User Methods of Core::Object

•  

  apply_rule (pattern)

  
  

  Executes the specified production rule. If the object does not possess enough initial properties to provide all required rule sources, or any of its preconditions are not satisfied, an exception is raised.

  Parameters

  String

  pattern

  : either a label (see prefer) or a rule header. The rule header must exactly match the definition in the rulefile, up to white spaces around property names. If the given pattern matches headers of several rules, or the given label is associated with several rules, the rule chain with smallest total weight (including the rules supplying the source properties) is chosen.

•  

  disable_rules (pattern)

  
  

  Temporarily disable production rules matching given pattern for an object. Rules are re-enabled after the complete execution of the current script or input expression in interactive mode.

  Works much like the user function disable_rules but only affecting the given object.

  Parameters

  String

  pattern

  : either a label (see prefer) or a rule header. The rule header must exactly match the definition in the rulefile, up to white spaces around property names. If the given pattern matches headers of several rules, or the given label is associated with several rules, they all will be disabled regardless their precoditions, weights, or other attributes.

•  

  dont_save ()

  
  

  Clears the `changed' flag in the object, so that it won't be saved in the XML file it stems from. This method is primarily designed for unit tests, but could be also useful in interactive mode if you want to revert all recent changes and reload the object from the data file.
•  

  get_schedule (request) → Core::RuleChain

  
  

  Compose an optimal chain of production rules providing all requested properties. The returned RuleChain object can be applied to the original object as well as to any other object with the same initial set of properties. If no feasible rule chain exists, `undef' is returned.

  To watch the rule scheduler at work, e.g. to see announcements about tried preconditions, you may temporarily increase the verbosity levels $Verbose::rules and $Verbose::scheduler.

  Parameters

  String

  request

  : name of a property with optional alternatives or a property path in dotted notation. Several requests may be listed.

  Returns

  Core::RuleChain

•  

  ceil ()

  
  

  UNDOCUMENTED
•  

  denominator ()

  
  

  UNDOCUMENTED
•  

  div () → Div

  
  

  Compute the quotient and remainder in one operation.

  Returns

  Div

•  

  div_exact ()

  
  

  Compute the ratio of two given integral numbers under the assumption that the dividend is a multiple of the divisor.
•  

  ext_gcd () → ExtGCD

  
  

  Compute the greatest common divisor of two numbers (a,b) and accompanying co-factors.

  Returns

  ExtGCD

•  

  fac () → Integer

  
  

  Compute the factorial of the given number.

  Returns

  Integer

•  

  floor ()

  
  

  UNDOCUMENTED
•  

  gcd ()

  
  

  Compute the greatest common divisor of two given integral numbers.
•  

  gcd ()

  
  

  Compute the greatest common divisor of the elements of the given vector.
•  

  isfinite ()

  
  

  Check whether the given number has a finite value.
•  

  isinf ()

  
  

  Check whether the given number has an infinite value. Return -1/+1 for infinity and 0 for all finite values.
•  

  is_one ()

  
  

  compare with the one (1) value of the corresponding data type
•  

  is_zero ()

  
  

  compare with the zero (0) value of the corresponding data type
•  

  lcm ()

  
  

  Compute the least common multiple of two given integral numbers.
•  

  lcm ()

  
  

  Compute the least common multiple of the elements of the given vector.
•  

  numerator ()

  
  

  UNDOCUMENTED

•  

  binomial ()

  
  

  UNDOCUMENTED
•  

  find_permutation ()

  
  

  UNDOCUMENTED
•  

  permutation_cycles ()

  
  

  UNDOCUMENTED
•  

  permutation_matrix ()

  
  

  UNDOCUMENTED

•  

  cast <Target> (object) → Object

  
  

  Change the type of the polymake object: only downcast (to a derived type) and upcast (to an ancestor type) are allowed.

  Type Parameters

  Target

  the desired new type

  Parameters

  Object

  object

  to be modified

  Returns

  Object

  the same object, but with modified type

•  

  convert_to ()

  
  

  convert to different scalar type if needed
•  

  convert_to ()

  
  

  Convert to different element type if needed.
•  

  convert_to ()

  
  

  Convert to different element type if needed.
•  

  dense (v) → Vector

  
  

  Return the input vector (which is already in dense form).

  Parameters

  Vector

  v

  Returns

  Vector

•  

  dense (m) → Matrix

  
  

  Return the input matrix (which is already in dense form).

  Parameters

  Matrix

  m

  Returns

  Matrix

•  

  dense <Element> (v) → Vector

  
  

  Convert to an equivalent dense vector of the same element type.

  Type Parameters

  Element

  Parameters

  SparseVector

  v

  Returns

  Vector

•  

  dense <Element> (m) → Matrix

  
  

  Convert to an equivalent dense matrix of the same element type.

  Type Parameters

  Element

  Parameters

  SparseMatrix

  m

  Returns

  Matrix

•  

  dense (m) → Matrix<Int>

  
  

  Convert to a dense 0/1 matrix.

  Parameters

  IncidenceMatrix

  m

  Returns

  Matrix<Int>

•  

  dense (s, dim) → Vector<Int>

  
  

  Convert to a dense 0/1 vector of a given dimension.

  Parameters

  Set	s
  Int	dim

  Returns

  Vector<Int>

•  

  index_matrix (m) → IncidenceMatrix

  
  

  Get the positions of non-zero entries of a sparse matrix.

  Parameters

  SparseMatrix

  m

  Returns

  IncidenceMatrix

•  

  indices (v) → Set<Int>

  
  

  Get the positions of non-zero entries of a sparse vector.

  Parameters

  SparseVector

  v

  Returns

  Set<Int>

•  

  repeat_row ()

  
  

  UNDOCUMENTED
•  

  toMatrix <Scalar> (A)

  
  

  UNDOCUMENTED

  Type Parameters

  Scalar

  Parameters

  IncidenceMatrix

  A

•  

  toVector <Scalar> (S)

  
  

  UNDOCUMENTED

  Type Parameters

  Scalar

  Parameters

  Set

  S

•  

  vector2col ()

  
  

  UNDOCUMENTED
•  

  vector2row ()

  
  

  UNDOCUMENTED

•  

  labeled (data, elem_labels) → String

  
  

  Prepares a vector for printing, prepends each element with a label and a colon.

  Parameters

  Vector	data	to be printed
  Array<String>	elem_labels	optional labels for elements; if data is a Set, or similar, each element will be replaced by its label.

  Returns

  String

•  

  latex (data, elem_labels) → String

  
  

  LaTeX output of a matrix.

  Parameters

  Matrix	data	to be printed
  Array<String>	elem_labels	optional labels for elements; if data is an IncidenceMatrix, Array<Set>, or similar, each element will be replaced by its label.

  Returns

  String

  to be used with \usepackage{amsmath}

•  

  numbered (data) → String

  
  

  Equivalent to labeled with omitted elem_labels argument.

  Parameters

  Vector

  data

  to be printed

  Returns

  String

•  

  rows_labeled (data, row_labels, elem_labels) → Array<String>

  
  

  Prepares a matrix for printing, prepends each row with a label and a colon.

  Parameters

  Matrix	data	to be printed
  Array<String>	row_labels	labels for the rows
  Array<String>	elem_labels	optional labels for elements; if data is an IncidenceMatrix, Array<Set>, or similar, each element will be replaced by its label.

  Returns

  Array<String>

  each string ending with end-of-line

•  

  rows_labeled (graph, elem_labels) → Array<String>

  
  

  Like above, but specialized for Graphs (defined for convenience: a PTL Graph is not a container)

  Parameters

  Graph	graph	to be printed
  Array<String>	elem_labels	labels for the elements

  Returns

  Array<String>

  each string ending with end-of-line

•  

  rows_numbered (data) → Array<String>

  
  

  Equivalent to rows_labeled with omitted row_labels argument. Formerly called "numbered".

  Parameters

  Matrix

  data

  to be printed

  Returns

  Array<String>

  each string ending with end-of-line

•  

  adjacency_matrix ()

  
  

  Returns the adjacency matrix of graph nodes. For a normal graph, it will be a kind of IncidenceMatrix, for multigraph, it will be a SparseMatrix<Int>, with entries encoding the number of parallel edges between two nodes.
•  

  edges ()

  
  

  Returns the sequence of all edges of a graph. The edges will appear in ascending order of their tail and head nodes. In the Undirected case, the edges will appear once, ordered by the larger index of their incident nodes.
•  

  induced_subgraph (graph, set)

  
  

  Creates an induced subgraph for the given subset of nodes.

  Parameters

  Graph	graph
  Set	set	indices of selected nodes

•  

  nodes ()

  
  

  Returns the sequence of all valid nodes of a graph.
•  

  node_edge_incidences <Coord> (graph) → SparseMatrix

  
  

  UNDOCUMENTED

  Type Parameters

  Coord

  coordinate type for the resulting matrix

  Parameters

  Graph

  graph

  Returns

  SparseMatrix

  node-edge incidence matrix of a graph

•  

  eliminate_denominators (v) → Vector<Integer>

  
  

  scale a vector with the least common multiple of the denominators of its coordinates

  Parameters

  Vector

  v

  Returns

  Vector<Integer>

•  

  eliminate_denominators_entire (v) → Matrix<Integer>

  
  

  scale entire matrix with the least common multiple of the denominators of its coordinates

  Parameters

  Matrix

  v

  Returns

  Matrix<Integer>

•  

  eliminate_denominators_entire_affine (v) → Matrix<Integer>

  
  

  scale entire matrix with the least common multiple of the denominators of its coordinates (ignore first column)

  Parameters

  Matrix

  v

  Returns

  Matrix<Integer>

•  

  eliminate_denominators_in_rows (v) → Matrix<Integer>

  
  

  scale a matrix rowwise with the least common multiple of the denominators of its coordinates

  Parameters

  Matrix

  v

  Returns

  Matrix<Integer>

•  

  is_integral (v) → bool

  
  

  checks whether all coordinates of a rational vector are integral

  Parameters

  Vector

  v

  Returns

  bool

•  

  is_integral (m) → bool

  
  

  checks whether all coordinates of a rational matrix are integral

  Parameters

  Matrix

  m

  Returns

  bool

•  

  primitive (v) → Vector<Integer>

  
  

  scales the vector to a primitive integral vector

  Parameters

  Vector

  v

  Returns

  Vector<Integer>

•  

  primitive (M) → Matrix<Integer>

  
  

  scales each row of the matrix to a primitive integral vector

  Parameters

  Matrix

  M

  Returns

  Matrix<Integer>

•  

  primitive_affine (v) → Vector<Integer>

  
  

  scales the affine part of a vector to a primitive integral vector

  Parameters

  Vector

  v

  Returns

  Vector<Integer>

•  

  primitive_affine (M) → Matrix<Integer>

  
  

  scales the affine part of each row of the matrix to a primitive integral vector

  Parameters

  Matrix

  M

  Returns

  Matrix<Integer>

•  

  anti_diag (d) → SparseMatrix

  
  

  Produces a SparseMatrix from its anti-diagonal.

  Parameters

  Vector

  d

  the anti-diagonal entries

  Returns

  SparseMatrix

•  

  anti_diag ()

  
  
•  

  basis (A) → Pair<Set<Int>, Set<Int>>

  
  

  Computes subsets of the rows and columns of A that form a basis for the linear space spanned by A.

  Parameters

  Matrix

  A

  Returns

  Pair<Set<Int>, Set<Int>>

  The first set corresponds to the rows, the second to the columns.

•  

  basis_affine (A) → Pair<Set<Int>, Set<Int>>

  
  

  Does the same as basis ignoring the first column of the matrix.

  Parameters

  Matrix

  A

  Returns

  Pair<Set<Int>, Set<Int>>

  The first set corresponds to the rows, the second to the columns.

•  

  basis_cols (A) → Set<Int>

  
  

  Computes a subset of the columns of A that form a basis for the linear space spanned by A.

  Parameters

  Matrix

  A

  Returns

  Set<Int>

•  

  basis_rows (A) → Set<Int>

  
  

  Computes a subset of the rows of A that form a basis for the linear space spanned by A.

  Parameters

  Matrix

  A

  Returns

  Set<Int>

•  

  det (A) → Int

  
  

  Computes the determinant of a matrix using Gauss elimination.

  Parameters

  Matrix

  A

  Returns

  Int

  det(A)

•  

  diag ()

  
  

  UNDOCUMENTED
•  

  diag (d) → SparseMatrix

  
  

  Produces a SparseMatrix from its diagonal.

  Parameters

  Vector

  d

  the diagonal entries

  Returns

  SparseMatrix

•  

  equal_bases (A1, A2) → Bool

  
  

  Check whether both matrices are bases of the same linear subspace.

  Parameters

  Matrix	A1
  Matrix	A2

  Returns

  Bool

•  

  inv (A) → Matrix

  
  

  Computes the inverse A^-1 of an invertible matrix A using Gauss elimination. Seems not to work for sparse matrices (produces seg fault).

  Parameters

  Matrix

  A

  Returns

  Matrix

•  

  lineality_space ()

  
  

  UNDOCUMENTED
•  

  lin_solve (A, b)

  
  

  Computes the solution of the system Ax = b

  Parameters

  Matrix	A	must be invertible
  Vector	b

•  

  normalized (A) → Matrix

  
  

  Computes the inverse A^-1 of an invertible matrix A using Gauss elimination. Seems not to work for sparse matrices (produces seg fault).

  Parameters

  Matrix

  A

  Returns

  Matrix

•  

  null_space ()

  
  

  UNDOCUMENTED
•  

  null_space ()

  
  
•  

  ones_vector <Element> (d)

  
  

  Creates a vector with all elements equal to 1.

  Type Parameters

  Element

  type of the vector elements, defaults to Rational.

  Parameters

  Int

  d

  vector dimension. If omitted, a vector of dimension 0 is created, which can adjust itself when involved in a block matrix operation.

•  

  rank (A) → Int

  
  

  Computes the rank of a matrix.

  Parameters

  Matrix

  A

  Returns

  Int

•  

  reduce ()

  
  

  UNDOCUMENTED
•  

  smith_normal_form (M, inv)

  
  

  Compute the Smith normal form of a given matrix M.

  Parameters

  Matrix	M	, must be of integer type
  Bool	inv	, optional, if true, compute the inverse of the companion matrices

•  

  transpose ()

  
  

  UNDOCUMENTED
•  

  transpose (A) → Matrix

  
  

  Computes the transpose A^T of a matrix A, i.e., (a^T)_ij = a_ji.

  Parameters

  Matrix

  A

  Returns

  Matrix

•  

  unit_matrix <Element> (d) → SparseMatrix<Element>

  
  

  Creates a unit matrix of given dimension

  Type Parameters

  Element

  Parameters

  Int

  d

  dimension of the matrix

  Returns

  SparseMatrix<Element>

•  

  unit_vector <Element> (d, pos) → SparseVector<Element>

  
  

  Creates a SparseVector of given length d with a one entry at position pos and zeroes elsewhere.

  Type Parameters

  Element

  Parameters

  Int	d	the dimension of the vector
  Int	pos	the position of the 1

  Returns

  SparseVector<Element>

•  

  zero_vector <Element> (d)

  
  

  Creates a vector with all elements equal to zero.

  Type Parameters

  Element

  type of the vector elements, defaults to Rational.

  Parameters

  Int

  d

  vector dimension. If omitted, a vector of dimension 0 is created, which can adjust itself when involved in a block matrix operation.

•  

  sum_of_square_roots_naive (a) → Map<Rational, Rational>

  
  

  Make a naive attempy to sum the square roots of the entries of the input array.

  Parameters

  Array<Rational>

  a

  list of rational numbers (other coefficents are not implemented).

  Returns

  Map<Rational, Rational>

  coefficient_of_sqrt a map collecting the coefficients of various roots encountered in the sum. For example, {(3 1/2),(5 7)} represents sqrt{3}/2 + 7 sqrt{5}. If the output is not satisfactory, please use a symbolic algebra package.

•  

  print_constraints (M, are_eqs) → bool

  
  

  Write the rows of a matrix M as inequalities (are_eqs=0) or equations (are_eqs=1) in a readable way. It is possible to specify labels for the coordinates via an optional array coord_labels.

  Parameters

  Matrix<Scalar>	M	the matrix whose rows are to be written
  Int	are_eqs	optional; needs to be set to 1 if the rows represent equations instead of inequalities

  Options

  Array<String>

  coord_labels

  changes the labels of the coordinates from 'x1,...,xn' to the specified names

  Returns

  bool

•  

  perturb_matrix (M, eps, not_hom) → Matrix

  
  

  Perturb a given matrix M by adding a random matrix. The random matrix consists of vectors that are uniformly distributed on the unit sphere. Optionally, the random matrix can be scaled by a factor eps.

  Parameters

  Matrix	M
  Float	eps	the factor by which the random matrix is multiplied default value: 1
  Bool	not_hom	if set to 1, the first column will also be perturbed; otherwise the first columns of the input matrix M and the perturbed one coincide (useful for working with homogenized coordinates); default value: 0 (homogen. coords)

  Options

  Int

  seed

  controls the outcome of the random number generator; fixing a seed number guarantees the same outcome.

  Returns

  Matrix

•  

  pluecker (V) → Vector

  
  

  Compute vector of maximal minors of a matrix. WARNING: interpretation different in tropical::lifted_pluecker

  Parameters

  Matrix

  V

  Returns

  Vector

•  

  incl ()

  
  

  UNDOCUMENTED
•  

  range ()

  
  

  UNDOCUMENTED
•  

  scalar2set ()

  
  

  UNDOCUMENTED
•  

  select_subset ()

  
  

  UNDOCUMENTED
•  

  sequence ()

  
  

  UNDOCUMENTED

•  

  average (array)

  
  

  Returns the average value of the array elements.

  Parameters

  Array

  array

•  

  bounding_box (m) → Matrix

  
  

  Compute a column-wise bounding box for the give matrix

  Parameters

  Matrix

  m

  Returns

  Matrix

  : row(0) contains lower bounds, row(1) contains upper bounds

•  

  fibonacci (m)

  
  

  Returns the first m Fibonacci numbers.

  Parameters

  Int

  m

•  

  histogram (data) → Map<Element, Int>

  
  

  Produce a histogram of a container: each different element value is mapped on the number of its occurences.

  Parameters

  Container<Element>

  data

  Returns

  Map<Element, Int>

•  

  maximum (array)

  
  

  Returns the maximal element of an array.

  Parameters

  Array

  array

•  

  minimum (array)

  
  

  Returns the minimal element of an array.

  Parameters

  Array

  array

•  

  compose (vis_obj ...) → Visual::Container

  
  

  Create a composite drawing of several objects.

  Parameters

  Visual::Object

  vis_obj ...

  objects to be drawn together

  Options

  String	Title	name of the whole drawing; per default the name of the first Object is taken.
  any	decorations	to be applied to all components as default values.

  Returns

  Visual::Container

  if called in void context, immediately starts the preferred rendering program.

•  

  compose (vis_container, vis_obj ...) → Visual::Container

  
  

  Add new objects to a composite drawing.

  Parameters

  Visual::Container	vis_container	drawing produced by some visualization function
  Visual::Object	vis_obj ...	objects to be added

  Options

  String	Title	new name for the drawing
  any	decorations	to be applied to all components as default values.

  Returns

  Visual::Container

  if called in void context, immediately starts the preferred rendering program.

•  

  geomview (vis_obj ...)

  
  

  Run geomview to display given visual objects.

  Parameters

  Visual::Object

  vis_obj ...

  objects to display

  Options

  String

  File

  "filename" or "AUTO" Store the objects in a gcl (geomview control language) file instead of starting the interactive GUI. The geometric data in OFF format is embedded in the Lisp-style commands, but can be easily extracted using any text editor, if needed. The .gcl suffix is automatically added to the file name. Specify AUTO if you want the filename be automatically derived from the drawing title. You can also use any expression allowed for the open function, including "-" for terminal output, "&HANDLE" for an already opened file handle, or "| program" for a pipe.

•  

  postscript (vis_obj ...)

  
  

  Create a Postscript (tm) drawing with the given visual objects.

  Parameters

  Visual::Object

  vis_obj ...

  objects to draw

  Options

  String

  File

  "filename" or "AUTO" Store the drawing in a file without starting the viewer. The .ps suffix is automatically added to the file name. Specify AUTO if you want the filename be automatically derived from the drawing title. You can also use any expression allowed for the open function, including "-" for terminal output, "&HANDLE" for an already opened file handle, or "| program" for a pipe.

•  

  povray (vis_obj ...)

  
  

  Run POVRAY to display given visual objects.

  Parameters

  Visual::Object

  vis_obj ...

  objects to display

  Options

  String

  File

  "filename" or "AUTO" Store the object description in a POVRAY source file without actual rendering. The .pov suffix is automatically added to the file name. Specify AUTO if you want the filename be automatically derived from the drawing title. You can also use any expression allowed for the open function, including "-" for terminal output, "&HANDLE" for an already opened file handle, or "| program" for a pipe.

•  

  sketch (vis_obj ...)

  
  

  Produce a Sketch input file with given visual objects.

  Parameters

  Visual::Object

  vis_obj ...

  objects to display

  Options

  String

  File

  "filename" or "AUTO" For the file name you can use any expression allowed for the open function, including "-" for terminal output, "&HANDLE" for an already opened file handle, or "| program" for a pipe. Real file names are automatically completed with the .sk suffix if needed. An automatically generated file name is displayed in the verbose mode.

•  

  static (vis_obj) → Visual::Object

  
  

  Suppress creation of dynamic (interactive) scenes.

  Parameters

  Visual::Object

  vis_obj

  drawing, e.g. created by VISUAL_GRAPH or SCHLEGEL.

  Returns

  Visual::Object

  if called in void context, immediately starts the preferred rendering program.

User Methods of Array

•  

  size ()

  
  

  UNDOCUMENTED

User Methods of boost_dynamic_bitset

•  

  size ()

  
  

  UNDOCUMENTED

User Methods of FacetList

•  

  insertMax ()

  
  

  UNDOCUMENTED
•  

  insertMin ()

  
  

  UNDOCUMENTED
•  

  size ()

  
  

  UNDOCUMENTED

User Methods of Graph

•  

  add_edge (tail_node, head_node)

  
  

  In a multigraph, creates a new edge connecting two given nodes. In a normal graph, creates a new edge only if the nodes were not connected yet. Returns the number of the (new) edge.

  Parameters

  Int	tail_node
  Int	head_node

•  

  add_node ()

  
  

  Adds a new node without incident edes, returns its index.
•  

  adjacent_nodes (node)

  
  

  Returns the set of indices of nodes adjacent to node.

  Parameters

  Int

  node

•  

  all_edges (tail_node, head_node)

  
  

  Returns an iterator visiting all (parallel) edges connecting two given nodes.

  Parameters

  Int	tail_node
  Int	head_node

•  

  contract_edge (node1, node2)

  
  

  Reconnects all edges from node2 to node1, deleting the edge(s) between them and, finally, deleting node2.

  Parameters

  Int	node1
  Int	node2

•  

  degree (node)

  
  

  Returns the number of edges incident to node.

  Parameters

  Int

  node

•  

  delete_all_edges (tail_node, head_node)

  
  

  Deletes all edges in a multigraph connecting two given nodes.

  Parameters

  Int	tail_node
  Int	head_node

•  

  delete_edge (tail_node, head_node)

  
  

  Deletes the edge connecting two given nodes, if there was one. In a multigraph, deletes one arbitrary edge from the parallel bundle.

  Parameters

  Int	tail_node
  Int	head_node

•  

  delete_edge (iterator)

  
  

  Delete the edge in a multigraph pointed to by the given iterator

  Parameters

  Iterator

  iterator

  as returned by all_edges.

•  

  delete_node (node)

  
  

  Deletes all edges incident to the given node and marks it as invalid. The numeration of other nodes stays unchanged.

  Parameters

  Int

  node

•  

  dim ()

  
  

  Returns the maximal node index + 1. If the graph does not have gaps caused by node deletion, the result is equivalent to nodes().
•  

  edge (tail_node, head_node)

  
  

  Returns the number of the edge connecting two given nodes. The edge is created if was not there. In a multigraph, an arbitrary edge from the parallel bundle will be picked.

  Parameters

  Int	tail_node
  Int	head_node

•  

  edges ()

  
  

  Returns the total number of edges.
•  

  edge_exists (tail_node, head_node)

  
  

  Checks whether two given nodes are connected by (at least) one edge.

  Parameters

  Int	tail_node
  Int	head_node

•  

  has_gaps ()

  
  

  Returns true if some nodes have been deleted since the last squeeze operation.
•  

  invalid_node (node)

  
  

  Returns true if the given node index is either out of valid range or points to a formerly deleted node.

  Parameters

  Int

  node

•  

  in_adjacent_nodes (node)

  
  

  Returns the set of indices of the nodes that have an edge heading to node.

  Parameters

  Int

  node

•  

  in_degree (node)

  
  

  Returns the number of edges heading to node.

  Parameters

  Int

  node

•  

  in_edges (node)

  
  

  Returns a sequence of edges heading to (in Directed case) or incident to (in Undirected case) node.

  Parameters

  Int

  node

•  

  nodes ()

  
  

  Returns the total number of nodes.
•  

  node_exists (node)

  
  

  Checks whether the node with given index exists

  Parameters

  Int

  node

•  

  node_out_of_range (node)

  
  

  Returns true if the given node index is out of valid range.

  Parameters

  Int

  node

•  

  out_adjacent_nodes (node)

  
  

  Returns the set of indices of the nodes with an edge arriving from node.

  Parameters

  Int

  node

•  

  out_degree (node)

  
  

  Returns the number of edges leaving node.

  Parameters

  Int

  node

•  

  out_edges (node)

  
  

  Returns a sequence of edges leaving (in Directed case) or incident to (in Undirected case) node.

  Parameters

  Int

  node

•  

  squeeze ()

  
  

  Renumbers the valid nodes as to eliminate all gaps left after deleting.
•  

  squeeze_isolated ()

  
  

  Deletes all nodes of degree 0, then renumbers the remaining nodes without gaps.

User Methods of IncidenceMatrix

•  

  col ()

  
  

  UNDOCUMENTED
•  

  cols ()

  
  

  UNDOCUMENTED
•  

  minor ()

  
  

  UNDOCUMENTED
•  

  row ()

  
  

  UNDOCUMENTED
•  

  rows ()

  
  

  UNDOCUMENTED
•  

  squeeze ()

  
  

  UNDOCUMENTED
•  

  squeeze_cols ()

  
  

  UNDOCUMENTED
•  

  squeeze_rows ()

  
  

  UNDOCUMENTED

User Methods of Set

•  

  size ()

  
  

  UNDOCUMENTED

•  

  all_rows_or_cols

  UNDOCUMENTED
•  

  Matrix

  UNDOCUMENTED

  User Methods of Matrix

  •  

    anti_diagonal () → Vector<Element>

    
    

    Returns the anti-diagonal of the matrix.

    Returns

    Vector<Element>

  •  

    col ()

    
    

    UNDOCUMENTED
  •  

    cols () → Int

    
    

    Returns the number of columns.

    Returns

    Int

  •  

    diagonal ()

    
    

    UNDOCUMENTED
  •  

    div_exact (a) → Matrix

    
    

    Divides every entry by a (assuming that every entry is divisible by a).

    Parameters

    Int

    a

    Returns

    Matrix

  •  

    minor ()

    
    

    UNDOCUMENTED
  •  

    row ()

    
    

    UNDOCUMENTED
  •  

    rows () → Int

    
    

    Returns the number of rows.

    Returns

    Int

•  

  Monomial <Coefficient, Exponent>

  UNDOCUMENTED

  Type Parameters

  Coefficient	default: Rational
  Exponent	default: Int

•  

  Polynomial <Coefficient, Exponent>

  UNDOCUMENTED

  Type Parameters

  Coefficient	default: Rational
  Exponent	default: Int

  User Methods of Polynomial

  •  

    coefficients_as_vector ()

    
    

    UNDOCUMENTED
  •  

    get_ring ()

    
    

    UNDOCUMENTED
  •  

    lc () → Int

    
    

    The leading coefficient.

    Returns

    Int

  •  

    lm_exp () → Int

    
    

    The exponent of the leading monomial.

    Returns

    Int

  •  

    monomials_as_matrix ()

    
    

    UNDOCUMENTED
  •  

    trivial ()

    
    

    UNDOCUMENTED
•  

  Ring <Coefficient, Exponent>

  UNDOCUMENTED

  Type Parameters

  Coefficient	default: Rational
  Exponent	default: Int

  User Methods of Ring

  •  

    names ()

    
    

    UNDOCUMENTED
  •  

    n_vars ()

    
    

    UNDOCUMENTED
  •  

    variable ()

    
    

    UNDOCUMENTED
  •  

    variables ()

    
    

    UNDOCUMENTED
•  

  SparseIterator

  UNDOCUMENTED
•  

  SparseMatrix <Element, Sym>

  A SparseMatrix is encoded as a list of SparseVectors (its rows).

  Use dense to convert this into its dense form.

  You can create a new SparseMatrix by entering its entries row by row, e.g.:

  $A = new SparseMatrix<Int>(<< '.');

  (5) (1 1)

  (5) (4 2)

  (5)

  (5) (0 3) (1 -1)

  .

  Type Parameters

  Element
  Sym	one of Symmetric or NonSymmetric, default: NonSymmetric

  User Methods of SparseMatrix

  •  

    squeeze ()

    
    

    Removes empty rows and columns. The remaining rows and columns are renumbered without gaps.
  •  

    squeeze_cols ()

    
    

    Removes empty columns. The remaining columns are renumbered without gaps.
  •  

    squeeze_rows ()

    
    

    Removes empty rows. The remaining rows are renumbered without gaps.
•  

  SparseVector <Element>

  A SparseVector is encoded as a sequence (l) (p₁ v₁) ... (p_k v_k), where l is the length of the vector and each pair (p_i v_i) denotes an entry with value v_i at position p_i. All other entries are zero.

  Use dense to convert this into its dense form.

  You can create a new SparseVector by entering its encoding as described above, e.g.:

  $v = new SparseVector<Int>(<< '.');

  (6) (1 1) (2 2)

  .

  Type Parameters

  Element

  User Methods of SparseVector

  •  

    size () → Int

    
    

    The number of non-zero entries.

    Returns

    Int

•  

  Term <Coefficient, Exponent>

  UNDOCUMENTED

  Type Parameters

  Coefficient	default: Rational
  Exponent	default: Int

•  

  UniMonomial <Coefficient, Exponent>

  A class for univariate monomials.

  Type Parameters

  Coefficient	default: Rational
  Exponent	default: Int

•  

  UniPolynomial <Coefficient, Exponent>

  A class for univariate polynomials.

  Type Parameters

  Coefficient	default: Rational
  Exponent	default: Int

  User Methods of UniPolynomial

  •  

    deg ()

    
    

    UNDOCUMENTED
  •  

    lc () → Int

    
    

    The leading coefficient.

    Returns

    Int

  •  

    low_deg ()

    
    

    UNDOCUMENTED
•  

  UniTerm <Coefficient, Exponent>

  A class for univariate terms.

  Type Parameters

  Coefficient	default: Rational
  Exponent	default: Int

•  

  Vector <Element>

  A type for vectors with entries of type Element.

  You can perform algebraic operations such as addition or scalar multiplication.

  You can create a new Vector by entering its entries, e.g.:

  $v = new Vector<Int>(1,2,3);

  Type Parameters

  Element

  User Methods of Vector

  •  

    dim () → Int

    
    

    The length of the vector

    Returns

    Int

  •  

    div_exact (a) → Vector

    
    

    Divides every entry by a (assuming that every entry is divisible by a).

    Parameters

    Int

    a

    Returns

    Vector

  •  

    slice ()

    
    

    UNDOCUMENTED
  •  

    slice ()

    
    

    UNDOCUMENTED

•  

  Div

  Complete result of an integral division, quotient and remainder
•  

  ExtGCD

  Complete result of calculation of the greatest common divisor of two numbers (a,b): g=gcd(a,b) g==a*p+b*q a==g*k1 b==g*k2

•  

  SmithNormalForm

  Complete result of the Smith normal form computation. Contains the following fields: SparseMatrix<Scalar> form: the Smith normal form S of the given matrix M List<Pair<Scalar, Int>> torsion: absolute values of the entries greater than 1 of the diagonal together with their multiplicity Int rank: rank of M SparseMatrix<Scalar> left_companion, right_companion: unimodular matrices L and R such that M = LSR in normal case, or S = LMR in inverted case (as specified in the call to [smith_normal_form] function).

•  

  Flexible

  This is a pseudo-type for documentation purposes only. Many options of visualization functions modifying the appearance of some set of graphical elements like points, edges, facets, etc. accept a wide range of possible values, allowing for different grades of flexibility (and complexity):

  SCALAR the same attribute value is applied to all elements ARRAY each element gets its individual attribute value HASH elements found in the hash get their individual attribute values, for the rest the appropriate default applies SUB a piece of code computing the attribute value for the given element

  Unless specified explicitly in the detailed option description, the indices, keys, or subroutine arguments used for retrieval of the attribute values are just the zero-based ordinal numbers of the elements.
•  

  HSV

  A color described as a Hue-Saturation-Value triple. Is convertible to and from an RGB representation.
•  

  RGB

  A color described as a Red-Green-Blue triple. Can be constructed from a list of three integral values from the range 0..255, or a list of three floating-point values from the range 0..1, or one of symbolic names listed in the system-wide file rgb.txt.

•  

  Visual::PointSet::decorations

  Common attributes modifying the appearance of PointSets and all visual objects derived thereof. Please be aware that no one visualization program interfaced to polymake supports all of them. Unsupported options are normally ignored.

  Options

  String	Title	the name of the drawing
  String	Name	the name of this visual object in the drawing
  Bool	Hidden	if set to true, the visual object is not rendered (useful for interactive visualization programs allowing for switching details on and off) String if set to "hidden", no point labels are displayed
  enum	PointLabels	("hidden"), String alias for PointLabels
  enum	VertexLabels	("hidden"),
  Flexible<RGB>	PointColor	color of the spheres or rectangles representing the points
  Flexible<RGB>	VertexColor	alias for PointColor
  Flexible<Float>	PointThickness	scaling factor for the size of the spheres or rectangles representing the points
  Flexible<Float>	VertexThickness	alias for PointThickness
  Flexible<RGB>	PointBorderColor	color of the border line of rectangles representing the points
  Flexible<Float>	VertexBorderColor	alias for PointBorderColor
  Flexible<Float>	PointBorderThickness	scaling factor for the thickness of the border line of rectangles representing the points
  Flexible<Float>	VertexBorderThickness	alias for PointBorderThickness
  Flexible<String>	PointStyle	if set to "hidden", neither point nor its label is rendered
  Flexible<String>	VertexStyle	alias for PointStyle
  Vector<Float>	ViewPoint	ViewPoint for Sketch visualization
  Vector<Float>	ViewDirection	ViewDirection for Sketch visualization
  Vector<Float>	ViewUp	ViewUp for Sketch visualization
  Float	Scale	scale for Sketch visualization

•  

  Visual::Polygon::decorations

  Attributes modifying the appearance of filled polygons.

  imports from: Visual::PointSet::decorations
  
  

  Options

  RGB	FacetColor	filling color of the polygon
  Float	FacetTransparency	transparency factor of the polygon between 0 (opaque) and 1 (completely translucent)
  String	FacetStyle	if set to "hidden", the inner area of the polygon is not rendered
  RGB	EdgeColor	color of the boundary lines
  Float	EdgeThickness	scaling factor for the thickness of the boundary lines
  String	EdgeStyle	if set to "hidden", the boundary lines are not rendered

•  

  Visual::Polygons::decorations

  Attributes modifying the appearance of a set of polygons (like a polygonal surface).

  imports from: Visual::PointSet::decorations
  
  

  Options

  Flexible<RGB>	FacetColor	filling color of the polygons
  Flexible<Float>	FacetTransparency	transparency factor of the polygons between 0 (opaque) and 1 (completely translucent)
  Flexible<String>	FacetStyle	if set to "hidden", the inner area of the polygons are not rendered at all
  String	FacetLabels	if set to "hidden", the facet labels are not displayed (in the most cases this is the default behavior)
  RGB	EdgeColor	color of the boundary lines
  Float	EdgeThickness	scaling factor for the thickness of the boundary lines
  String	EdgeStyle	if set to "hidden", the boundary lines are not rendered

•  

  Visual::Wire::decorations

  Attributes modifying the appearance of "wire frameworks". Unlike the rest, the flexible edge attributes are retrieved using the edge iterator as an index/key/argument.

  imports from: Visual::PointSet::decorations
  
  

  Options

  Flexible<RGB>	EdgeColor	color of the lines representing the edges
  Flexible<Float>	EdgeThickness	scaling factor for the thickness of the lines representing the edges
  EdgeMap<String>	EdgeLabels	textual labels to be placed along the edges
  Flexible<String>	EdgeStyle	if set to "hidden", neither the edge nor its label is rendered