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Timestamp:
Dec 16, 2014, 3:43:21 PM (9 years ago)
Author:
Hans Schoenemann <hannes@…>
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(u'spielwiese', 'fe61d9c35bf7c61f2b6cbf1b56e25e2f08d536cc')
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fce947c9e6c3e8c6d5a622c7f6b0d724580993cc
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a2e4470c6e9a666de8ab7b706370c15e13092f76
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  • IntegerProgramming/toric_ideal.hlp

    ra2e447 r75f460  
    1 USAGE: toric_ideal [options] matrix_file 
    2                  
    3              
    4              
     1USAGE: toric_ideal [options] matrix_file
     2
     3
     4
    55DESCRIPTION:
    6        
     6
    77toric_ideal is a program for computing the toric ideal of a matrix A.
    88This ideal is always given by the reduced Groebner basis with respect
    99to a given term ordering which is computed via BuchbergerŽs
    10 algorithm. 
     10algorithm.
    1111For this purpose, we can use six different algorithms:
    1212The algorithm of Conti/Traverso (ct) computes the toric ideal of an
    1313extended matrix. This ideal can be used later for solving integer
    1414programming problems for given right hand vectors. By eliminating
    15 auxiliary variables, we can get the toric ideal of the original matrix 
     15auxiliary variables, we can get the toric ideal of the original matrix
    1616from it. The same is true for the "positive" variant of the
    1717Conti-Traverso-algorithm (pct) which can only be applied if A has
    … …  
    2323are the algorithms of Pottier (pt), Bigatti/La Scala/Robbiano (blr),
    2424Hosten/Sturmfels (hs) and Di Biase/Urbanke (du). The last two seem to
    25 be the fastest in the actual implementation. 
    26          
    27                                
    28                            
     25be the fastest in the actual implementation.
     26
     27
     28
    2929FILE FORMAT:
    30          
     30
    3131The input file has to be a MATRIX file. toric_ideal produces a
    32 GROEBNER file named like the MATRIX file with extensions replaced by     
    33            
    34         .GB.<alg> 
    35            
     32GROEBNER file named like the MATRIX file with extensions replaced by
     33
     34        .GB.<alg>
     35
    3636where GB stands for GROEBNER and <alg> is the abbreviation for the
    3737used algorithm as above.
    38          
     38
    3939A MATRIX file should look as follows:
    40        
    41        
    42   MATRIX             
    43            
     40
     41
     42  MATRIX
     43
    4444  columns:
    4545  <number of columns>
    46          
     46
    4747  cost vector:
    4848  <coefficients of the cost vector>
    49        
     49
    5050  rows:
    5151  <number of rows>
    52          
     52
    5353  matrix:
    5454  <matrix coefficients>
    55            
    56   positive row space vector:             
    57   <coefficients of row space vector>                                 
    58          
    59              
    60 The last two lines are only needed when toric_ideal is called with the 
     55
     56  positive row space vector:
     57  <coefficients of row space vector>
     58
     59
     60The last two lines are only needed when toric_ideal is called with the
    6161algorithms of Hosten/Sturmfels or Bigatti/La Scala/Robbiano, i.e. the
    62 options 
    63        
     62options
     63
    6464        -alg hs
    6565or
    6666        -alg blr
    67          
    68 The other algorithms ignore these lines. 
    69          
     67
     68The other algorithms ignore these lines.
     69
    7070Example:
    71      
    72      
     71
     72
    7373  MATRIX
    74      
     74
    7575  columns:
    7676  3
    77          
     77
    7878  cost vector:
    7979  1 1 1
    80        
     80
    8181  rows:
    8282  2
    83        
     83
    8484  matrix:
    8585  1 2 3
    8686  4 5 6
    87          
     87
    8888  positive row space vector:
    8989  1 2 3
    90          
    91          
     90
     91
    9292A GROEBNER file looks as follows:
    93        
    94              
    95   GROEBNER 
    96            
     93
     94
     95  GROEBNER
     96
    9797  computed with algorithm:
    9898  <algorithm name abbreviation>       (* abbreviations as above *)
    99   from file(s):                       (* the following four lines are 
     99  from file(s):                       (* the following four lines are
    100100  <name of respective MATRIX file>       optional *)
    101   computation time:                     
    102   <computation time in seconds>         
    103            
     101  computation time:
     102  <computation time in seconds>
     103
    104104  term ordering:
    105105  elimination block
    … …  
    109109  weighted block
    110110  <number of weighted variables>
    111   <W_LEX / W_REV_LEX                  (* number of weighted variables   
     111  <W_LEX / W_REV_LEX                  (* number of weighted variables
    112112  / W_DEG_LEX / W_DEG_REV_LEX>           should always be >0 *)
    113113  <weight_vector>
    114          
     114
    115115  size:
    116116  <number of elements>
    117            
     117
    118118  Groebner basis:
    119   <basis elements>                   
    120            
    121   <settings for the Buchberger 
    122    algorithm and compiler settings>  (* optional *)             
    123          
    124            
     119  <basis elements>
     120
     121  <settings for the Buchberger
     122   algorithm and compiler settings>  (* optional *)
     123
     124
    125125The Groebner basis consists always of binomials of the form x^a - x^b
    126126where x^a and x^b are relatively prime. Such a binomial can be
    127 represented by the vector a-b. The basis elements in the GROEBNER file 
     127represented by the vector a-b. The basis elements in the GROEBNER file
    128128are given by the coefficients of this vector representation.
    129129The settings for BuchbergerŽs algorithm and the compiler flags are
    130 produced when the GROEBNER file is generated by a call of toric_ideal 
     130produced when the GROEBNER file is generated by a call of toric_ideal
    131131with the verbose output option
    132          
    133         -v, --verbose 
    134          
     132
     133        -v, --verbose
     134
    135135Example (not belonging to the example above):
    136            
    137        
     136
     137
    138138  GROEBNER
    139          
     139
    140140  computed with algorithm:
    141141  du
    142          
    143   term ordering:       
     142
     143  term ordering:
    144144  elimination block:
    145145  0
    … …  
    148148  W_LEX
    149149  1 2 3
    150          
     150
    151151  size:
    152152  1
    153            
     153
    154154  Groebner basis:
    155   2 3 -2                            (*  x^2 * y^3 - z^2  *) 
    156            
    157          
    158              
    159 OPTIONS: 
    160          
    161  -alg       <alg>,     
     155  2 3 -2                            (*  x^2 * y^3 - z^2  *)
     156
     157
     158
     159OPTIONS:
     160
     161 -alg       <alg>,
    162162--algorithm <alg>         algorithm to use for computing the toric
    163                           ideal; <alg> may be 
     163                          ideal; <alg> may be
    164164             ct           for Conti/Traverso,
    165165             pct          for the positive Conti/Traverso,
    … …  
    169169             du           for Di Biase/Urbanke,
    170170             blr          for Bigatti-LaScal-Robbiano.
    171              
     171
    172172 -p         <number>      percentage of new generators to cause an
    173                           autoreduction during BuchbergerŽs algorithm; 
     173                          autoreduction during BuchbergerŽs algorithm;
    174174                          <number> may be an arbitrary float, a
    175175                          negative value allows no intermediate
    176176                          autoreductions
    177                           default is 
    178                           -p 12.0 
     177                          default is
     178                          -p 12.0
    179179
    180180 -S [RP] [M] [B] [M] [2]  criteria to use in BuchbergerŽs algorithm
    … …  
    185185             B            Gebauer-Moeller criterion B
    186186             2            BuchbergerŽs second criterion
    187                           default is 
    188                           -S RP M B             
    189                  
    190  -v,         
    191 --verbose                 verbose output mode; writes the settings for 
     187                          default is
     188                          -S RP M B
     189
     190 -v,
     191--verbose                 verbose output mode; writes the settings for
    192192                          BuchbergerŽs algorithm and the compiler
    193                           flags into the output GROEBNER file 
    194                      
    195 -V <number>, 
     193                          flags into the output GROEBNER file
     194
     195-V <number>,
    196196--version <number>        version of BuchbergerŽs algorithm to use;
    197197                          <number> may be 1, 1a, 2 or 3
    198198                          default is
    199                           -V 1 
    200        
    201 
     199                          -V 1
     200
     201
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