{2}}+{{\left(-{{12}\ {x_{n - 1}}}+{9 \ {x_{n - 2}}}\right)}\ {x_{n + 1}}}- \ \ \displaystyle {{x_{n - 1}}^{2}}+{{x_{n - 2}}\ {x_{n - 1}}}$

(3)

Type: Polynomial(Integer)

fricas

factor denom(r(n))

$\label{eq4}\begin{array}{@{}l} \displaystyle 4 \ {\left({x_{n - 1}}-{x_{n - 2}}\right)}\ {\left({x_{n + 2}}-{x_{n + 1}}\right)}\ {\left({x_{n}}-{x_{n + 1}}\right)}\ \cdot \ \ \displaystyle {\left({x_{n}}-{x_{n - 1}}\right)}$

(4)

Type: Factored(Polynomial(Integer))

Poirier to Hall

1-D

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C:List Symbol := [subscript('x,[0])]

$\label{eq5}\left[{x_{0}}\right]$

(5)

Type: List(Symbol)

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d := #C

$\label{eq6}1$

(6)

Type: PositiveInteger?

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J := matrix [[D(x(C(1),t),C(1))]]

$\label{eq7}\left[ \begin{array}{c} {{x_{, 1}}\left({{x_{0}}, \: t}\right)}$

(7)

Type: Matrix(Expression(Integer))

fricas

K := inverse J

$\label{eq8}\left[ \begin{array}{c} {\frac{1}{{x_{, 1}}\left({{x_{0}}, \: t}\right)}}$

(8)

Type: Union(Matrix(Expression(Integer)),...)

fricas

Req18a:=[reduce(+,[reduce(+,[reduce(+,[reduce(+,[ (1/4)*D( K(k,i)*K(m,j)*D( D( K(l,j),C(k) ),C(l) ),C(m) ) for m in 1..d]) for k in 1..d]) for j in 1..d]) for l in 1..d]) for i in 1..d]

$\label{eq9}\left[ \frac{-{{{{x_{, 1}}\left({{x_{0}}, \: t}\right)}^{2}}\ {{x_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}+{8 \ {{x_{, 1}}\left({{x_{0}}, \: t}\right)}\ {{x_{{, 1}{, 1}}}\left({{x_{0}}, \: t}\right)}\ {{x_{{{, 1}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}-{{10}\ {{{x_{{, 1}{, 1}}}\left({{x_{0}}, \: t}\right)}^{3}}}}{4 \ {{{x_{, 1}}\left({{x_{0}}, \: t}\right)}^{6}}}\right]$

(9)

Type: List(Expression(Integer))

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diff1(n)==subscript('x,[n])-subscript('x,[n-1])

Type: Void

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diff1(n)

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Compiling function diff1 with type Variable(n) -> Polynomial(Integer
      )

$\label{eq10}{x_{n}}-{x_{n - 1}}$

(10)

Type: Polynomial(Integer)

fricas

Req18b:=eval(Req18a,D(x(C(1),t),C(1))=diff1(n))

$\label{eq11}\left[ \frac{{{\left(-{{x_{n}}^{2}}+{2 \ {x_{n - 1}}\ {x_{n}}}-{{x_{n - 1}}^{2}}\right)}\ {{x_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}+{{\left({8 \ {x_{n}}}-{8 \ {x_{n - 1}}}\right)}\ {{x_{{, 1}{, 1}}}\left({{x_{0}}, \: t}\right)}\ {{x_{{{, 1}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}-{{10}\ {{{x_{{, 1}{, 1}}}\left({{x_{0}}, \: t}\right)}^{3}}}}{{4 \ {{x_{n}}^{6}}}-{{24}\ {x_{n - 1}}\ {{x_{n}}^{5}}}+{{60}\ {{x_{n - 1}}^{2}}\ {{x_{n}}^{4}}}-{{80}\ {{x_{n - 1}}^{3}}\ {{x_{n}}^{3}}}+{{60}\ {{x_{n - 1}}^{4}}\ {{x_{n}}^{2}}}-{{24}\ {{x_{n - 1}}^{5}}\ {x_{n}}}+{4 \ {{x_{n - 1}}^{6}}}}\right]$

(11)

Type: List(Expression(Integer))

fricas

diff2(n)==diff1(n)-diff1(n-1)

Type: Void

fricas

diff2(n)

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Compiling function diff1 with type Polynomial(Integer) -> Polynomial
      (Integer)

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Compiling function diff2 with type Variable(n) -> Polynomial(Integer
      )

$\label{eq12}{x_{n}}-{2 \ {x_{n - 1}}}+{x_{n - 2}}$

(12)

Type: Polynomial(Integer)

fricas

Req18c:=eval(Req18b,D(x(C(1),t),[C(1),C(1)])=diff2(n));

Type: List(Expression(Integer))

fricas

numer Req18c.1

$\label{eq13}\begin{array}{@{}l} \displaystyle {{\left(-{{x_{n}}^{2}}+{2 \ {x_{n - 1}}\ {x_{n}}}-{{x_{n - 1}}^{2}}\right)}\ {{x_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {8 \ {{x_{n}}^{2}}}+{{\left(-{{24}\ {x_{n - 1}}}+{8 \ {x_{n - 2}}}\right)}\ {x_{n}}}+ \ \ \displaystyle {{16}\ {{x_{n - 1}}^{2}}}-{8 \ {x_{n - 2}}\ {x_{n - 1}}}$

(13)

Type: SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer)))

fricas

diff3(n)==diff2(n+1)-diff2(n)

Type: Void

fricas

diff3(n)

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Compiling function diff2 with type Polynomial(Integer) -> Polynomial
      (Integer)

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Compiling function diff3 with type Variable(n) -> Polynomial(Integer
      )

$\label{eq14}-{3 \ {x_{n}}}+{x_{n + 1}}+{3 \ {x_{n - 1}}}-{x_{n - 2}}$

(14)

Type: Polynomial(Integer)

fricas

Req18d:=eval(Req18c,D(x(C(1),t),[C(1),C(1),C(1)])=diff3(n));

Type: List(Expression(Integer))

fricas

numer Req18d.1

$\label{eq15}\begin{array}{@{}l} \displaystyle {{\left(-{{x_{n}}^{2}}+{2 \ {x_{n - 1}}\ {x_{n}}}-{{x_{n - 1}}^{2}}\right)}\ {{x_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \: t}\right)}}-{{34}\ {{x_{n}}^{3}}}+ \ \ \displaystyle {{\left({8 \ {x_{n + 1}}}+{{156}\ {x_{n - 1}}}-{{62}\ {x_{n - 2}}}\right)}\ {{x_{n}}^{2}}}+ \ \ \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left(-{{24}\ {x_{n - 1}}}+{8 \ {x_{n - 2}}}\right)}\ {x_{n + 1}}}-{{240}\ {{x_{n - 1}}^{2}}}+ \ \ \displaystyle {{192}\ {x_{n - 2}}\ {x_{n - 1}}}-{{38}\ {{x_{n - 2}}^{2}}}$

(15)

Type: SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer)))

fricas

diff4(n)==diff3(n+1)-diff3(n)

Type: Void

fricas

diff4(n)

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Compiling function diff3 with type Polynomial(Integer) -> Polynomial
      (Integer)

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Compiling function diff4 with type Variable(n) -> Polynomial(Integer
      )

$\label{eq16}{6 \ {x_{n}}}+{x_{n + 2}}-{4 \ {x_{n + 1}}}-{4 \ {x_{n - 1}}}+{x_{n - 2}}$

(16)

Type: Polynomial(Integer)

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Req18e:=eval(Req18d,D(x(C(1),t),[C(1),C(1),C(1),C(1)])=diff4(n));

Type: List(Expression(Integer))

fricas

--numer(Req18e.1-r(n))
factor numer Req18e.1

$\label{eq17}\begin{array}{@{}l} \displaystyle -{\left({ \begin{array}{@{}l} \displaystyle {{40}\ {{x_{n}}^{3}}}+{{\left({ \begin{array}{@{}l} \displaystyle {x_{n + 2}}-{{12}\ {x_{n + 1}}}- \ \ \displaystyle {{172}\ {x_{n - 1}}}+{{63}\ {x_{n - 2}}}$

(17)

Type: Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer))))

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factor denom Req18e.1

$\label{eq18}4 \ {{\left({x_{n}}-{x_{n - 1}}\right)}^{6}}$

(18)

Type: Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer))))

2-D

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x := operator('x)

$\label{eq19}x$

(19)

Type: BasicOperator?

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y := operator('y)

$\label{eq20}y$

(20)

Type: BasicOperator?

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C:List Symbol := [subscript('x,[0]),subscript('y,[0])]

$\label{eq21}\left[{x_{0}}, \:{y_{0}}\right]$

(21)

Type: List(Symbol)

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d := #C

$\label{eq22}2$

(22)

Type: PositiveInteger?

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J := matrix [[D(x(C(1),C(2),t),C(1)), D(x(C(1),C(2),t),C(2))], 
             [D(y(C(1),C(2),t),C(1)), D(y(C(1),C(2),t),C(2))]]

$\label{eq23}\left[ \begin{array}{cc} {{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}&{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)} \ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}&{{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}$

(23)

Type: Matrix(Expression(Integer))

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K := inverse J

$\label{eq24}\left[ \begin{array}{cc} -{\frac{{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}{{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}-{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}}}&{\frac{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}{{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}-{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}}} \ {\frac{{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}{{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}-{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}}}& -{\frac{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}{{{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}-{{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\ {{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}}}}$

(24)

Type: Union(Matrix(Expression(Integer)),...)

fricas

Req18:=[reduce(+,[reduce(+,[reduce(+,[reduce(+,[
 (1/4)*D( K(k,i)*K(m,j)*D( D( K(l,j),C(k) ),C(l) ),C(m) )
   for m in 1..d]) for k in 1..d]) for j in 1..d]) for l in 1..d]) for i in 1..d];

Type: List(Expression(Integer))

fricas

#Req18

$\label{eq25}2$

(25)

Type: PositiveInteger?

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kernels Req18.1

$\label{eq26}\begin{array}{@{}l} \displaystyle \left[{{y_{{{{, 1}{, 1}}{, 1}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{{, 1}{, 1}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{{, 1}{, 2}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right. \ \ \displaystyle \left.{{y_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{{, 2}{, 2}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{{, 1}{, 1}}{, 1}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right. \ \ \displaystyle \left.{{x_{{{{, 1}{, 1}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{{, 1}{, 2}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{{, 1}{, 1}}{, 1}}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right. \ \ \displaystyle \left.{{x_{{{{, 2}{, 2}}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{, 1}{, 1}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{, 1}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right. \ \ \displaystyle \left.{{y_{{{, 1}{, 1}}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{{, 2}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{, 1}{, 1}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right. \ \ \displaystyle \left.{{x_{{{, 1}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{, 1}{, 1}}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{{, 2}{, 2}}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right. \ \ \displaystyle \left.{{y_{{, 1}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{, 1}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{{, 2}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right. \ \ \displaystyle \left.{{x_{{, 1}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{, 1}{, 1}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{{, 2}{, 2}}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \: \right. \ \ \displaystyle \left.{{y_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{y_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{, 1}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}, \:{{x_{, 2}}\left({{x_{0}}, \:{y_{0}}, \: t}\right)}\right]$

(26)

Type: List(Kernel(Expression(Integer)))

Subject: Be Bold !!

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