Normaliz also allows the combination of different kinds of input, e.g.
multiple constraint types, or additional data like a grading.
<P/>
Suppose that you have a 3 by 3 <Q>square</Q> of nonnegative integers such that
the 3 numbers in all rows, all columns, and both diagonals sum to the same
constant <M>M</M>. Sometimes such squares are called magic and <M>M</M> is the
magic constant. This leads to a linear system of equations. The magic constant
is a natural choice for the grading. Additionally we force here the 4 corner
to have even value by adding congruences.
<P/>
<#Include Label="Demo_example_3x3magiceven">
<P/>
</Section>
<Section Label="Chapter_Examples_Section_Using_the_dual_mode">
<Heading>Using the dual mode</Heading>
For solving systems of equations and inequalities it is often faster to use
the dual Normaliz algorithm. We demonstrate how to use it with an
inhomogeneous system of equations and inequalities.
<P/>
The input consists of a system of 8 inhomogeneous equations in R^3. The first
row of the matrix M encodes the inequality <M>8x + 8y + 8z + 7 \geq 0</M>.
Additionally we say that <M>x, y, z</M> should be non-negative by giving the
sign vector and use the total degree.
<#Include Label="example_dual">
<P/>
As result we get the Hilbert basis of the cone of the solutions to the
homogeneous system and the module generators which are the base
solutions to the inhomogeneous system.
<P/>
</Section>
</Chapter>
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