slackFromGaleCircuits -- computes the slack matrix of a polytope from a Gale transform of the polytope

Synopsis

Usage:

M = slackFromGaleCircuits(G, Tolerance => 12)

M = slackFromGaleCircuits G
Inputs:
- G, a matrix, matrix of the form matrix(RR, {...}) whose columns are the vectors of a Gale transform
Optional inputs:
- Tolerance => ..., default value 14, specifies the tolerance to compute the slack matrix of a polytope from a Gale transform of a polytope
Outputs:
- SP, a matrix, the slack matrix of the polytope whose Gale transform is G

Description

This function computes the slack matrix of a polytope starting from a matrix whose columns correspond to the vertices of a Gale transform of the polytope. The second argument Tolerance, which is optional, is a positive integer that expresses the tolerance used in the computations, i.e. if an entry of a matrix is less than 10^(-Tolerance), then it will be treated as zero. The default value for the tolerance is 14.

i1 : G = matrix(RR, {{0, 1, -1, 0, -1, 1}, {1, 0, -1, -1, 0, 1}});

                2         6
o1 : Matrix RR    <-- RR
              53        53

i2 : M = slackFromGaleCircuits G
-- warning: experimental computation over inexact field begun, results not reliable

o2 = | 1 0 0 1 0 |
     | 0 1 0 1 0 |
     | 0 0 1 1 0 |
     | 1 0 0 0 1 |
     | 0 1 0 0 1 |
     | 0 0 1 0 1 |

                6         5
o2 : Matrix RR    <-- RR
              53        53

i3 : G = matrix(RR, {{1, 1, -1, -1, 0, 0}, {1, 1, 0, 0, -1, -1}});

                2         6
o3 : Matrix RR    <-- RR
              53        53

i4 : M = slackFromGaleCircuits G
-- warning: experimental computation over inexact field begun, results not reliable

o4 = | 1 0 1 0 1 0 1 0 |
     | 0 1 0 1 0 1 0 1 |
     | 1 1 0 0 1 1 0 0 |
     | 0 0 1 1 0 0 1 1 |
     | 1 1 1 1 0 0 0 0 |
     | 0 0 0 0 1 1 1 1 |

                6         8
o4 : Matrix RR    <-- RR
              53        53

i5 : G = matrix(RR, {{1, 0, 1, 0, 2, -2, -2}, {0, 1, 0, 1, 2, -2, -2}});

                2         7
o5 : Matrix RR    <-- RR
              53        53

i6 : M = slackFromGaleCircuits G
-- warning: experimental computation over inexact field begun, results not reliable

o6 = | 0 0 2 0 2 0 2 0 2 0 |
     | 0 0 2 2 0 0 2 2 0 0 |
     | 0 0 0 2 0 2 0 2 0 2 |
     | 0 0 0 0 2 2 0 0 2 2 |
     | 1 1 0 0 0 0 0 0 0 0 |
     | 1 0 1 1 1 1 0 0 0 0 |
     | 0 1 0 0 0 0 1 1 1 1 |

                7         10
o6 : Matrix RR    <-- RR
              53        53

i7 : G = matrix(RR, {{1, -4, 6, -4, 1, 0, 0}, {4, -15, 20, -10, 0, 1, 0}, {10, -36, 45, -20, 0, 0, 1}});

                3         7
o7 : Matrix RR    <-- RR
              53        53

i8 : M = slackFromGaleCircuits G
-- warning: experimental computation over inexact field begun, results not reliable

o8 = | 20 0    36 6       1  0   0  0       0        0   |
     | 10 2.5  15 1.66667 0  0   1  .166667 .0277778 0   |
     | 4  3    4  0       0  .4  1  .111111 0        0   |
     | 1  2.25 0  0       .5 .9  .5 0       0        .05 |
     | 0  1    0  .666667 1  1.2 0  0       .111111  .2  |
     | 0  0    1  1       1  1   0  .277778 .416667  .5  |
     | 0  0    0  0       0  0   1  1       1        1   |

                7         10
o8 : Matrix RR    <-- RR
              53        53

i9 : G = matrix(RR, {{-sqrt(10+2*sqrt(5))*(sqrt(5)+1)/(4*(sqrt(5)-1)), sqrt(10+2*sqrt(5))*(sqrt(5)+1)/(4*(sqrt(5)-1)), -4*sqrt(5)/(sqrt(10+2*sqrt(5))*(sqrt(5)+1)+(sqrt(5)-5)*sqrt(10-2*sqrt(5))), 4*sqrt(5)/(sqrt(10+2*sqrt(5))*(sqrt(5)+1)+(sqrt(5)-5)*sqrt(10-2*sqrt(5))), -sqrt(10-2*sqrt(5))/4, sqrt(10-2*sqrt(5))/4, -sqrt(10+2*sqrt(5))/4, sqrt(10+2*sqrt(5))/4, -sqrt(10-2*sqrt(5))/4, sqrt(10+2*sqrt(5))/4, -sqrt(10+2*sqrt(5))/4, 0}, {(sqrt(5)+1)/4, (sqrt(5)+1)/4, (-5-sqrt(5))/(6*sqrt(5)-10), (-5-sqrt(5))/(6*sqrt(5)-10), (sqrt(5)+1)/4, (sqrt(5)+1)/4, -(sqrt(5)-1)/4, -(sqrt(5)-1)/4, -(sqrt(5)+1)/4, (sqrt(5)-1)/4, (sqrt(5)-1)/4, 0}, {1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1}});

                3         12
o9 : Matrix RR    <-- RR
              53        53

i10 : M = slackFromGaleCircuits G
-- warning: experimental computation over inexact field begun, results not reliable

o10 = | 0 0 0 .381966 0       .618034 0       0       0       0       0      
      | 0 0 0 .618034 .381966 0       0       .618034 0       0       0      
      | 0 0 0 0       0       0       .381966 .381966 0       0       .276393
      | 0 0 0 0       0       .381966 0       0       .381966 .276393 0      
      | 0 0 0 0       .618034 0       .618034 0       0       .723607 0      
      | 1 0 0 0       0       0       0       0       .618034 0       .723607
      | 0 1 0 0       0       0       0       0       0       0       0      
      | 0 0 1 0       0       0       0       0       0       0       0      
      | 1 0 0 1       1       0       0       0       0       0       0      
      | 0 1 0 0       0       1       1       0       0       0       0      
      | 0 0 1 0       0       0       0       1       1       0       0      
      | 0 0 0 0       0       0       0       0       0       1       1      
      -----------------------------------------------------------------------
      0       .276393 .381966 .236068 .381966 0       .618034 0       .236068
      .276393 0       .236068 0       0       0       0       0       .381966
      0       0       .381966 .381966 .236068 0       0       0       0      
      0       0       0       0       0       .381966 0       0       .381966
      0       0       0       0       0       1.61803 0       1.61803 0      
      0       0       0       .381966 0       0       0       0       0      
      .723607 0       0       0       0       0       0       0       0      
      0       .723607 0       0       .381966 0       1       1       0      
      0       0       0       0       0       1       .618034 1.61803 0      
      0       0       1       1       1       1       1       1       0      
      0       0       0       0       0       0       0       0       1      
      1       1       0       0       0       0       0       0       0      
      -----------------------------------------------------------------------
      0       0       0 0       0       .276393 .447214 0       .276393
      .236068 .381966 1 0       0       .447214 .276393 .276393 0      
      0       0       0 0       1       .276393 0       .276393 0      
      .381966 .236068 0 .618034 0       0       .276393 0       .276393
      .381966 0       0 1       0       0       0       .447214 0      
      0       0       0 0       1.61803 0       0       0       .447214
      0       .381966 1 0       0       0       0       0       0      
      0       0       0 0       0       0       0       0       0      
      0       0       1 0       0       0       0       0       0      
      0       0       0 .618034 1.61803 0       0       0       0      
      1       1       1 1       1       0       0       0       0      
      0       0       0 0       0       1       1       1       1      
      -----------------------------------------------------------------------
      0       0       0       0       0       |
      0       0       0       0       0       |
      0       .105573 0       0       0       |
      .105573 0       0       0       0       |
      0       .447214 .276393 0       .723607 |
      .447214 0       0       .276393 0       |
      .447214 0       .276393 .447214 0       |
      0       .447214 .447214 .276393 .723607 |
      0       0       0       0       .447214 |
      0       0       0       0       0       |
      0       0       0       0       0       |
      1       1       1       1       1       |

                 12         34
o10 : Matrix RR     <-- RR
               53         53

Ways to use slackFromGaleCircuits :

slackFromGaleCircuits(Matrix)

For the programmer

The object slackFromGaleCircuits is a method function with options.

slackFromGaleCircuits -- computes the slack matrix of a polytope from a Gale transform of the polytope

Synopsis

Description

See also

Ways to use slackFromGaleCircuits :

For the programmer