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ExampleXDistributedPolynomial

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Editor: hemmecke Time: 2015/06/11 00:13:45 GMT+0
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Demonstration of XDistributedPolynomial

  The following example is an attempt to answer
  https://groups.google.com/d/msg/fricas-devel/UxxF97W5tXA/EYbkeCBZYJsJ
  (see also the mail following this post)

\begin{axiom}
Q ==> Fraction Integer
V ==> OrderedVariableList(['y11,'y12,'y21,'y22])
v := enumerate()$V
X ==> XDistributedPolynomial(V, Q)
z11: X := v.1
z12: X := v.2
z21: X := v.3
z22: X := v.4
q: Matrix X := matrix [[z11, z12],[z21,z22]]
K := kroneckerProduct(q, q)
MM: Matrix(Q) :=matrix([[1,0,0,1], [0,1,1,0], [0,1,-1,0], [1,0,0,-1]])
T := MM*K*inverse(MM)

r1 := (z11*z11+z21*z21-1)
r2 := (z11*z11+z12*z12-1)
r3 := (z12*z12+z22*z22-1)
r4 := (z21*z21+z22*z22-1)

r5 := (z11*z12+z21*z22)
r6 := (z11*z21+z12*z22)
r7 := (z12*z11+z22*z21)
r8 := (z21*z11+z22*z12);

sigma ==> 1111;
r9 := z11*z22+z12*z21-sigma

ra := (z11*z22-z22*z11)
rb := (z12*z21-z21*z12)

rc := (z11*z12+z12*z11)
rd := (z21*z22+z22*z21)
re := (z11*z21+z21*z11)
rf := (z22*z12+z12*z22)

t11 := 2*T(1,1) - r3 - r1
t12 := 2*T(1,2) - r7 - r5
t21 := 2*T(2,1) - r8 - r6
t22 := 2*T(2,2) + ra + rb - 2*r9
\end{axiom}

Demonstration of XDistributedPolynomial

The following example is an attempt to answer https://groups.google.com/d/msg/fricas-devel/UxxF97W5tXA/EYbkeCBZYJsJ (see also the mail following this post)

fricas

Q ==> Fraction Integer

Type: Void

fricas

V ==> OrderedVariableList(['y11,'y12,'y21,'y22])

Type: Void

fricas

v := enumerate()$V

$\label{eq1}\left[ y 11, \: y 12, \: y 21, \: y 22 \right]$

(1)

Type: List(OrderedVariableList([y11,y12,y21,y22]))

fricas

X ==> XDistributedPolynomial(V, Q)

Type: Void

fricas

z11: X := v.1

$\label{eq2}y 11$

(2)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

z12: X := v.2

$\label{eq3}y 12$

(3)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

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z21: X := v.3

$\label{eq4}y 21$

(4)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

z22: X := v.4

$\label{eq5}y 22$

(5)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

q: Matrix X := matrix [[z11, z12],[z21,z22]]

$\label{eq6}\left[ \begin{array}{cc} y 11 & y 12 \ y 21 & y 22$

(6)

Type: Matrix(XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer)))

fricas

K := kroneckerProduct(q, q)

$\label{eq7}\left[ \begin{array}{cccc} {{y 11}^{2}}&{y 11 \ y 12}&{y 12 \ y 11}&{{y 12}^{2}} \ {y 11 \ y 21}&{y 11 \ y 22}&{y 12 \ y 21}&{y 12 \ y 22} \ {y 21 \ y 11}&{y 21 \ y 12}&{y 22 \ y 11}&{y 22 \ y 12} \ {{y 21}^{2}}&{y 21 \ y 22}&{y 22 \ y 21}&{{y 22}^{2}}$

(7)

Type: Matrix(XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer)))

fricas

MM: Matrix(Q) :=matrix([[1,0,0,1], [0,1,1,0], [0,1,-1,0], [1,0,0,-1]])

$\label{eq8}\left[ \begin{array}{cccc} 1 & 0 & 0 & 1 \ 0 & 1 & 1 & 0 \ 0 & 1 & - 1 & 0 \ 1 & 0 & 0 & - 1$

(8)

Type: Matrix(Fraction(Integer))

fricas

T := MM*K*inverse(MM)

$\label{eq9}\left[ \begin{array}{cccc} {{{1 \over 2}\ {{y 22}^{2}}}+{{1 \over 2}\ {{y 21}^{2}}}+{{1 \over 2}\ {{y 12}^{2}}}+{{1 \over 2}\ {{y 11}^{2}}}}&{{{1 \over 2}\ y 22 \ y 21}+{{1 \over 2}\ y 21 \ y 22}+{{1 \over 2}\ y 12 \ y 11}+{{1 \over 2}\ y 11 \ y 12}}&{-{{1 \over 2}\ y 22 \ y 21}+{{1 \over 2}\ y 21 \ y 22}-{{1 \over 2}\ y 1 2 \ y 11}+{{1 \over 2}\ y 11 \ y 12}}&{-{{1 \over 2}\ {{y 22}^{2}}}+{{1 \over 2}\ {{y 21}^{2}}}-{{1 \over 2}\ {{y 12}^{2}}}+{{1 \over 2}\ {{y 11}^{2}}}} \ {{{1 \over 2}\ y 22 \ y 12}+{{1 \over 2}\ y 21 \ y 11}+{{1 \over 2}\ y 12 \ y 22}+{{1 \over 2}\ y 11 \ y 21}}&{{{1 \over 2}\ y 22 \ y 11}+{{1 \over 2}\ y 21 \ y 12}+{{1 \over 2}\ y 12 \ y 21}+{{1 \over 2}\ y 11 \ y 22}}&{-{{1 \over 2}\ y 22 \ y 11}+{{1 \over 2}\ y 21 \ y 12}-{{1 \over 2}\ y 1 2 \ y 21}+{{1 \over 2}\ y 11 \ y 22}}&{-{{1 \over 2}\ y 2 2 \ y 12}+{{1 \over 2}\ y 21 \ y 11}-{{1 \over 2}\ y 12 \ y 22}+{{1 \over 2}\ y 11 \ y 21}} \ {-{{1 \over 2}\ y 22 \ y 12}-{{1 \over 2}\ y 21 \ y 11}+{{1 \over 2}\ y 12 \ y 22}+{{1 \over 2}\ y 11 \ y 21}}&{-{{1 \over 2}\ y 22 \ y 11}-{{1 \over 2}\ y 21 \ y 12}+{{1 \over 2}\ y 12 \ y 21}+{{1 \over 2}\ y 11 \ y 22}}&{{{1 \over 2}\ y 22 \ y 11}-{{1 \over 2}\ y 21 \ y 12}-{{1 \over 2}\ y 1 2 \ y 21}+{{1 \over 2}\ y 11 \ y 22}}&{{{1 \over 2}\ y 22 \ y 12}-{{1 \over 2}\ y 21 \ y 11}-{{1 \over 2}\ y 12 \ y 22}+{{1 \over 2}\ y 11 \ y 21}} \ {-{{1 \over 2}\ {{y 22}^{2}}}-{{1 \over 2}\ {{y 21}^{2}}}+{{1 \over 2}\ {{y 12}^{2}}}+{{1 \over 2}\ {{y 11}^{2}}}}&{-{{1 \over 2}\ y 22 \ y 21}-{{1 \over 2}\ y 21 \ y 22}+{{1 \over 2}\ y 12 \ y 11}+{{1 \over 2}\ y 11 \ y 12}}&{{{1 \over 2}\ y 22 \ y 21}-{{1 \over 2}\ y 21 \ y 22}-{{1 \over 2}\ y 12 \ y 11}+{{1 \over 2}\ y 11 \ y 12}}&{{{1 \over 2}\ {{y 22}^{2}}}-{{1 \over 2}\ {{y 21}^{2}}}-{{1 \over 2}\ {{y 12}^{2}}}+{{1 \over 2}\ {{y 11}^{2}}}}$

(9)

Type: Matrix(XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer)))

fricas

r1 := (z11*z11+z21*z21-1)

$\label{eq10}- 1 +{{y 21}^{2}}+{{y 11}^{2}}$

(10)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

r2 := (z11*z11+z12*z12-1)

$\label{eq11}- 1 +{{y 12}^{2}}+{{y 11}^{2}}$

(11)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

r3 := (z12*z12+z22*z22-1)

$\label{eq12}- 1 +{{y 22}^{2}}+{{y 12}^{2}}$

(12)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

r4 := (z21*z21+z22*z22-1)

$\label{eq13}- 1 +{{y 22}^{2}}+{{y 21}^{2}}$

(13)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

r5 := (z11*z12+z21*z22)

$\label{eq14}{y 21 \ y 22}+{y 11 \ y 12}$

(14)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

r6 := (z11*z21+z12*z22)

$\label{eq15}{y 12 \ y 22}+{y 11 \ y 21}$

(15)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

r7 := (z12*z11+z22*z21)

$\label{eq16}{y 22 \ y 21}+{y 12 \ y 11}$

(16)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

r8 := (z21*z11+z22*z12);

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

sigma ==> 1111;

Type: Void

fricas

r9 := z11*z22+z12*z21-sigma

$\label{eq17}-{1111}+{y 12 \ y 21}+{y 11 \ y 22}$

(17)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

ra := (z11*z22-z22*z11)

$\label{eq18}-{y 22 \ y 11}+{y 11 \ y 22}$

(18)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

rb := (z12*z21-z21*z12)

$\label{eq19}-{y 21 \ y 12}+{y 12 \ y 21}$

(19)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

rc := (z11*z12+z12*z11)

$\label{eq20}{y 12 \ y 11}+{y 11 \ y 12}$

(20)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

rd := (z21*z22+z22*z21)

$\label{eq21}{y 22 \ y 21}+{y 21 \ y 22}$

(21)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

re := (z11*z21+z21*z11)

$\label{eq22}{y 21 \ y 11}+{y 11 \ y 21}$

(22)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

rf := (z22*z12+z12*z22)

$\label{eq23}{y 22 \ y 12}+{y 12 \ y 22}$

(23)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

t11 := 2*T(1,1) - r3 - r1

$\label{eq24}2$

(24)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

t12 := 2*T(1,2) - r7 - r5

$\label{eq25}0$

(25)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

t21 := 2*T(2,1) - r8 - r6

$\label{eq26}0$

(26)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))

fricas

t22 := 2*T(2,2) + ra + rb - 2*r9

$\label{eq27}2222$

(27)

Type: XDistributedPolynomial(OrderedVariableList([y11,y12,y21,y22]),Fraction(Integer))