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<title>GAP (NormalizInterface) - Chapter 3: Examples</title>
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<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a>&java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 57

3 <span">java.lang.StringIndexOutOfBoundsException: Range [95, 94) out of bounds for length 105
<div class="ContSect"> <a href="chap3.html#X7BD5B55C802805B4">3.1 Generators</a>

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<div class="ContSect"> <a href="chap3.html#X872A3FB37D6D7DFE">3.2 System of equations</a>

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<div class=<a id"X7A489A5D79DA9E5C"name="</>/>
</java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
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<div class="ContSect"> <a href="chap3.html#X7B191A1778012C1D">3.4 Combined input</a>

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< =""class"ocss"&bsp<span<href"hap3.#X8406B01578B99D14">3. < ="eading>Using mode/span></a>

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<h3>3 Examples</</iv>

<a id="X7BD5B55C802805B4/span>

<h4>3.1 Generators</h4>

<class="example"><pre>
gap> 
< Normaliz cone>
gap></java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 7
alse
gap> NmzHasConeProperty(C,"SupportHyperplanes");
false
gap> NmzConeProperty(C,"HilbertBasis");java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[[
gap> NmzHasConeProperty(C,"SupportHyperplanes");
rue
gap> NmzConeProperty(C,"SupportHyperplanes");</java.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 23
[ [-, 2 ] [3,-1 ]
</pre></div>

<a id="X872A3FB37D6D7DFE" name="X872A3FB37D6D7DFE"></a>

<h4>3.2 System of equations</java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 5

<div class="example"><pre>
gap> 1 1],[1,2 ] [, 3 ] [2,1 ]
true
gap> NmzHilbertBasis(D);
[ ] ]
gap> NmzHilbertSeries(D);
[ t<pre>/div>
gap> NmzHasConeProperty(D,"SupportHyperplanes");
true
gap> NmzSupportHyperplanes(D);
[ 0,1,0 , [ 1 0, 0 ]]
gap> NmzEquations(D);
[[ 1,2,-3 ] ]
</pre></div>

<a id="X8570F3C87B5B7DD7" name="span class="GAPprompt">gap> mzComputeD,"DualMode","HilbertSeries"]);

<h4>3.3 System of inhomogeneous equations</h4>

<div class="example"><pre>
gap> P := NmzCone(<pan class==GAPprompt">gap> NmzHilbertSeries(D);
<spanclass="GAPprompt">gap> NmzHasConeProperty(D,"SupportHyperplanes");
<java.lang.StringIndexOutOfBoundsException: Range [82, 5) out of bounds for length 91
false
gap> NmzIsInhomogeneous(P);
rue
gap> NmzHilbertBasis(P);
[1,1,1,0 ],[ , 0,1,0 ], [0, 3,2 0] ]
gap> NmzModuleGenerators(P);
[ [ 0, 1, 1, 1 ], <pre><div
</pre></div>

<a id="X7B191A1778012C1D" name="X7B191A1778012C1D"></a>

<h4>3.4 Combined input</h4>

Normaliz also allows the combination of different kinds of input, e.g. multiplespan class="GAPprompt">gap> P := NmzCone(["inhom_equations&ta Normaliz cone>

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

<div class="example"><pre>
<spantrue
> [ [1, 1, 1, -1, -1, -1, 0, 0, 0],
>
 <pan class="GAPinput"> [0, 1, 1, -1, 0, 0, -1, 0, 0],
> [1, 0, 1, 0, -1, 0, 0, -1, 0],
> [1, 1, 0, 0, 0, -1, 0, 0, -1],
<="GAPprompt>gt</pan> [0, 1, 1, 0, -1, 0, 0, 0, -1],
> [1, 1, 0, 0, -1, 0, -1, 0, 0] ],
> "congruences",
> [ [1, 0, 0, 0, 0, 0, 0, 0, 0, 2],
></gap></s class"GAPinput">Magic3x3even := NmzCone(["equations",
> [0, 0, 0, 0, 0, 0, 1, 0, 0, 2],
> [0, 0, 0, 0, 0, 0, 0, 0, 1, 2] ],
> "grading",
> [ [1, 1, 1, 0, 0, 0, 0, 0, 0] ] ] );<<pan class="GAPprompt">> [1, 0, 1, 0, -1, 0, 0, -1, 1]
&;
gap><<panclass="APprompt">> [ [1, 0, 0, 0, 0, 0, 0, 0, 0, 2],
[ [ 0, 4, 2, 4, 2, 0, 2, 0, 4 ], [ 2, 0, 4, 4, 2, 0, 0, 4, 2 ],
 [ 2, 2, 2, span ="APprompt">gt; [0 , 0, 0 0 0 0,0,1 2]],</pan>
 [ 4, 0, 2, 0, 2, 4, 2, 4, 0 ], [ 2, 3, 4, 5, 3, 1, 2, 3, 4 ],
 [ 2, 5, 2, 3, 3, 3, 4, 1, 4 ], [ 4, 1, 4, 3, 3, 3, 2, 5, 2 ],
 [ 4, 3, 2, 1, 3, 5, 4, 3, 2 ] ]
gap> NmzHilbertQuasiPolynomial(Magic3x3even);
[ 1/2*t^2+t+1, 1/2*t^2-1/2 ]
java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 12

<a id="X8406B01578B99D14" name="X8406B01578B99D14"[t^3*^2-t+1 [[1, ] 2,2 ]]

<h4>3.5 Using the dual mode</h4>

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.

 input consistsa 8 inhomogeneous in R3. first row of matrixM the inequality x, y, z should be non-negative by giving the sign vector and use the total degree.

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<class&;/><span=GAPinput>M = [span>
> [ 8, 8, 8, 7 ],
gt; 
> [ 0, 1, 0, 7 ],
gt;/pan> gt; [ 8, 48, 8, 17 ],
> [ 1, 6, 1, 34 ],
> [ 2,-12, -2, 37 ],
> [ 4,-24, -4, 14 ]
> ];;
<pan="GAPprompt">gap> D := NmzCone(["inhom_inequalities", M,
> "signs", [[1,1,1]], class="APprompt">> <spanclass"APinput" [8,48, 8,17 ]> "grading", [[1,1,1]]]);
<a Normaliz cone>
gap> > 
true
gap> NmzHilbertBasis(D);
[ spanclass="GAPprompt">> "signs", [[1,1,1]],
gap> NmzModuleGenerators(D);
[ [ 0, 0, 0, 1 ], [ 0span class="GAPprompt">gap> NmzCompute(D,["DualMode","HilbertBasis","ModuleGenerators"]);
</pre></div>

As result we get the Hilbert basis ofspan class=GAPprompt"gapgt;/span>NmzModuleGenerators(D);

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