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# Calculations in idempotent algebras

## 12.1 Solving systems of linear algebraic inequalities

## 12.2 The solution of the Bellman equation

## 12.3 The solution Bellman inequality

### The homogeneous Bellman inequality

### The inhomogeneous Bellman inequality

## 12.4 Finding the shortest path between the vertices of the graph

\section {Tropical algebras} You can work in the following tropical algebras :

SEMIFIELDS 1) On the set of integers $ { mathbb Z} $ we define: $ZMaxPlus$, $ZMinPlus$. 2) On the set of numbers ${\mathbb R}$ we define: $RMaxPlus$, $RMinPlus$, $RMaxMult$, $RMinMult$. 3) On the set of numbers ${\mathbb R}64$ we define: $R64MaxPlus$, $R64MinPlus$, $R64MaxMult$, $R64MinMult$.

SEMIRINGS 1) On the set of numbers ${\mathbb Z}$ we define:\ $ZMaxMin$, $ZMinMax$, $ZMaxMult$, $ZMinMult$. 2) On the set of numbers ${\mathbb R}$ we define: $RMaxMin$, $RMinMax$. 3) On the set of numbers ${\mathbb R}64$ we define: $R64MaxMin$, $R64MinMax$.

Examples of tropical algebras:

SPACE = ZMaxPlus [x, y, z];

SPACE = R64MinMult [u, v];

SPACE = RMaxMin [u, v].

An example of a simple problem in a semiring $ZMaxMult$.

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In the remaining sections of this chapter we have given some examples of problems that are solved in the tropical algebra, which are semi-fields.

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The command $\backslash solveLAITropic(A,b)$ enables us to find a particular solution of the system of inequalities

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\subsection {The homogeneous Bellman equation} The command $\backslash BellmanEquation(A)$ enables us to find a solution of the homogeneous Bellman equation $Ax = x$.

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\subsection {The inhomogeneous Bellman equation} The command $\backslash BellmanEquation(A,b)$ enables us to find a solution of the inhomogeneous Bellman equation $Ax\oplus b=x$.

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The command $\backslash BellmanInequality(A)$ enables us to find a solution of the homogeneous Bellman inequality $Ax\leq x$.

The command $\backslash BellmanInequality(A, b)$ enables us to find a solution of the inhomogeneous Bellman inequality $Ax\oplus b\leq x$.

\subsection {Calculation of the table of shortest distances for all vertices of the graph} Let $A=(x_{ij})$ be matrix of distances between adjacent vertices. We put $x_{ii}$=0 $\forall i$ and we put $x_{ij}=\infty$, if there is no edge connecting vertices i and j. The command $\backslash searchLeastDistances(A)$ allows you to find the smallest distance between all the nodes of the graph. This results in a matrix of shortest paths between all vertices.

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\subsection {Calculation of the shortest distances between two vertices of the graph} Let $A=(x_{ij})$ be matrix of distances between adjacent vertices. We put $x_{ii}$=0 $\forall i$ and we put $x_{ij}=\infty$, if there is no edge connecting vertices $i$ and $j$.

The command $\backslash findTheShortestPath(A, i, j)$ allows you to find the shortest path between nodes $i$ and $j$.

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