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DetlefA
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DetlefA
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MontyPythagoras
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Hallo Detlef,
vielleicht erst einmal ein paar grundsätzliche Überlegungen vorab. Sagen wir, das Netz hat $n\times n$ Knoten. Dann hast Du insgesamt $2n(n-1)$ unbekannte Widerstände (ob Widerstände oder Leitwerte ist ja unter dem Strich egal). Außerdem hast Du $4(n-1)$ Widerstände am Rand und ebenso viele Knotenpunkte, zwischen denen Du eine Messung vornehmen kannst. Wenn wir bei Deiner Skizze bleiben, kannst Du z.B. eine Messung zwischen den Knoten 11 und 20 machen, und dieser Messung gewissermaßen einen Gesamt-Widerstand (oder Leitwert) zuordnen. Bei $4n-4$ äußeren Knoten sind theoretisch
$$(4n-5)+(4n-6)+(4n-7)+\dots+2+1==\frac12(4n-4)(4n-5)=2(n-1)(4n-5)$$Messwerte möglich. Das sind deutlich mehr als genug, aber man hat auch die Qual der Wahl, zwischen welchen Knoten man sinnvollerweise misst.
Ich weiß nicht, ob Du dir dessen überhaupt bewusst bist, aber Dein vor einiger Zeit mal gepostetes Problem
Linknichtlineares Gleichungssystem mit 4 Unbekannten
entspricht genau diesem Thema mit dem kleinstmöglichen Netz der Größe $2\times 2$. Betrachte hier nur mal das Mininetz "1-2-7-6". In Deinem alten Problem hattest Du vier Messungen vorliegen, nämlich "1-2", "2-7", "7-6" und "6-1", wobei z.B. der zu "1-2" parallele Widerstand der in Reihe geschalteten Widerstände "1-6-7-2" leicht auszurechnen ist.
Was ich damit sagen will: Dein altes $2\times 2$-Problem hatte schon eine recht komplizierte Lösung. Ich habe wenig Hoffnung, dass man dieses Problem für allgemeine $n\times n$-Netze einfach lösen kann. Selbst beim Berechnungsalgorithmus muss man wohl ein nichtlineares System aus $2n(n-1)$ Gleichungen lösen. Dass man alle bis auf eine Variable eliminieren kann, halte ich hier für aussichtslos.

Ciao,

Thomas



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DetlefA
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Hallo Thomas,

an das alte Problem entsinne ich mich selbstverständlich, für das damals vorgestellte minimale 2x2 Problem hattest Du ja eine sehr große analytische Lösung gefunden. Es wundert mich auch, welch einfache Fragestellungen große aber analytisch lösbare Probleme generieren. Das hat auch funktioniert, die berechneten R's waren korrekt und sind verifiziert.

Für das vorgestellte Problem ist eine analytische Lösung sicher nicht machbar, 5x5 ist auch nur ein Beispiel, 100x100 ist technisch interessant.
Eine numerische Lösung des nichtlinearen Gleichungssystems ist angestrebt.

Meßwerte kann ich in der Tat viele generieren, das nährt auch die Hoffnung, dass eine Lösung überhaupt möglich ist.

So 'schlimm' nichtlinear ist es ja auch nicht, es hat eine Struktur: Ich kenne beliebig viele Summen mit vier Summanden, die jeweils das Produkt aus zwei Unbekannten sind. Das riecht danach, dass es da was Schlaues geben muss. Kann die 2x2 Lösung nicht 'Induktionsstart' sein?

Cheers
Detlef



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StefanVogel
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Hallo Detlef,
das Gleichungssystem ist aufgestellt, Lösbarkeit untersucht, Programme zum Lösen von Gleichungssystemen gibt es auch. Da bleibt eigentlich nur noch die Dateneingabe übrig. Ich habe das mit

https://cocalc.com
 --> Welcome to CoCalc!
   --> Jupyter notebook
     --> Sagemath 9.1

und der Funktion solve() versucht.

An jeden einzelnen Randknoten wird der Reihe nach ein konstanter Strom zugeführt und dann die Spannungen an allen Randknoten gemessen. Der rechte untere Knotenpunkt ist mit Masse verbunden. Dort soll der eingeprägte Strom abfließen. Gemessen werden die Spannungen bezüglich diesem Massepunkt.

Als Testbeispiel habe ich eine Schaltung mit 4*4 Knotenpunkten verwendet, mit Leitwerten alle gleich 1/224.

<math>
\begin{tikzpicture} \\(ist keine optimale TikZ-Eingabe)
\foreach \i/\j in {
0/3,
0/2, 1/3,
0/1, 1/2, 2/3,
}
{
\draw[gray,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- +(1,0)
(\i,\j) -- +(0,-0.25)
+(-0.1,-0.25) -- +(-0.1,-0.75) -- +(0.1,-0.75) -- +(0.1,-0.25) -- +(-0.1,-0.25)
+(0,-0.75) -- +(0,-1) ;
\fill[gray,thick] (\i,\j) circle (1.5pt) +(1,0) circle (1.5pt) +(0,-1) circle (1.5pt);
}

\foreach \i/\j in {
0/0, 1/1, 2/2,
1/0, 2/1,
2/0
}
{
\draw[gray,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- ++(1,0)
-- +(0,0.25)
+(-0.1,0.25) -- +(-0.1,0.75) -- +(0.1,0.75) -- +(0.1,0.25) -- +(-0.1,0.25)
+(0,0.75) -- +(0,1) ;
\fill[gray] (\i,\j) circle (1.5pt) +(1,0) circle (1.5pt) +(0,1) circle (1.5pt);
}
\end{tikzpicture}
</math>


Variablenbezeichnungen:
(i,j) Knotenpunkt in Zeile i, Spalte j, beginnend mit (1,1) in der linken oberen Ecke.
Uzisjzksl gemessene Spannung in Randknoten (i,j), wenn in Randpunkt (k,l) ein Strom 1 zufeführt wird.
Gzisjr Leitwert, der von Knoten (i,j) nach rechts abzweigt.
Gzisjr Leitwert, der von Knoten (i,j) nach unten abzweigt.

Zuerst gebe ich die gemessenen Spannungen ein. Das sind bei diesem Beispiel berechnete Werte, die ich durch Invertieren der Leitwertmatrix (alle Leitwerte=1/224) erhalten habe:

Uz1s1z1s1=416; Uz1s2z1s1=304; Uz1s3z1s1=240; Uz1s4z1s1=208; Uz2s1z1s1=304; Uz3s1z1s1=240; Uz4s1z1s1=208; 
Uz1s1z1s2=304; Uz1s2z1s2=349; Uz1s3z1s2=263; Uz1s4z1s2=224; Uz2s1z1s2=259; Uz3s1z1s2=217; Uz4s1z1s2=192; 
Uz1s1z1s3=240; Uz1s2z1s3=263; Uz1s3z1s3=327; Uz1s4z1s3=266; Uz2s1z1s3=217; Uz3s1z1s3=189; Uz4s1z1s3=170; 
Uz1s1z1s4=208; Uz1s2z1s4=224; Uz1s3z1s4=266; Uz1s4z1s4=362; Uz2s1z1s4=192; Uz3s1z1s4=170; Uz4s1z1s4=154; 
Uz1s1z2s1=304; Uz1s2z2s1=259; Uz1s3z2s1=217; Uz1s4z2s1=192; Uz2s1z2s1=349; Uz3s1z2s1=263; Uz4s1z2s1=224; 
Uz1s1z3s1=240; Uz1s2z3s1=217; Uz1s3z3s1=189; Uz1s4z3s1=170; Uz2s1z3s1=263; Uz3s1z3s1=327; Uz4s1z3s1=266; 
Uz1s1z4s1=208; Uz1s2z4s1=192; Uz1s3z4s1=170; Uz1s4z4s1=154; Uz2s1z4s1=224; Uz3s1z4s1=266; Uz4s1z4s1=362; 
Uz1s1z4s2=176; Uz1s2z4s2=167; Uz1s3z4s2=151; Uz1s4z4s2=138; Uz2s1z4s2=185; Uz3s1z4s2=205; Uz4s1z4s2=234; 
Uz1s1z4s3=112; Uz1s2z4s3=109; Uz1s3z4s3=103; Uz1s4z4s3=96;  Uz2s1z4s3=115; Uz3s1z4s3=121; Uz4s1z4s3=128; 
Uz1s1z4s4=0;   Uz1s2z4s4=0;   Uz1s3z4s4=0;   Uz1s4z4s4=0;   Uz2s1z4s4=0;   Uz3s1z4s4=0;   Uz4s1z4s4=0; 
Uz1s1z2s4=176; Uz1s2z2s4=185; Uz1s3z2s4=205; Uz1s4z2s4=234; Uz2s1z2s4=167; Uz3s1z2s4=151; Uz4s1z2s4=138; 
Uz1s1z3s4=112; Uz1s2z3s4=115; Uz1s3z3s4=121; Uz1s4z3s4=128; Uz2s1z3s4=109; Uz3s1z3s4=103; Uz4s1z3s4=96; 
 
Uz4s2z1s1=176; Uz4s3z1s1=112; Uz4s4z1s1=0;   Uz2s4z1s1=176; Uz3s4z1s1=112; 
Uz4s2z1s2=167; Uz4s3z1s2=109; Uz4s4z1s2=0;   Uz2s4z1s2=185; Uz3s4z1s2=115; 
Uz4s2z1s3=151; Uz4s3z1s3=103; Uz4s4z1s3=0;   Uz2s4z1s3=205; Uz3s4z1s3=121; 
Uz4s2z1s4=138; Uz4s3z1s4=96;  Uz4s4z1s4=0;   Uz2s4z1s4=234; Uz3s4z1s4=128; 
Uz4s2z2s1=185; Uz4s3z2s1=115; Uz4s4z2s1=0;   Uz2s4z2s1=167; Uz3s4z2s1=109; 
Uz4s2z3s1=205; Uz4s3z3s1=121; Uz4s4z3s1=0;   Uz2s4z3s1=151; Uz3s4z3s1=103; 
Uz4s2z4s1=234; Uz4s3z4s1=128; Uz4s4z4s1=0;   Uz2s4z4s1=138; Uz3s4z4s1=96; 
Uz4s2z4s2=263; Uz4s3z4s2=135; Uz4s4z4s2=0;   Uz2s4z4s2=125; Uz3s4z4s2=89; 
Uz4s2z4s3=135; Uz4s3z4s3=157; Uz4s4z4s3=0;   Uz2s4z4s3=89;  Uz3s4z4s3=67; 
Uz4s2z4s4=0;   Uz4s3z4s4=0;   Uz4s4z4s4=0;   Uz2s4z4s4=0;   Uz3s4z4s4=0; 
Uz4s2z2s4=125; Uz4s3z2s4=89;  Uz4s4z2s4=0;   Uz2s4z2s4=263; Uz3s4z2s4=135; 
Uz4s2z3s4=89;  Uz4s3z3s4=67;  Uz4s4z3s4=0;   Uz2s4z3s4=135; Uz3s4z3s4=157; 

Uz4s2z3s1=205 bedeutet, an Knoten (3,1) wird der konstante Strom zugeführt und dann in Knoten (4,2) die Spannung 205 gemessen.

Als nächstes gebe ich die gesuchten Größen ein: die Leitwerte, und auch die inneren Knotenspannungen, weil diese als Zwischenergebnisse benötigt werden. Gz2s1r ist der von Knoten (2,1) nach rechts abzweigende Leitwert und Gz2s1u der von (2,1) nach unten abzweigt.

#gesuchte Leitwerte:
var ('                   Gz1s1r      Gz1s2r      Gz1s3r            ');
var ('             Gz1s1u      Gz1s2u      Gz1s3u      Gz1s4u      ');
var ('                   Gz2s1r      Gz2s2r      Gz2s3r            ');
var ('             Gz2s1u      Gz2s2u      Gz2s3u      Gz2s4u      ');
var ('                   Gz3s1r      Gz3s2r      Gz3s3r            ');
var ('             Gz3s1u      Gz3s2u      Gz3s3u      Gz3s4u      ');
var ('                   Gz4s1r      Gz4s2r      Gz4s3r            ');
 
#gesuchte Knotenspannungen:
var (' Uz2s2z1s1 Uz2s3z1s1 ');
var (' Uz2s2z1s2 Uz2s3z1s2 ');
var (' Uz2s2z1s3 Uz2s3z1s3 ');
var (' Uz2s2z1s4 Uz2s3z1s4 ');
var (' Uz2s2z2s4 Uz2s3z2s4 ');
var (' Uz2s2z3s4 Uz2s3z3s4 ');
var (' Uz2s2z4s4 Uz2s3z4s4 ');
var (' Uz2s2z4s3 Uz2s3z4s3 ');
var (' Uz2s2z4s2 Uz2s3z4s2 ');
var (' Uz2s2z4s1 Uz2s3z4s1 ');
var (' Uz2s2z3s1 Uz2s3z3s1 ');
var (' Uz2s2z2s1 Uz2s3z2s1 ');
 
var (' Uz3s2z1s1 Uz3s3z1s1 ');
var (' Uz3s2z1s2 Uz3s3z1s2 ');
var (' Uz3s2z1s3 Uz3s3z1s3 ');
var (' Uz3s2z1s4 Uz3s3z1s4 ');
var (' Uz3s2z2s4 Uz3s3z2s4 ');
var (' Uz3s2z3s4 Uz3s3z3s4 ');
var (' Uz3s2z4s4 Uz3s3z4s4 ');
var (' Uz3s2z4s3 Uz3s3z4s3 ');
var (' Uz3s2z4s2 Uz3s3z4s2 ');
var (' Uz3s2z4s1 Uz3s3z4s1 ');
var (' Uz3s2z3s1 Uz3s3z3s1 ');
var (' Uz3s2z2s1 Uz3s3z2s1 ');

Jetzt beginnt die Berechnung mit der Bestimmung der beiden von Knoten (1,1) ausgehenden Leitwerte.

<math>
\begin{tikzpicture}
\definecolor{Violet}{rgb}{0.93,0.51,0.93}

\foreach \i/\j in {
0/3
}
{
\draw[gray,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- +(1,0)
(\i,\j) -- +(0,-0.25)
+(-0.1,-0.25) -- +(-0.1,-0.75) -- +(0.1,-0.75) -- +(0.1,-0.25) -- +(-0.1,-0.25)
+(0,-0.75) -- +(0,-1) ;
\fill[gray,thick] (\i,\j) circle (1.5pt) +(1,0) circle (1.5pt) +(0,-1) circle (1.5pt);
}

\foreach \i/\j in {
0/3
}
{
\draw[Violet,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- +(1,0)
(\i,\j) -- +(0,-0.25)
+(-0.1,-0.25) -- +(-0.1,-0.75) -- +(0.1,-0.75) -- +(0.1,-0.25) -- +(-0.1,-0.25)
+(0,-0.75) -- +(0,-1) ;
\fill[Violet] (\i,\j) circle (1.5pt);
}

\node[left] at (0,3) {(1,1)}

\end{tikzpicture}
</math>

Für diese zwei Variablen stehen insgesamt 12 Gleichungen zur Verfügung. Ich gebe alle Gleichungen ein, weil die Randspannungen in diesem Beispiel exakt gegeben sind. Bei realen Meßwerten würde man die leere Menge als Lösungsmenge erhalten. Da muss man dann Gleichungen entfernen. Anfangs hatte ich auch versucht, nur bestimmte Gleichungen einzugeben, aber keine allgemein brauchbare Auswahl gefunden.

s=solve([
  Gz1s1r*(Uz1s1z1s1-Uz1s2z1s1)+Gz1s1u*(Uz1s1z1s1-Uz2s1z1s1)==1,
  Gz1s1r*(Uz1s1z1s2-Uz1s2z1s2)+Gz1s1u*(Uz1s1z1s2-Uz2s1z1s2)==0,
  Gz1s1r*(Uz1s1z1s3-Uz1s2z1s3)+Gz1s1u*(Uz1s1z1s3-Uz2s1z1s3)==0,
  Gz1s1r*(Uz1s1z1s4-Uz1s2z1s4)+Gz1s1u*(Uz1s1z1s4-Uz2s1z1s4)==0,
  Gz1s1r*(Uz1s1z2s4-Uz1s2z2s4)+Gz1s1u*(Uz1s1z2s4-Uz2s1z2s4)==0,
  Gz1s1r*(Uz1s1z3s4-Uz1s2z3s4)+Gz1s1u*(Uz1s1z3s4-Uz2s1z3s4)==0,
  Gz1s1r*(Uz1s1z4s4-Uz1s2z4s4)+Gz1s1u*(Uz1s1z4s4-Uz2s1z4s4)==0,
  Gz1s1r*(Uz1s1z4s3-Uz1s2z4s3)+Gz1s1u*(Uz1s1z4s3-Uz2s1z4s3)==0,
  Gz1s1r*(Uz1s1z4s2-Uz1s2z4s2)+Gz1s1u*(Uz1s1z4s2-Uz2s1z4s2)==0,
  Gz1s1r*(Uz1s1z4s1-Uz1s2z4s1)+Gz1s1u*(Uz1s1z4s1-Uz2s1z4s1)==0,
  Gz1s1r*(Uz1s1z3s1-Uz1s2z3s1)+Gz1s1u*(Uz1s1z3s1-Uz2s1z3s1)==0,
  Gz1s1r*(Uz1s1z2s1-Uz1s2z2s1)+Gz1s1u*(Uz1s1z2s1-Uz2s1z2s1)==0
  ],
  Gz1s1r, Gz1s1u,
  solution_dict=True);
s;
Gz1s1r=s[0][Gz1s1r]; print('Gz1s1r==',Gz1s1r,';');
Gz1s1u=s[0][Gz1s1u]; print('Gz1s1u==',Gz1s1u,';');
 
Gestartet wird die Berechnung immer mit Shift+Enter, auch bei den vorhergehenden Eingaben. Lösung ist wie erwartet

Gz1s1r==1/224
Gz1s1u==1/224

Dann der nächste Rechenschritt, die von Knoten (1,2) und (2,1) ausgehenden Leitwerte

<math>
\begin{tikzpicture}
\definecolor{Violet}{rgb}{0.93,0.51,0.93}

\foreach \i/\j in {
0/3,
0/2, 1/3
}
{
\draw[gray,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- +(1,0)
(\i,\j) -- +(0,-0.25)
+(-0.1,-0.25) -- +(-0.1,-0.75) -- +(0.1,-0.75) -- +(0.1,-0.25) -- +(-0.1,-0.25)
+(0,-0.75) -- +(0,-1) ;
\fill[gray,thick] (\i,\j) circle (1.5pt) +(1,0) circle (1.5pt) +(0,-1) circle (1.5pt);
}

\foreach \i/\j in {
0/2, 1/3
}
{
\draw[Violet,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- +(1,0)
(\i,\j) -- +(0,-0.25)
+(-0.1,-0.25) -- +(-0.1,-0.75) -- +(0.1,-0.75) -- +(0.1,-0.25) -- +(-0.1,-0.25)
+(0,-0.75) -- +(0,-1) ;
\fill[Violet] (\i,\j) circle (1.5pt);
}

\node[left] at (0,2) {(2,1)}

\end{tikzpicture}
</math>

#Berechne die von Knotenpunkt (1,2), (2,1) ausgehenden Leitwerte Gz1s2r Gz1s2u Gz2s1r Gz2s1u :
s=solve([
  Gz1s2r*(Uz1s2z1s1-Uz1s3z1s1)+Gz1s1r*(Uz1s2z1s1-Uz1s1z1s1)+Gz1s2u*(Uz1s2z1s1-Uz2s2z1s1)==0,
  Gz1s2r*(Uz1s2z1s2-Uz1s3z1s2)+Gz1s1r*(Uz1s2z1s2-Uz1s1z1s2)+Gz1s2u*(Uz1s2z1s2-Uz2s2z1s2)==1,
  Gz1s2r*(Uz1s2z1s3-Uz1s3z1s3)+Gz1s1r*(Uz1s2z1s3-Uz1s1z1s3)+Gz1s2u*(Uz1s2z1s3-Uz2s2z1s3)==0,
  Gz1s2r*(Uz1s2z1s4-Uz1s3z1s4)+Gz1s1r*(Uz1s2z1s4-Uz1s1z1s4)+Gz1s2u*(Uz1s2z1s4-Uz2s2z1s4)==0,
  Gz1s2r*(Uz1s2z2s4-Uz1s3z2s4)+Gz1s1r*(Uz1s2z2s4-Uz1s1z2s4)+Gz1s2u*(Uz1s2z2s4-Uz2s2z2s4)==0,
  Gz1s2r*(Uz1s2z3s4-Uz1s3z3s4)+Gz1s1r*(Uz1s2z3s4-Uz1s1z3s4)+Gz1s2u*(Uz1s2z3s4-Uz2s2z3s4)==0,
  Gz1s2r*(Uz1s2z4s4-Uz1s3z4s4)+Gz1s1r*(Uz1s2z4s4-Uz1s1z4s4)+Gz1s2u*(Uz1s2z4s4-Uz2s2z4s4)==0,
  Gz1s2r*(Uz1s2z4s3-Uz1s3z4s3)+Gz1s1r*(Uz1s2z4s3-Uz1s1z4s3)+Gz1s2u*(Uz1s2z4s3-Uz2s2z4s3)==0,
  Gz1s2r*(Uz1s2z4s2-Uz1s3z4s2)+Gz1s1r*(Uz1s2z4s2-Uz1s1z4s2)+Gz1s2u*(Uz1s2z4s2-Uz2s2z4s2)==0,
  Gz1s2r*(Uz1s2z4s1-Uz1s3z4s1)+Gz1s1r*(Uz1s2z4s1-Uz1s1z4s1)+Gz1s2u*(Uz1s2z4s1-Uz2s2z4s1)==0,
  Gz1s2r*(Uz1s2z3s1-Uz1s3z3s1)+Gz1s1r*(Uz1s2z3s1-Uz1s1z3s1)+Gz1s2u*(Uz1s2z3s1-Uz2s2z3s1)==0,
  Gz1s2r*(Uz1s2z2s1-Uz1s3z2s1)+Gz1s1r*(Uz1s2z2s1-Uz1s1z2s1)+Gz1s2u*(Uz1s2z2s1-Uz2s2z2s1)==0,
  Gz2s1r*(Uz2s1z1s1-Uz2s2z1s1)+Gz1s1u*(Uz2s1z1s1-Uz1s1z1s1)+Gz2s1u*(Uz2s1z1s1-Uz3s1z1s1)==0,
  Gz2s1r*(Uz2s1z1s2-Uz2s2z1s2)+Gz1s1u*(Uz2s1z1s2-Uz1s1z1s2)+Gz2s1u*(Uz2s1z1s2-Uz3s1z1s2)==0,
  Gz2s1r*(Uz2s1z1s3-Uz2s2z1s3)+Gz1s1u*(Uz2s1z1s3-Uz1s1z1s3)+Gz2s1u*(Uz2s1z1s3-Uz3s1z1s3)==0,
  Gz2s1r*(Uz2s1z1s4-Uz2s2z1s4)+Gz1s1u*(Uz2s1z1s4-Uz1s1z1s4)+Gz2s1u*(Uz2s1z1s4-Uz3s1z1s4)==0,
  Gz2s1r*(Uz2s1z2s4-Uz2s2z2s4)+Gz1s1u*(Uz2s1z2s4-Uz1s1z2s4)+Gz2s1u*(Uz2s1z2s4-Uz3s1z2s4)==0,
  Gz2s1r*(Uz2s1z3s4-Uz2s2z3s4)+Gz1s1u*(Uz2s1z3s4-Uz1s1z3s4)+Gz2s1u*(Uz2s1z3s4-Uz3s1z3s4)==0,
  Gz2s1r*(Uz2s1z4s4-Uz2s2z4s4)+Gz1s1u*(Uz2s1z4s4-Uz1s1z4s4)+Gz2s1u*(Uz2s1z4s4-Uz3s1z4s4)==0,
  Gz2s1r*(Uz2s1z4s3-Uz2s2z4s3)+Gz1s1u*(Uz2s1z4s3-Uz1s1z4s3)+Gz2s1u*(Uz2s1z4s3-Uz3s1z4s3)==0,
  Gz2s1r*(Uz2s1z4s2-Uz2s2z4s2)+Gz1s1u*(Uz2s1z4s2-Uz1s1z4s2)+Gz2s1u*(Uz2s1z4s2-Uz3s1z4s2)==0,
  Gz2s1r*(Uz2s1z4s1-Uz2s2z4s1)+Gz1s1u*(Uz2s1z4s1-Uz1s1z4s1)+Gz2s1u*(Uz2s1z4s1-Uz3s1z4s1)==0,
  Gz2s1r*(Uz2s1z3s1-Uz2s2z3s1)+Gz1s1u*(Uz2s1z3s1-Uz1s1z3s1)+Gz2s1u*(Uz2s1z3s1-Uz3s1z3s1)==0,
  Gz2s1r*(Uz2s1z2s1-Uz2s2z2s1)+Gz1s1u*(Uz2s1z2s1-Uz1s1z2s1)+Gz2s1u*(Uz2s1z2s1-Uz3s1z2s1)==1
  ],
  Gz1s2r, Gz1s2u, Gz2s1r, Gz2s1u,
  Uz2s2z1s1, Uz2s2z1s2, Uz2s2z1s3, Uz2s2z1s4, Uz2s2z2s4, Uz2s2z3s4, Uz2s2z4s4, Uz2s2z4s3, Uz2s2z4s2, Uz2s2z4s1, Uz2s2z3s1, Uz2s2z2s1,
  solution_dict=True);
s;
Gz1s2r=s[0][Gz1s2r]; print('Gz1s2r==',Gz1s2r,';');
Gz1s2u=s[0][Gz1s2u]; print('Gz1s2u==',Gz1s2u,';');
Gz2s1r=s[0][Gz2s1r]; print('Gz2s1r==',Gz2s1r,';');
Gz2s1u=s[0][Gz2s1u]; print('Gz2s1u==',Gz2s1u,';');
Uz2s2z1s1=s[0][Uz2s2z1s1]; print('Uz2s2z1s1==',Uz2s2z1s1,';');
Uz2s2z1s2=s[0][Uz2s2z1s2]; print('Uz2s2z1s2==',Uz2s2z1s2,';');
Uz2s2z1s3=s[0][Uz2s2z1s3]; print('Uz2s2z1s3==',Uz2s2z1s3,';');
Uz2s2z1s4=s[0][Uz2s2z1s4]; print('Uz2s2z1s4==',Uz2s2z1s4,';');
Uz2s2z2s4=s[0][Uz2s2z2s4]; print('Uz2s2z2s4==',Uz2s2z2s4,';');
Uz2s2z3s4=s[0][Uz2s2z3s4]; print('Uz2s2z3s4==',Uz2s2z3s4,';');
Uz2s2z4s4=s[0][Uz2s2z4s4]; print('Uz2s2z4s4==',Uz2s2z4s4,';');
Uz2s2z4s3=s[0][Uz2s2z4s3]; print('Uz2s2z4s3==',Uz2s2z4s3,';');
Uz2s2z4s2=s[0][Uz2s2z4s2]; print('Uz2s2z4s2==',Uz2s2z4s2,';');
Uz2s2z4s1=s[0][Uz2s2z4s1]; print('Uz2s2z4s1==',Uz2s2z4s1,';');
Uz2s2z3s1=s[0][Uz2s2z3s1]; print('Uz2s2z3s1==',Uz2s2z3s1,';');
Uz2s2z2s1=s[0][Uz2s2z2s1]; print('Uz2s2z2s1==',Uz2s2z2s1,';');

Da werden auch jede Menge Knotenspannungen mit berechnet. So geht es dann weiter, die von (1,3), (2,2), (3,1) ausgehenden Leitwerte, dann die von (1,4), (2,3), (3,2), (4,1) ausgehenden Leitwerte, also immer parallel zur Diagonale (1,4) --- (4,1).

<math>
\begin{tikzpicture}
\definecolor{Violet}{rgb}{0.93,0.51,0.93}

\foreach \i/\j in {
0/3,
0/2, 1/3,
0/1, 1/2, 2/3
}
{
\draw[gray,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- +(1,0)
(\i,\j) -- +(0,-0.25)
+(-0.1,-0.25) -- +(-0.1,-0.75) -- +(0.1,-0.75) -- +(0.1,-0.25) -- +(-0.1,-0.25)
+(0,-0.75) -- +(0,-1) ;
\fill[gray,thick] (\i,\j) circle (1.5pt) +(1,0) circle (1.5pt) +(0,-1) circle (1.5pt);
}

\foreach \i/\j in {
0/1, 1/2, 2/3
}
{
\draw[Violet,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- +(1,0)
(\i,\j) -- +(0,-0.25)
+(-0.1,-0.25) -- +(-0.1,-0.75) -- +(0.1,-0.75) -- +(0.1,-0.25) -- +(-0.1,-0.25)
+(0,-0.75) -- +(0,-1) ;
\fill[Violet] (\i,\j) circle (1.5pt);
}

\node[left] at (0,1) {(3,1)}

\end{tikzpicture}
</math>


#Berechne die von Knotenpunkt (1,3), (2,2), (3,1) ausgehenden Leitwerte Gz1s3r Gz1s3u Gz2s2r Gz2s2u Gz3s1r Gz3s1u :
s=solve([
  Gz1s3r*(Uz1s3z1s1-Uz1s4z1s1)+Gz1s2r*(Uz1s3z1s1-Uz1s2z1s1)+Gz1s3u*(Uz1s3z1s1-Uz2s3z1s1)==0,
  Gz1s3r*(Uz1s3z1s2-Uz1s4z1s2)+Gz1s2r*(Uz1s3z1s2-Uz1s2z1s2)+Gz1s3u*(Uz1s3z1s2-Uz2s3z1s2)==0,
  Gz1s3r*(Uz1s3z1s3-Uz1s4z1s3)+Gz1s2r*(Uz1s3z1s3-Uz1s2z1s3)+Gz1s3u*(Uz1s3z1s3-Uz2s3z1s3)==1,
  Gz1s3r*(Uz1s3z1s4-Uz1s4z1s4)+Gz1s2r*(Uz1s3z1s4-Uz1s2z1s4)+Gz1s3u*(Uz1s3z1s4-Uz2s3z1s4)==0,
  Gz1s3r*(Uz1s3z2s4-Uz1s4z2s4)+Gz1s2r*(Uz1s3z2s4-Uz1s2z2s4)+Gz1s3u*(Uz1s3z2s4-Uz2s3z2s4)==0,
  Gz1s3r*(Uz1s3z3s4-Uz1s4z3s4)+Gz1s2r*(Uz1s3z3s4-Uz1s2z3s4)+Gz1s3u*(Uz1s3z3s4-Uz2s3z3s4)==0,
  Gz1s3r*(Uz1s3z4s4-Uz1s4z4s4)+Gz1s2r*(Uz1s3z4s4-Uz1s2z4s4)+Gz1s3u*(Uz1s3z4s4-Uz2s3z4s4)==0,
  Gz1s3r*(Uz1s3z4s3-Uz1s4z4s3)+Gz1s2r*(Uz1s3z4s3-Uz1s2z4s3)+Gz1s3u*(Uz1s3z4s3-Uz2s3z4s3)==0,
  Gz1s3r*(Uz1s3z4s2-Uz1s4z4s2)+Gz1s2r*(Uz1s3z4s2-Uz1s2z4s2)+Gz1s3u*(Uz1s3z4s2-Uz2s3z4s2)==0,
  Gz1s3r*(Uz1s3z4s1-Uz1s4z4s1)+Gz1s2r*(Uz1s3z4s1-Uz1s2z4s1)+Gz1s3u*(Uz1s3z4s1-Uz2s3z4s1)==0,
  Gz1s3r*(Uz1s3z3s1-Uz1s4z3s1)+Gz1s2r*(Uz1s3z3s1-Uz1s2z3s1)+Gz1s3u*(Uz1s3z3s1-Uz2s3z3s1)==0,
  Gz1s3r*(Uz1s3z2s1-Uz1s4z2s1)+Gz1s2r*(Uz1s3z2s1-Uz1s2z2s1)+Gz1s3u*(Uz1s3z2s1-Uz2s3z2s1)==0,
  Gz2s2r*(Uz2s2z1s1-Uz2s3z1s1)+Gz1s2u*(Uz2s2z1s1-Uz1s2z1s1)+Gz2s1r*(Uz2s2z1s1-Uz2s1z1s1)+Gz2s2u*(Uz2s2z1s1-Uz3s2z1s1)==0,
  Gz2s2r*(Uz2s2z1s2-Uz2s3z1s2)+Gz1s2u*(Uz2s2z1s2-Uz1s2z1s2)+Gz2s1r*(Uz2s2z1s2-Uz2s1z1s2)+Gz2s2u*(Uz2s2z1s2-Uz3s2z1s2)==0,
  Gz2s2r*(Uz2s2z1s3-Uz2s3z1s3)+Gz1s2u*(Uz2s2z1s3-Uz1s2z1s3)+Gz2s1r*(Uz2s2z1s3-Uz2s1z1s3)+Gz2s2u*(Uz2s2z1s3-Uz3s2z1s3)==0,
  Gz2s2r*(Uz2s2z1s4-Uz2s3z1s4)+Gz1s2u*(Uz2s2z1s4-Uz1s2z1s4)+Gz2s1r*(Uz2s2z1s4-Uz2s1z1s4)+Gz2s2u*(Uz2s2z1s4-Uz3s2z1s4)==0,
  Gz2s2r*(Uz2s2z2s4-Uz2s3z2s4)+Gz1s2u*(Uz2s2z2s4-Uz1s2z2s4)+Gz2s1r*(Uz2s2z2s4-Uz2s1z2s4)+Gz2s2u*(Uz2s2z2s4-Uz3s2z2s4)==0,
  Gz2s2r*(Uz2s2z3s4-Uz2s3z3s4)+Gz1s2u*(Uz2s2z3s4-Uz1s2z3s4)+Gz2s1r*(Uz2s2z3s4-Uz2s1z3s4)+Gz2s2u*(Uz2s2z3s4-Uz3s2z3s4)==0,
  Gz2s2r*(Uz2s2z4s4-Uz2s3z4s4)+Gz1s2u*(Uz2s2z4s4-Uz1s2z4s4)+Gz2s1r*(Uz2s2z4s4-Uz2s1z4s4)+Gz2s2u*(Uz2s2z4s4-Uz3s2z4s4)==0,
  Gz2s2r*(Uz2s2z4s3-Uz2s3z4s3)+Gz1s2u*(Uz2s2z4s3-Uz1s2z4s3)+Gz2s1r*(Uz2s2z4s3-Uz2s1z4s3)+Gz2s2u*(Uz2s2z4s3-Uz3s2z4s3)==0,
  Gz2s2r*(Uz2s2z4s2-Uz2s3z4s2)+Gz1s2u*(Uz2s2z4s2-Uz1s2z4s2)+Gz2s1r*(Uz2s2z4s2-Uz2s1z4s2)+Gz2s2u*(Uz2s2z4s2-Uz3s2z4s2)==0,
  Gz2s2r*(Uz2s2z4s1-Uz2s3z4s1)+Gz1s2u*(Uz2s2z4s1-Uz1s2z4s1)+Gz2s1r*(Uz2s2z4s1-Uz2s1z4s1)+Gz2s2u*(Uz2s2z4s1-Uz3s2z4s1)==0,
  Gz2s2r*(Uz2s2z3s1-Uz2s3z3s1)+Gz1s2u*(Uz2s2z3s1-Uz1s2z3s1)+Gz2s1r*(Uz2s2z3s1-Uz2s1z3s1)+Gz2s2u*(Uz2s2z3s1-Uz3s2z3s1)==0,
  Gz2s2r*(Uz2s2z2s1-Uz2s3z2s1)+Gz1s2u*(Uz2s2z2s1-Uz1s2z2s1)+Gz2s1r*(Uz2s2z2s1-Uz2s1z2s1)+Gz2s2u*(Uz2s2z2s1-Uz3s2z2s1)==0,
  Gz3s1r*(Uz3s1z1s1-Uz3s2z1s1)+Gz2s1u*(Uz3s1z1s1-Uz2s1z1s1)+Gz3s1u*(Uz3s1z1s1-Uz4s1z1s1)==0,
  Gz3s1r*(Uz3s1z1s2-Uz3s2z1s2)+Gz2s1u*(Uz3s1z1s2-Uz2s1z1s2)+Gz3s1u*(Uz3s1z1s2-Uz4s1z1s2)==0,
  Gz3s1r*(Uz3s1z1s3-Uz3s2z1s3)+Gz2s1u*(Uz3s1z1s3-Uz2s1z1s3)+Gz3s1u*(Uz3s1z1s3-Uz4s1z1s3)==0,
  Gz3s1r*(Uz3s1z1s4-Uz3s2z1s4)+Gz2s1u*(Uz3s1z1s4-Uz2s1z1s4)+Gz3s1u*(Uz3s1z1s4-Uz4s1z1s4)==0,
  Gz3s1r*(Uz3s1z2s4-Uz3s2z2s4)+Gz2s1u*(Uz3s1z2s4-Uz2s1z2s4)+Gz3s1u*(Uz3s1z2s4-Uz4s1z2s4)==0,
  Gz3s1r*(Uz3s1z3s4-Uz3s2z3s4)+Gz2s1u*(Uz3s1z3s4-Uz2s1z3s4)+Gz3s1u*(Uz3s1z3s4-Uz4s1z3s4)==0,
  Gz3s1r*(Uz3s1z4s4-Uz3s2z4s4)+Gz2s1u*(Uz3s1z4s4-Uz2s1z4s4)+Gz3s1u*(Uz3s1z4s4-Uz4s1z4s4)==0,
  Gz3s1r*(Uz3s1z4s3-Uz3s2z4s3)+Gz2s1u*(Uz3s1z4s3-Uz2s1z4s3)+Gz3s1u*(Uz3s1z4s3-Uz4s1z4s3)==0,
  Gz3s1r*(Uz3s1z4s2-Uz3s2z4s2)+Gz2s1u*(Uz3s1z4s2-Uz2s1z4s2)+Gz3s1u*(Uz3s1z4s2-Uz4s1z4s2)==0,
  Gz3s1r*(Uz3s1z4s1-Uz3s2z4s1)+Gz2s1u*(Uz3s1z4s1-Uz2s1z4s1)+Gz3s1u*(Uz3s1z4s1-Uz4s1z4s1)==0,
  Gz3s1r*(Uz3s1z3s1-Uz3s2z3s1)+Gz2s1u*(Uz3s1z3s1-Uz2s1z3s1)+Gz3s1u*(Uz3s1z3s1-Uz4s1z3s1)==1,
  Gz3s1r*(Uz3s1z2s1-Uz3s2z2s1)+Gz2s1u*(Uz3s1z2s1-Uz2s1z2s1)+Gz3s1u*(Uz3s1z2s1-Uz4s1z2s1)==0
  ],
  Gz1s3r, Gz1s3u, Gz2s2r, Gz2s2u, Gz3s1r, Gz3s1u,
  Uz2s3z1s1, Uz2s3z1s2, Uz2s3z1s3, Uz2s3z1s4, Uz2s3z2s4, Uz2s3z3s4, Uz2s3z4s4, Uz2s3z4s3, Uz2s3z4s2, Uz2s3z4s1, Uz2s3z3s1, Uz2s3z2s1,
  Uz3s2z1s1, Uz3s2z1s2, Uz3s2z1s3, Uz3s2z1s4, Uz3s2z2s4, Uz3s2z3s4, Uz3s2z4s4, Uz3s2z4s3, Uz3s2z4s2, Uz3s2z4s1, Uz3s2z3s1, Uz3s2z2s1,
  solution_dict=True);
s;
Gz1s3r=s[0][Gz1s3r]; print('Gz1s3r==',Gz1s3r,';');
Gz1s3u=s[0][Gz1s3u]; print('Gz1s3u==',Gz1s3u,';');
Gz2s2r=s[0][Gz2s2r]; print('Gz2s2r==',Gz2s2r,';');
Gz2s2u=s[0][Gz2s2u]; print('Gz2s2u==',Gz2s2u,';');
Gz3s1r=s[0][Gz3s1r]; print('Gz3s1r==',Gz3s1r,';');
Gz3s1u=s[0][Gz3s1u]; print('Gz3s1u==',Gz3s1u,';');
Uz2s3z1s1=s[0][Uz2s3z1s1]; print('Uz2s3z1s1==',Uz2s3z1s1,';');
Uz2s3z1s2=s[0][Uz2s3z1s2]; print('Uz2s3z1s2==',Uz2s3z1s2,';');
Uz2s3z1s3=s[0][Uz2s3z1s3]; print('Uz2s3z1s3==',Uz2s3z1s3,';');
Uz2s3z1s4=s[0][Uz2s3z1s4]; print('Uz2s3z1s4==',Uz2s3z1s4,';');
Uz2s3z2s4=s[0][Uz2s3z2s4]; print('Uz2s3z2s4==',Uz2s3z2s4,';');
Uz2s3z3s4=s[0][Uz2s3z3s4]; print('Uz2s3z3s4==',Uz2s3z3s4,';');
Uz2s3z4s4=s[0][Uz2s3z4s4]; print('Uz2s3z4s4==',Uz2s3z4s4,';');
Uz2s3z4s3=s[0][Uz2s3z4s3]; print('Uz2s3z4s3==',Uz2s3z4s3,';');
Uz2s3z4s2=s[0][Uz2s3z4s2]; print('Uz2s3z4s2==',Uz2s3z4s2,';');
Uz2s3z4s1=s[0][Uz2s3z4s1]; print('Uz2s3z4s1==',Uz2s3z4s1,';');
Uz2s3z3s1=s[0][Uz2s3z3s1]; print('Uz2s3z3s1==',Uz2s3z3s1,';');
Uz2s3z2s1=s[0][Uz2s3z2s1]; print('Uz2s3z2s1==',Uz2s3z2s1,';');
Uz3s2z1s1=s[0][Uz3s2z1s1]; print('Uz3s2z1s1==',Uz3s2z1s1,';');
Uz3s2z1s2=s[0][Uz3s2z1s2]; print('Uz3s2z1s2==',Uz3s2z1s2,';');
Uz3s2z1s3=s[0][Uz3s2z1s3]; print('Uz3s2z1s3==',Uz3s2z1s3,';');
Uz3s2z1s4=s[0][Uz3s2z1s4]; print('Uz3s2z1s4==',Uz3s2z1s4,';');
Uz3s2z2s4=s[0][Uz3s2z2s4]; print('Uz3s2z2s4==',Uz3s2z2s4,';');
Uz3s2z3s4=s[0][Uz3s2z3s4]; print('Uz3s2z3s4==',Uz3s2z3s4,';');
Uz3s2z4s4=s[0][Uz3s2z4s4]; print('Uz3s2z4s4==',Uz3s2z4s4,';');
Uz3s2z4s3=s[0][Uz3s2z4s3]; print('Uz3s2z4s3==',Uz3s2z4s3,';');
Uz3s2z4s2=s[0][Uz3s2z4s2]; print('Uz3s2z4s2==',Uz3s2z4s2,';');
Uz3s2z4s1=s[0][Uz3s2z4s1]; print('Uz3s2z4s1==',Uz3s2z4s1,';');
Uz3s2z3s1=s[0][Uz3s2z3s1]; print('Uz3s2z3s1==',Uz3s2z3s1,';');
Uz3s2z2s1=s[0][Uz3s2z2s1]; print('Uz3s2z2s1==',Uz3s2z2s1,';');
 

<math>
\begin{tikzpicture}
\definecolor{Violet}{rgb}{0.93,0.51,0.93}

\foreach \i/\j in {
0/3,
0/2, 1/3,
0/1, 1/2, 2/3
}
{
\draw[gray,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- +(1,0)
(\i,\j) -- +(0,-0.25)
+(-0.1,-0.25) -- +(-0.1,-0.75) -- +(0.1,-0.75) -- +(0.1,-0.25) -- +(-0.1,-0.25)
+(0,-0.75) -- +(0,-1) ;
\fill[gray,thick] (\i,\j) circle (1.5pt) +(1,0) circle (1.5pt) +(0,-1) circle (1.5pt);
}

\foreach \i/\j in {
0/0, 1/1, 2/2
}
{
\draw[gray,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- ++(1,0)
-- +(0,0.25)
+(-0.1,0.25) -- +(-0.1,0.75) -- +(0.1,0.75) -- +(0.1,0.25) -- +(-0.1,0.25)
+(0,0.75) -- +(0,1) ;
\fill[gray] (\i,\j) circle (1.5pt) +(1,0) circle (1.5pt) +(0,1) circle (1.5pt);
}

\foreach \i/\j in {
0/0, 1/1, 2/2
}
{
\draw[Violet,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- ++(1,0)
-- +(0,0.25)
+(-0.1,0.25) -- +(-0.1,0.75) -- +(0.1,0.75) -- +(0.1,0.25) -- +(-0.1,0.25)
+(0,0.75) -- +(0,1) ;
\fill[Violet] (\i,\j) circle (1.5pt) +(1,1) circle (1.5pt);
}

\node[left] at (0,0) {(4,1)}


\end{tikzpicture}
</math>



#Berechne die von Knotenpunkt (1,4), (2,3), (3,2), (4,1) ausgehenden Leitwerte Gz1s4u Gz2s3r Gz2s3u Gz3s2r Gz3s2u Gz4s1r :
s=solve([
  0+Gz1s3r*(Uz1s4z1s1-Uz1s3z1s1)+Gz1s4u*(Uz1s4z1s1-Uz2s4z1s1)==0,
  0+Gz1s3r*(Uz1s4z1s2-Uz1s3z1s2)+Gz1s4u*(Uz1s4z1s2-Uz2s4z1s2)==0,
  0+Gz1s3r*(Uz1s4z1s3-Uz1s3z1s3)+Gz1s4u*(Uz1s4z1s3-Uz2s4z1s3)==0,
  0+Gz1s3r*(Uz1s4z1s4-Uz1s3z1s4)+Gz1s4u*(Uz1s4z1s4-Uz2s4z1s4)==1,
  0+Gz1s3r*(Uz1s4z2s4-Uz1s3z2s4)+Gz1s4u*(Uz1s4z2s4-Uz2s4z2s4)==0,
  0+Gz1s3r*(Uz1s4z3s4-Uz1s3z3s4)+Gz1s4u*(Uz1s4z3s4-Uz2s4z3s4)==0,
  0+Gz1s3r*(Uz1s4z4s4-Uz1s3z4s4)+Gz1s4u*(Uz1s4z4s4-Uz2s4z4s4)==0,
  0+Gz1s3r*(Uz1s4z4s3-Uz1s3z4s3)+Gz1s4u*(Uz1s4z4s3-Uz2s4z4s3)==0,
  0+Gz1s3r*(Uz1s4z4s2-Uz1s3z4s2)+Gz1s4u*(Uz1s4z4s2-Uz2s4z4s2)==0,
  0+Gz1s3r*(Uz1s4z4s1-Uz1s3z4s1)+Gz1s4u*(Uz1s4z4s1-Uz2s4z4s1)==0,
  0+Gz1s3r*(Uz1s4z3s1-Uz1s3z3s1)+Gz1s4u*(Uz1s4z3s1-Uz2s4z3s1)==0,
  0+Gz1s3r*(Uz1s4z2s1-Uz1s3z2s1)+Gz1s4u*(Uz1s4z2s1-Uz2s4z2s1)==0,
  Gz2s3r*(Uz2s3z1s1-Uz2s4z1s1)+Gz1s3u*(Uz2s3z1s1-Uz1s3z1s1)+Gz2s2r*(Uz2s3z1s1-Uz2s2z1s1)+Gz2s3u*(Uz2s3z1s1-Uz3s3z1s1)==0,
  Gz2s3r*(Uz2s3z1s2-Uz2s4z1s2)+Gz1s3u*(Uz2s3z1s2-Uz1s3z1s2)+Gz2s2r*(Uz2s3z1s2-Uz2s2z1s2)+Gz2s3u*(Uz2s3z1s2-Uz3s3z1s2)==0,
  Gz2s3r*(Uz2s3z1s3-Uz2s4z1s3)+Gz1s3u*(Uz2s3z1s3-Uz1s3z1s3)+Gz2s2r*(Uz2s3z1s3-Uz2s2z1s3)+Gz2s3u*(Uz2s3z1s3-Uz3s3z1s3)==0,
  Gz2s3r*(Uz2s3z1s4-Uz2s4z1s4)+Gz1s3u*(Uz2s3z1s4-Uz1s3z1s4)+Gz2s2r*(Uz2s3z1s4-Uz2s2z1s4)+Gz2s3u*(Uz2s3z1s4-Uz3s3z1s4)==0,
  Gz2s3r*(Uz2s3z2s4-Uz2s4z2s4)+Gz1s3u*(Uz2s3z2s4-Uz1s3z2s4)+Gz2s2r*(Uz2s3z2s4-Uz2s2z2s4)+Gz2s3u*(Uz2s3z2s4-Uz3s3z2s4)==0,
  Gz2s3r*(Uz2s3z3s4-Uz2s4z3s4)+Gz1s3u*(Uz2s3z3s4-Uz1s3z3s4)+Gz2s2r*(Uz2s3z3s4-Uz2s2z3s4)+Gz2s3u*(Uz2s3z3s4-Uz3s3z3s4)==0,
  Gz2s3r*(Uz2s3z4s4-Uz2s4z4s4)+Gz1s3u*(Uz2s3z4s4-Uz1s3z4s4)+Gz2s2r*(Uz2s3z4s4-Uz2s2z4s4)+Gz2s3u*(Uz2s3z4s4-Uz3s3z4s4)==0,
  Gz2s3r*(Uz2s3z4s3-Uz2s4z4s3)+Gz1s3u*(Uz2s3z4s3-Uz1s3z4s3)+Gz2s2r*(Uz2s3z4s3-Uz2s2z4s3)+Gz2s3u*(Uz2s3z4s3-Uz3s3z4s3)==0,
  Gz2s3r*(Uz2s3z4s2-Uz2s4z4s2)+Gz1s3u*(Uz2s3z4s2-Uz1s3z4s2)+Gz2s2r*(Uz2s3z4s2-Uz2s2z4s2)+Gz2s3u*(Uz2s3z4s2-Uz3s3z4s2)==0,
  Gz2s3r*(Uz2s3z4s1-Uz2s4z4s1)+Gz1s3u*(Uz2s3z4s1-Uz1s3z4s1)+Gz2s2r*(Uz2s3z4s1-Uz2s2z4s1)+Gz2s3u*(Uz2s3z4s1-Uz3s3z4s1)==0,
  Gz2s3r*(Uz2s3z3s1-Uz2s4z3s1)+Gz1s3u*(Uz2s3z3s1-Uz1s3z3s1)+Gz2s2r*(Uz2s3z3s1-Uz2s2z3s1)+Gz2s3u*(Uz2s3z3s1-Uz3s3z3s1)==0,
  Gz2s3r*(Uz2s3z2s1-Uz2s4z2s1)+Gz1s3u*(Uz2s3z2s1-Uz1s3z2s1)+Gz2s2r*(Uz2s3z2s1-Uz2s2z2s1)+Gz2s3u*(Uz2s3z2s1-Uz3s3z2s1)==0,
  Gz3s2r*(Uz3s2z1s1-Uz3s3z1s1)+Gz2s2u*(Uz3s2z1s1-Uz2s2z1s1)+Gz3s1r*(Uz3s2z1s1-Uz3s1z1s1)+Gz3s2u*(Uz3s2z1s1-Uz4s2z1s1)==0,
  Gz3s2r*(Uz3s2z1s2-Uz3s3z1s2)+Gz2s2u*(Uz3s2z1s2-Uz2s2z1s2)+Gz3s1r*(Uz3s2z1s2-Uz3s1z1s2)+Gz3s2u*(Uz3s2z1s2-Uz4s2z1s2)==0,
  Gz3s2r*(Uz3s2z1s3-Uz3s3z1s3)+Gz2s2u*(Uz3s2z1s3-Uz2s2z1s3)+Gz3s1r*(Uz3s2z1s3-Uz3s1z1s3)+Gz3s2u*(Uz3s2z1s3-Uz4s2z1s3)==0,
  Gz3s2r*(Uz3s2z1s4-Uz3s3z1s4)+Gz2s2u*(Uz3s2z1s4-Uz2s2z1s4)+Gz3s1r*(Uz3s2z1s4-Uz3s1z1s4)+Gz3s2u*(Uz3s2z1s4-Uz4s2z1s4)==0,
  Gz3s2r*(Uz3s2z2s4-Uz3s3z2s4)+Gz2s2u*(Uz3s2z2s4-Uz2s2z2s4)+Gz3s1r*(Uz3s2z2s4-Uz3s1z2s4)+Gz3s2u*(Uz3s2z2s4-Uz4s2z2s4)==0,
  Gz3s2r*(Uz3s2z3s4-Uz3s3z3s4)+Gz2s2u*(Uz3s2z3s4-Uz2s2z3s4)+Gz3s1r*(Uz3s2z3s4-Uz3s1z3s4)+Gz3s2u*(Uz3s2z3s4-Uz4s2z3s4)==0,
  Gz3s2r*(Uz3s2z4s4-Uz3s3z4s4)+Gz2s2u*(Uz3s2z4s4-Uz2s2z4s4)+Gz3s1r*(Uz3s2z4s4-Uz3s1z4s4)+Gz3s2u*(Uz3s2z4s4-Uz4s2z4s4)==0,
  Gz3s2r*(Uz3s2z4s3-Uz3s3z4s3)+Gz2s2u*(Uz3s2z4s3-Uz2s2z4s3)+Gz3s1r*(Uz3s2z4s3-Uz3s1z4s3)+Gz3s2u*(Uz3s2z4s3-Uz4s2z4s3)==0,
  Gz3s2r*(Uz3s2z4s2-Uz3s3z4s2)+Gz2s2u*(Uz3s2z4s2-Uz2s2z4s2)+Gz3s1r*(Uz3s2z4s2-Uz3s1z4s2)+Gz3s2u*(Uz3s2z4s2-Uz4s2z4s2)==0,
  Gz3s2r*(Uz3s2z4s1-Uz3s3z4s1)+Gz2s2u*(Uz3s2z4s1-Uz2s2z4s1)+Gz3s1r*(Uz3s2z4s1-Uz3s1z4s1)+Gz3s2u*(Uz3s2z4s1-Uz4s2z4s1)==0,
  Gz3s2r*(Uz3s2z3s1-Uz3s3z3s1)+Gz2s2u*(Uz3s2z3s1-Uz2s2z3s1)+Gz3s1r*(Uz3s2z3s1-Uz3s1z3s1)+Gz3s2u*(Uz3s2z3s1-Uz4s2z3s1)==0,
  Gz3s2r*(Uz3s2z2s1-Uz3s3z2s1)+Gz2s2u*(Uz3s2z2s1-Uz2s2z2s1)+Gz3s1r*(Uz3s2z2s1-Uz3s1z2s1)+Gz3s2u*(Uz3s2z2s1-Uz4s2z2s1)==0,
  Gz4s1r*(Uz4s1z1s1-Uz4s2z1s1)+Gz3s1u*(Uz4s1z1s1-Uz3s1z1s1)==0,
  Gz4s1r*(Uz4s1z1s2-Uz4s2z1s2)+Gz3s1u*(Uz4s1z1s2-Uz3s1z1s2)==0,
  Gz4s1r*(Uz4s1z1s3-Uz4s2z1s3)+Gz3s1u*(Uz4s1z1s3-Uz3s1z1s3)==0,
  Gz4s1r*(Uz4s1z1s4-Uz4s2z1s4)+Gz3s1u*(Uz4s1z1s4-Uz3s1z1s4)==0,
  Gz4s1r*(Uz4s1z2s4-Uz4s2z2s4)+Gz3s1u*(Uz4s1z2s4-Uz3s1z2s4)==0,
  Gz4s1r*(Uz4s1z3s4-Uz4s2z3s4)+Gz3s1u*(Uz4s1z3s4-Uz3s1z3s4)==0,
  Gz4s1r*(Uz4s1z4s4-Uz4s2z4s4)+Gz3s1u*(Uz4s1z4s4-Uz3s1z4s4)==0,
  Gz4s1r*(Uz4s1z4s3-Uz4s2z4s3)+Gz3s1u*(Uz4s1z4s3-Uz3s1z4s3)==0,
  Gz4s1r*(Uz4s1z4s2-Uz4s2z4s2)+Gz3s1u*(Uz4s1z4s2-Uz3s1z4s2)==0,
  Gz4s1r*(Uz4s1z4s1-Uz4s2z4s1)+Gz3s1u*(Uz4s1z4s1-Uz3s1z4s1)==1,
  Gz4s1r*(Uz4s1z3s1-Uz4s2z3s1)+Gz3s1u*(Uz4s1z3s1-Uz3s1z3s1)==0,
  Gz4s1r*(Uz4s1z2s1-Uz4s2z2s1)+Gz3s1u*(Uz4s1z2s1-Uz3s1z2s1)==0
  ],
  Gz1s4u, Gz2s3r, Gz2s3u, Gz3s2r, Gz3s2u, Gz4s1r,
  Uz3s3z1s1, Uz3s3z1s2, Uz3s3z1s3, Uz3s3z1s4, Uz3s3z2s4, Uz3s3z3s4, Uz3s3z4s4, Uz3s3z4s3, Uz3s3z4s2, Uz3s3z4s1, Uz3s3z3s1, Uz3s3z2s1,
  solution_dict=True);
s;
Gz1s4u=s[0][Gz1s4u]; print('Gz1s4u==',Gz1s4u,';');
Gz2s3r=s[0][Gz2s3r]; print('Gz2s3r==',Gz2s3r,';');
Gz2s3u=s[0][Gz2s3u]; print('Gz2s3u==',Gz2s3u,';');
Gz3s2r=s[0][Gz3s2r]; print('Gz3s2r==',Gz3s2r,';');
Gz3s2u=s[0][Gz3s2u]; print('Gz3s2u==',Gz3s2u,';');
Gz4s1r=s[0][Gz4s1r]; print('Gz4s1r==',Gz4s1r,';');
Uz3s3z1s1=s[0][Uz3s3z1s1]; print('Uz3s3z1s1==',Uz3s3z1s1,';');
Uz3s3z1s2=s[0][Uz3s3z1s2]; print('Uz3s3z1s2==',Uz3s3z1s2,';');
Uz3s3z1s3=s[0][Uz3s3z1s3]; print('Uz3s3z1s3==',Uz3s3z1s3,';');
Uz3s3z1s4=s[0][Uz3s3z1s4]; print('Uz3s3z1s4==',Uz3s3z1s4,';');
Uz3s3z2s4=s[0][Uz3s3z2s4]; print('Uz3s3z2s4==',Uz3s3z2s4,';');
Uz3s3z3s4=s[0][Uz3s3z3s4]; print('Uz3s3z3s4==',Uz3s3z3s4,';');
Uz3s3z4s4=s[0][Uz3s3z4s4]; print('Uz3s3z4s4==',Uz3s3z4s4,';');
Uz3s3z4s3=s[0][Uz3s3z4s3]; print('Uz3s3z4s3==',Uz3s3z4s3,';');
Uz3s3z4s2=s[0][Uz3s3z4s2]; print('Uz3s3z4s2==',Uz3s3z4s2,';');
Uz3s3z4s1=s[0][Uz3s3z4s1]; print('Uz3s3z4s1==',Uz3s3z4s1,';');
Uz3s3z3s1=s[0][Uz3s3z3s1]; print('Uz3s3z3s1==',Uz3s3z3s1,';');
Uz3s3z2s1=s[0][Uz3s3z2s1]; print('Uz3s3z2s1==',Uz3s3z2s1,';');
 

<math>
\begin{tikzpicture}
\definecolor{Violet}{rgb}{0.93,0.51,0.93}

\foreach \i/\j in {
0/3,
0/2, 1/3,
0/1, 1/2, 2/3
}
{
\draw[gray,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- +(1,0)
(\i,\j) -- +(0,-0.25)
+(-0.1,-0.25) -- +(-0.1,-0.75) -- +(0.1,-0.75) -- +(0.1,-0.25) -- +(-0.1,-0.25)
+(0,-0.75) -- +(0,-1) ;
\fill[gray,thick] (\i,\j) circle (1.5pt) +(1,0) circle (1.5pt) +(0,-1) circle (1.5pt);
}

\foreach \i/\j in {
0/0, 1/1, 2/2,
1/0, 2/1
}
{
\draw[gray,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- ++(1,0)
-- +(0,0.25)
+(-0.1,0.25) -- +(-0.1,0.75) -- +(0.1,0.75) -- +(0.1,0.25) -- +(-0.1,0.25)
+(0,0.75) -- +(0,1) ;
\fill[gray] (\i,\j) circle (1.5pt) +(1,0) circle (1.5pt) +(0,1) circle (1.5pt);
}

\foreach \i/\j in {
1/0, 2/1
}
{
\draw[Violet,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- ++(1,0)
-- +(0,0.25)
+(-0.1,0.25) -- +(-0.1,0.75) -- +(0.1,0.75) -- +(0.1,0.25) -- +(-0.1,0.25)
+(0,0.75) -- +(0,1) ;
\fill[Violet] (\i,\j) circle (1.5pt) +(1,1) circle (1.5pt);
}

\node[below] at (1,0) {(4,2)}

\end{tikzpicture}
</math>

#Berechne die von Knotenpunkt (2,4), (3,3), (4,2) ausgehenden Leitwerte Gz2s4u Gz3s3r Gz3s3u Gz4s2r :
s=solve([
  0+Gz1s4u*(Uz2s4z1s1-Uz1s4z1s1)+Gz2s3r*(Uz2s4z1s1-Uz2s3z1s1)+Gz2s4u*(Uz2s4z1s1-Uz3s4z1s1)==0,
  0+Gz1s4u*(Uz2s4z1s2-Uz1s4z1s2)+Gz2s3r*(Uz2s4z1s2-Uz2s3z1s2)+Gz2s4u*(Uz2s4z1s2-Uz3s4z1s2)==0,
  0+Gz1s4u*(Uz2s4z1s3-Uz1s4z1s3)+Gz2s3r*(Uz2s4z1s3-Uz2s3z1s3)+Gz2s4u*(Uz2s4z1s3-Uz3s4z1s3)==0,
  0+Gz1s4u*(Uz2s4z1s4-Uz1s4z1s4)+Gz2s3r*(Uz2s4z1s4-Uz2s3z1s4)+Gz2s4u*(Uz2s4z1s4-Uz3s4z1s4)==0,
  0+Gz1s4u*(Uz2s4z2s4-Uz1s4z2s4)+Gz2s3r*(Uz2s4z2s4-Uz2s3z2s4)+Gz2s4u*(Uz2s4z2s4-Uz3s4z2s4)==1,
  0+Gz1s4u*(Uz2s4z3s4-Uz1s4z3s4)+Gz2s3r*(Uz2s4z3s4-Uz2s3z3s4)+Gz2s4u*(Uz2s4z3s4-Uz3s4z3s4)==0,
  0+Gz1s4u*(Uz2s4z4s4-Uz1s4z4s4)+Gz2s3r*(Uz2s4z4s4-Uz2s3z4s4)+Gz2s4u*(Uz2s4z4s4-Uz3s4z4s4)==0,
  0+Gz1s4u*(Uz2s4z4s3-Uz1s4z4s3)+Gz2s3r*(Uz2s4z4s3-Uz2s3z4s3)+Gz2s4u*(Uz2s4z4s3-Uz3s4z4s3)==0,
  0+Gz1s4u*(Uz2s4z4s2-Uz1s4z4s2)+Gz2s3r*(Uz2s4z4s2-Uz2s3z4s2)+Gz2s4u*(Uz2s4z4s2-Uz3s4z4s2)==0,
  0+Gz1s4u*(Uz2s4z4s1-Uz1s4z4s1)+Gz2s3r*(Uz2s4z4s1-Uz2s3z4s1)+Gz2s4u*(Uz2s4z4s1-Uz3s4z4s1)==0,
  0+Gz1s4u*(Uz2s4z3s1-Uz1s4z3s1)+Gz2s3r*(Uz2s4z3s1-Uz2s3z3s1)+Gz2s4u*(Uz2s4z3s1-Uz3s4z3s1)==0,
  0+Gz1s4u*(Uz2s4z2s1-Uz1s4z2s1)+Gz2s3r*(Uz2s4z2s1-Uz2s3z2s1)+Gz2s4u*(Uz2s4z2s1-Uz3s4z2s1)==0,
  Gz3s3r*(Uz3s3z1s1-Uz3s4z1s1)+Gz2s3u*(Uz3s3z1s1-Uz2s3z1s1)+Gz3s2r*(Uz3s3z1s1-Uz3s2z1s1)+Gz3s3u*(Uz3s3z1s1-Uz4s3z1s1)==0,
  Gz3s3r*(Uz3s3z1s2-Uz3s4z1s2)+Gz2s3u*(Uz3s3z1s2-Uz2s3z1s2)+Gz3s2r*(Uz3s3z1s2-Uz3s2z1s2)+Gz3s3u*(Uz3s3z1s2-Uz4s3z1s2)==0,
  Gz3s3r*(Uz3s3z1s3-Uz3s4z1s3)+Gz2s3u*(Uz3s3z1s3-Uz2s3z1s3)+Gz3s2r*(Uz3s3z1s3-Uz3s2z1s3)+Gz3s3u*(Uz3s3z1s3-Uz4s3z1s3)==0,
  Gz3s3r*(Uz3s3z1s4-Uz3s4z1s4)+Gz2s3u*(Uz3s3z1s4-Uz2s3z1s4)+Gz3s2r*(Uz3s3z1s4-Uz3s2z1s4)+Gz3s3u*(Uz3s3z1s4-Uz4s3z1s4)==0,
  Gz3s3r*(Uz3s3z2s4-Uz3s4z2s4)+Gz2s3u*(Uz3s3z2s4-Uz2s3z2s4)+Gz3s2r*(Uz3s3z2s4-Uz3s2z2s4)+Gz3s3u*(Uz3s3z2s4-Uz4s3z2s4)==0,
  Gz3s3r*(Uz3s3z3s4-Uz3s4z3s4)+Gz2s3u*(Uz3s3z3s4-Uz2s3z3s4)+Gz3s2r*(Uz3s3z3s4-Uz3s2z3s4)+Gz3s3u*(Uz3s3z3s4-Uz4s3z3s4)==0,
  Gz3s3r*(Uz3s3z4s4-Uz3s4z4s4)+Gz2s3u*(Uz3s3z4s4-Uz2s3z4s4)+Gz3s2r*(Uz3s3z4s4-Uz3s2z4s4)+Gz3s3u*(Uz3s3z4s4-Uz4s3z4s4)==0,
  Gz3s3r*(Uz3s3z4s3-Uz3s4z4s3)+Gz2s3u*(Uz3s3z4s3-Uz2s3z4s3)+Gz3s2r*(Uz3s3z4s3-Uz3s2z4s3)+Gz3s3u*(Uz3s3z4s3-Uz4s3z4s3)==0,
  Gz3s3r*(Uz3s3z4s2-Uz3s4z4s2)+Gz2s3u*(Uz3s3z4s2-Uz2s3z4s2)+Gz3s2r*(Uz3s3z4s2-Uz3s2z4s2)+Gz3s3u*(Uz3s3z4s2-Uz4s3z4s2)==0,
  Gz3s3r*(Uz3s3z4s1-Uz3s4z4s1)+Gz2s3u*(Uz3s3z4s1-Uz2s3z4s1)+Gz3s2r*(Uz3s3z4s1-Uz3s2z4s1)+Gz3s3u*(Uz3s3z4s1-Uz4s3z4s1)==0,
  Gz3s3r*(Uz3s3z3s1-Uz3s4z3s1)+Gz2s3u*(Uz3s3z3s1-Uz2s3z3s1)+Gz3s2r*(Uz3s3z3s1-Uz3s2z3s1)+Gz3s3u*(Uz3s3z3s1-Uz4s3z3s1)==0,
  Gz3s3r*(Uz3s3z2s1-Uz3s4z2s1)+Gz2s3u*(Uz3s3z2s1-Uz2s3z2s1)+Gz3s2r*(Uz3s3z2s1-Uz3s2z2s1)+Gz3s3u*(Uz3s3z2s1-Uz4s3z2s1)==0,
  Gz4s2r*(Uz4s2z1s1-Uz4s3z1s1)+Gz3s2u*(Uz4s2z1s1-Uz3s2z1s1)+Gz4s1r*(Uz4s2z1s1-Uz4s1z1s1)==0,
  Gz4s2r*(Uz4s2z1s2-Uz4s3z1s2)+Gz3s2u*(Uz4s2z1s2-Uz3s2z1s2)+Gz4s1r*(Uz4s2z1s2-Uz4s1z1s2)==0,
  Gz4s2r*(Uz4s2z1s3-Uz4s3z1s3)+Gz3s2u*(Uz4s2z1s3-Uz3s2z1s3)+Gz4s1r*(Uz4s2z1s3-Uz4s1z1s3)==0,
  Gz4s2r*(Uz4s2z1s4-Uz4s3z1s4)+Gz3s2u*(Uz4s2z1s4-Uz3s2z1s4)+Gz4s1r*(Uz4s2z1s4-Uz4s1z1s4)==0,
  Gz4s2r*(Uz4s2z2s4-Uz4s3z2s4)+Gz3s2u*(Uz4s2z2s4-Uz3s2z2s4)+Gz4s1r*(Uz4s2z2s4-Uz4s1z2s4)==0,
  Gz4s2r*(Uz4s2z3s4-Uz4s3z3s4)+Gz3s2u*(Uz4s2z3s4-Uz3s2z3s4)+Gz4s1r*(Uz4s2z3s4-Uz4s1z3s4)==0,
  Gz4s2r*(Uz4s2z4s4-Uz4s3z4s4)+Gz3s2u*(Uz4s2z4s4-Uz3s2z4s4)+Gz4s1r*(Uz4s2z4s4-Uz4s1z4s4)==0,
  Gz4s2r*(Uz4s2z4s3-Uz4s3z4s3)+Gz3s2u*(Uz4s2z4s3-Uz3s2z4s3)+Gz4s1r*(Uz4s2z4s3-Uz4s1z4s3)==0,
  Gz4s2r*(Uz4s2z4s2-Uz4s3z4s2)+Gz3s2u*(Uz4s2z4s2-Uz3s2z4s2)+Gz4s1r*(Uz4s2z4s2-Uz4s1z4s2)==1,
  Gz4s2r*(Uz4s2z4s1-Uz4s3z4s1)+Gz3s2u*(Uz4s2z4s1-Uz3s2z4s1)+Gz4s1r*(Uz4s2z4s1-Uz4s1z4s1)==0,
  Gz4s2r*(Uz4s2z3s1-Uz4s3z3s1)+Gz3s2u*(Uz4s2z3s1-Uz3s2z3s1)+Gz4s1r*(Uz4s2z3s1-Uz4s1z3s1)==0,
  Gz4s2r*(Uz4s2z2s1-Uz4s3z2s1)+Gz3s2u*(Uz4s2z2s1-Uz3s2z2s1)+Gz4s1r*(Uz4s2z2s1-Uz4s1z2s1)==0
  ],
  Gz2s4u, Gz3s3r, Gz3s3u, Gz4s2r,
  solution_dict=True);
s;
Gz2s4u=s[0][Gz2s4u]; print('Gz2s4u==',Gz2s4u,';');
Gz3s3r=s[0][Gz3s3r]; print('Gz3s3r==',Gz3s3r,';');
Gz3s3u=s[0][Gz3s3u]; print('Gz3s3u==',Gz3s3u,';');
Gz4s2r=s[0][Gz4s2r]; print('Gz4s2r==',Gz4s2r,';');
 

<math>
\begin{tikzpicture}
\definecolor{Violet}{rgb}{0.93,0.51,0.93}

\foreach \i/\j in {
0/3,
0/2, 1/3,
0/1, 1/2, 2/3
}
{
\draw[gray,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- +(1,0)
(\i,\j) -- +(0,-0.25)
+(-0.1,-0.25) -- +(-0.1,-0.75) -- +(0.1,-0.75) -- +(0.1,-0.25) -- +(-0.1,-0.25)
+(0,-0.75) -- +(0,-1) ;
\fill[gray,thick] (\i,\j) circle (1.5pt) +(1,0) circle (1.5pt) +(0,-1) circle (1.5pt);
}

\foreach \i/\j in {
0/0, 1/1, 2/2,
1/0, 2/1,
2/0
}
{
\draw[gray,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- ++(1,0)
-- +(0,0.25)
+(-0.1,0.25) -- +(-0.1,0.75) -- +(0.1,0.75) -- +(0.1,0.25) -- +(-0.1,0.25)
+(0,0.75) -- +(0,1) ;
\fill[gray] (\i,\j) circle (1.5pt) +(1,0) circle (1.5pt) +(0,1) circle (1.5pt);
}

\foreach \i/\j in {
2/0
}
{
\draw[Violet,thick] (\i,\j) -- +(0.25,0)
+(0.25,0.1) -- +(0.75,0.1) -- +(0.75,-0.1) -- +(0.25,-0.1) -- +(0.25,0.1)
+(0.75,0) -- ++(1,0)
-- +(0,0.25)
+(-0.1,0.25) -- +(-0.1,0.75) -- +(0.1,0.75) -- +(0.1,0.25) -- +(-0.1,0.25)
+(0,0.75) -- +(0,1) ;
\fill[Violet] (\i,\j) circle (1.5pt) +(1,1) circle (1.5pt);
}

\node[below] at (2,0) {(4,3)}

\end{tikzpicture}
</math>

#Berechne die von Knotenpunkt (3,4), (4,3) ausgehenden Leitwerte Gz3s4u Gz4s3r :
s=solve([
  0+Gz2s4u*(Uz3s4z1s1-Uz2s4z1s1)+Gz3s3r*(Uz3s4z1s1-Uz3s3z1s1)+Gz3s4u*(Uz3s4z1s1-Uz4s4z1s1)==0,
  0+Gz2s4u*(Uz3s4z1s2-Uz2s4z1s2)+Gz3s3r*(Uz3s4z1s2-Uz3s3z1s2)+Gz3s4u*(Uz3s4z1s2-Uz4s4z1s2)==0,
  0+Gz2s4u*(Uz3s4z1s3-Uz2s4z1s3)+Gz3s3r*(Uz3s4z1s3-Uz3s3z1s3)+Gz3s4u*(Uz3s4z1s3-Uz4s4z1s3)==0,
  0+Gz2s4u*(Uz3s4z1s4-Uz2s4z1s4)+Gz3s3r*(Uz3s4z1s4-Uz3s3z1s4)+Gz3s4u*(Uz3s4z1s4-Uz4s4z1s4)==0,
  0+Gz2s4u*(Uz3s4z2s4-Uz2s4z2s4)+Gz3s3r*(Uz3s4z2s4-Uz3s3z2s4)+Gz3s4u*(Uz3s4z2s4-Uz4s4z2s4)==0,
  0+Gz2s4u*(Uz3s4z3s4-Uz2s4z3s4)+Gz3s3r*(Uz3s4z3s4-Uz3s3z3s4)+Gz3s4u*(Uz3s4z3s4-Uz4s4z3s4)==1,
  0+Gz2s4u*(Uz3s4z4s4-Uz2s4z4s4)+Gz3s3r*(Uz3s4z4s4-Uz3s3z4s4)+Gz3s4u*(Uz3s4z4s4-Uz4s4z4s4)==0,
  0+Gz2s4u*(Uz3s4z4s3-Uz2s4z4s3)+Gz3s3r*(Uz3s4z4s3-Uz3s3z4s3)+Gz3s4u*(Uz3s4z4s3-Uz4s4z4s3)==0,
  0+Gz2s4u*(Uz3s4z4s2-Uz2s4z4s2)+Gz3s3r*(Uz3s4z4s2-Uz3s3z4s2)+Gz3s4u*(Uz3s4z4s2-Uz4s4z4s2)==0,
  0+Gz2s4u*(Uz3s4z4s1-Uz2s4z4s1)+Gz3s3r*(Uz3s4z4s1-Uz3s3z4s1)+Gz3s4u*(Uz3s4z4s1-Uz4s4z4s1)==0,
  0+Gz2s4u*(Uz3s4z3s1-Uz2s4z3s1)+Gz3s3r*(Uz3s4z3s1-Uz3s3z3s1)+Gz3s4u*(Uz3s4z3s1-Uz4s4z3s1)==0,
  0+Gz2s4u*(Uz3s4z2s1-Uz2s4z2s1)+Gz3s3r*(Uz3s4z2s1-Uz3s3z2s1)+Gz3s4u*(Uz3s4z2s1-Uz4s4z2s1)==0,
  Gz4s3r*(Uz4s3z1s1-Uz4s4z1s1)+Gz3s3u*(Uz4s3z1s1-Uz3s3z1s1)+Gz4s2r*(Uz4s3z1s1-Uz4s2z1s1)==0,
  Gz4s3r*(Uz4s3z1s2-Uz4s4z1s2)+Gz3s3u*(Uz4s3z1s2-Uz3s3z1s2)+Gz4s2r*(Uz4s3z1s2-Uz4s2z1s2)==0,
  Gz4s3r*(Uz4s3z1s3-Uz4s4z1s3)+Gz3s3u*(Uz4s3z1s3-Uz3s3z1s3)+Gz4s2r*(Uz4s3z1s3-Uz4s2z1s3)==0,
  Gz4s3r*(Uz4s3z1s4-Uz4s4z1s4)+Gz3s3u*(Uz4s3z1s4-Uz3s3z1s4)+Gz4s2r*(Uz4s3z1s4-Uz4s2z1s4)==0,
  Gz4s3r*(Uz4s3z2s4-Uz4s4z2s4)+Gz3s3u*(Uz4s3z2s4-Uz3s3z2s4)+Gz4s2r*(Uz4s3z2s4-Uz4s2z2s4)==0,
  Gz4s3r*(Uz4s3z3s4-Uz4s4z3s4)+Gz3s3u*(Uz4s3z3s4-Uz3s3z3s4)+Gz4s2r*(Uz4s3z3s4-Uz4s2z3s4)==0,
  Gz4s3r*(Uz4s3z4s4-Uz4s4z4s4)+Gz3s3u*(Uz4s3z4s4-Uz3s3z4s4)+Gz4s2r*(Uz4s3z4s4-Uz4s2z4s4)==0,
  Gz4s3r*(Uz4s3z4s3-Uz4s4z4s3)+Gz3s3u*(Uz4s3z4s3-Uz3s3z4s3)+Gz4s2r*(Uz4s3z4s3-Uz4s2z4s3)==1,
  Gz4s3r*(Uz4s3z4s2-Uz4s4z4s2)+Gz3s3u*(Uz4s3z4s2-Uz3s3z4s2)+Gz4s2r*(Uz4s3z4s2-Uz4s2z4s2)==0,
  Gz4s3r*(Uz4s3z4s1-Uz4s4z4s1)+Gz3s3u*(Uz4s3z4s1-Uz3s3z4s1)+Gz4s2r*(Uz4s3z4s1-Uz4s2z4s1)==0,
  Gz4s3r*(Uz4s3z3s1-Uz4s4z3s1)+Gz3s3u*(Uz4s3z3s1-Uz3s3z3s1)+Gz4s2r*(Uz4s3z3s1-Uz4s2z3s1)==0,
  Gz4s3r*(Uz4s3z2s1-Uz4s4z2s1)+Gz3s3u*(Uz4s3z2s1-Uz3s3z2s1)+Gz4s2r*(Uz4s3z2s1-Uz4s2z2s1)==0
  ],
  Gz3s4u, Gz4s3r, 
  solution_dict=True);
s;
Gz3s4u=s[0][Gz3s4u]; print('Gz3s4u==',Gz3s4u,';');
Gz4s3r=s[0][Gz4s3r]; print('Gz4s3r==',Gz4s3r,';');
 

Alle Leitwerte sind 1/224.

Bis jetzt habe ich nur dieses eine Beispiel gerechnet. Da waren genug Fehler drin. Ich weiß nicht, wie das Programm das löst.




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