polymake interactive shell の使い方

起動法

polymake

終わり方

polytope > exit;

変数は $変数名 (perl like)

polytope > $a=1;
polytope > $b=2;
polytope > print $a+$b;
3

多面体を定義 (C++ like)

polytope > $p = new Polytope<Rational>;

多面体の点を入力

polytope > $p->POINTS=<<".";
polytope (2)> 1 0 0 0
polytope (3)> 1 1 0 0
polytope (4)> 1 0 1 0
polytope (5)> 1 1 1 0
polytope (6)> 1 0 0 1
polytope (7)> 1 1 0 1
polytope (8)> 1 0 1 1
polytope (9)> 1 1 1 1 
polytope (10)> .

多面体の各メンバーにアクセス 頂点を出力

polytope > print $p->VERTICES;
1 0 0 0
1 1 0 0
1 0 1 0
1 1 1 0
1 0 0 1
1 1 0 1
1 0 1 1
1 1 1 1

ファセットの数を出力

polytope > print $p->N_FACETS;
6

ファセットを出力

polytope > print $p->FACETS;
0 0 0 1
0 1 0 0
0 0 1 0
1 0 0 -1
1 0 -1 0
1 -1 0 0

ファセットにのっている頂点を出力

polytope > print $p->VERTICES_IN_FACETS;
{0 1 2 3}
{0 2 4 6}
{0 1 4 5}
{4 5 6 7}
{2 3 6 7}
{1 3 5 7}

3 次元立方体(0 --> 座標値が 0 or 1)

polytope > $p = cube(3,0);
polytope > print $p->VERTICES;
1 0 0 0
1 1 0 0
1 0 1 0
1 1 1 0
1 0 0 1
1 1 0 1
1 0 1 1
1 1 1 1

4 次元立方体(0 --> 座標値が 0 or 1)

polytope > $p = cube(4,0);
polytope > print $p->VERTICES;
1 0 0 0 0
1 1 0 0 0
1 0 1 0 0
1 1 1 0 0
1 0 0 1 0
1 1 0 1 0
1 0 1 1 0
1 1 1 1 0
1 0 0 0 1
1 1 0 0 1
1 0 1 0 1
1 1 1 0 1
1 0 0 1 1
1 1 0 1 1
1 0 1 1 1
1 1 1 1 1

置換多面体

polytope > $p = permutahedron(2);
polytope > print $p->VERTICES;
1 1 2 3
1 1 3 2
1 2 1 3
1 2 3 1
1 3 1 2
1 3 2 1

Birkhoff 多面体

polytope > $p = birkhoff(2);
polytope > print $p->FACETS;
1 -1 0 0 0
0 1 0 0 0
polytope > print $p->VERTICES;
1 1 0 0 1
1 0 1 1 0

Minkowski 和

polytope > $p = cube(3,0);
polytope > $q = cube(3,0);
polytope > $r=minkowski_sum($p,$q);
polytope > print $r->VERTICES;
1 0 0 0
1 2 0 0
1 0 2 0
1 2 2 0
1 0 0 2
1 2 0 2
1 0 2 2
1 2 2 2

多面体の視覚化

polytope > $r=cube(3,0);
polytope > $r->VISUAL;

多倍長整数の取扱い(GMP)

polytope > $a = new Integer(1);
polytope > for ($i = new Integer(1); $i < 100; $i++) {
polytope (2)> $a = $a * $i;
polytope (3)> }
polytope > print $a;
933262154439441526816992388562667004907159682643816214685929638952175999932299156089414639761565182862536979208272237582511852109168640000000000000000000000

初期化は文字列で

polytope > $b = new Integer("38401382480340810324808132488123481832480123481302840124898312480324823141241000000000000000000");
polytope > print $b;
38401382480340810324808132488123481832480123481302840124898312480324823141241000000000000000000

昔のバージョンの使い方もできる ただし出力されるファイルが XML 形式になっている

knoppix@Microknoppix:~/polymake$ polymake cube.p FACETS
polymake: used package cddlib
  Implementation of the double description method of Motzkin et al.
  Copyright by Komei Fukuda.
  http://www.ifor.math.ethz.ch/~fukuda/cdd_home/cdd.html

FACETS
0 0 0 1
0 1 0 0
0 0 1 0
1 0 0 -1
1 0 -1 0
1 -1 0 0


knoppix@Microknoppix:~/polymake$ 
<?xml version="1.0" encoding="utf-8"?>
<?pm chk="4e321628"?>

<object name="cube.p" type="polytope::Polytope<;Rational>;" version="2.9.9" xmlns="http://www.math.tu-berlin.de/polymake/#3">
  <credit product="cddlib"><![CDATA[
Copyright by Komei Fukuda.
http://www.ifor.math.ethz.ch/~fukuda/cdd_home/cdd.html
]]></credit>
  <property name="POINTS">
    <m>
      <v>1 0 0 0</v>
      <v>1 1 0 0</v>
      <v>1 0 1 0</v>
      <v>1 1 1 0</v>
      <v>1 0 0 1</v>
      <v>1 1 0 1</v>
      <v>1 0 1 1</v>
      <v>1 1 1 1</v>
    </m>
  </property>
  <property name="N_POINTS" value="8" />
  <property name="FEASIBLE" value="true" />
  <property name="BOUNDED" value="true" />
  <property name="DIM" value="3" />
  <property name="AMBIENT_DIM" value="3" />
  <property name="FACETS">
    <m>
      <v>0 0 0 1</v>
      <v>0 1 0 0</v>
      <v>0 0 1 0</v>
      <v>1 0 0 -1</v>
      <v>1 0 -1 0</v>
      <v>1 -1 0 0</v>
    </m>
  </property>
  <property name="AFFINE_HULL">
    <m />
  </property>
</object>

perl 言語 配列 @変数名, 各要素はスカラーだから $変数名[インデックス] でアクセス

polytope > @a = (1,2,3,4,5);
polytope > print $a[0];
1
polytope > print $a[4];
5
polytope > 

ハッシュ %変数名, 各要素は $変数名{インデックス} でアクセス

polytope > %b = {};
polytope > $b{"zero"} = 0;
polytope > $b{"one"} = 1;
polytope > $b{"two"} = 2;
polytope > print $b{"two"};
2
polytope > print $b{"zero"};
0

Reference