application: common

This artificial application gathers functionality shared by many "real" applications. While most users can probably do without looking into this, you may find some useful functions here.

Objects

Core::Object

This is the common base class of all `big' objects in polymake. It is included in the online help because it provides several useful methods for scripting and interactive use.

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
list_names ()

Returns the list of names of multiple subobject instances. This method can be applied to any instance. For a normal (non-multiple) subobject or a top-level object just returns its name.
list_properties (deep)

Returns the list of names of all properties kept in the object.
Parameters
Bool deep
: recursively descend in all subobjects and list their properties in dotted notation.
remove (prop)

Remove the property prop from the object. The property must be mutable, multiple, or unambiguously reconstructible from the remaining properties.
Parameters
String prop
: property name or a path to a property in a subobject in dotted notation, Several properties may be removed at once.
remove (subobj)

Remove the multiple subobject instance(s) from the object.
Parameters
Object subobj
: multiple subobject instance. Several subobjects may be removed at once.
set_as_default ()

Makes the multiple subobject instance the default one. Physically this means moving it at the 0-th position in the instance list.
The instance can be selected by give() or PROPERTY_NAME access method, e.g.:
by current position: > $p->TRIANGULATION->[$i]->set_as_default;
by subobject name: > $p->TRIANGULATION("placing")->set_as_default;
by checking for a specific property: > $p->TRIANGULATION(sub { defined($_->lookup("WEIGHTS")) })->set_as_default;
by analyzing all instances and picking the best one: > for (@{$p->TRIANGULATION}) { assign_min($min_facets, $_->N_FACETS) and $t=$_ } $t->set_as_default;
set_as_default_now ()

Temporarily make the multiple subobject instance the default one. The change is automatically reverted at the end of the current user cycle. Usage as set_as_default.

Core::RuleChain
A rule chain to compute properties of an object.
User Methods of Core::RuleChain

apply($) (o)

Apply the rule chain to an object.
Parameters
Core::Object o

list ()

List the properties computed by the rule chain.
Visual::Container

Category: Visualization
The common base class of all visual objects composed of several simpler objects. Instances of such classes can carry default decoration attributes applied to all contained objects.
derived from: Visual::Object
Visual::Object

Category: Visualization
The common base class of all visualization artifacts produced by various user methods like VISUAL, VISUAL_GRAPH, SCHLEGEL, etc. Visual objects can be passed to functions explicitly calling visualization software like jreality() or povray().

User Functions

Arithmetic

These are functions that perform arithmetic computations.

ceil (a) → Rational

The ceiling function. Returns the smallest integral number not smaller than a.
Parameters
Rational a
Returns
Rational
denominator (a) → Integer

Returns the denominator of a in a reduced representation.
Parameters
Rational a
Returns
Integer
denominator (f) → Polynomial

Returns the denominator of a RationalFunction f.
Parameters
RationalFunction f
Returns
Polynomial
denominator (f) → Polynomial

Returns the denominator of a PuiseuxFraction f.
Parameters
PuiseuxFraction f
Returns
Polynomial
div (a, b) → Div
Compute the quotient and remainder of a and b in one operation.
Parameters
Int a
Int b
Returns
Div
Example:
$d = div(10,3);> print $d->quot; 3> print $d->rem; 1
div_exact (a, b) → Int
Computes the ratio of two given integral numbers under the assumption that the dividend is a multiple of the divisor.
Parameters
Int a
Int b
a divisor of a
Returns
Int
Example:
> print div_exact(10,5); 2
ext_gcd (a, b) → ExtGCD
Compute the greatest common divisor of two numbers (a,b) and accompanying co-factors.
Parameters
Int a
Int b
Returns
ExtGCD
Example:
> $GCD = ext_gcd(15,6); The GCD of the numbers can then be accessed like this:> print $GCD->g; 3 The ExtGCD type also stores the Bezout coefficients (thus integers p and q such that g=a*p+b*q)...> print $GCD->p; 1 print $GCD->q; -2 ...and the quotients k1 of a and k2 of b by g.> print $GCD->k1; 5> print $GCD->k2; 2
fac (n) → Integer

Computes the factorial n! = n·(n-1)·(n-2)·...·2·1.
Parameters
Int n
>=0
Returns
Integer
n!
floor (a) → Rational
The floor function. Returns the smallest integral number not larger than a.
Parameters
Rational a
Returns
Rational
Example:
> print floor(1.8); 1
gcd (a, b) → Int
Computes the greatest common divisor of two integers.
Parameters
Int a
Int b
Returns
Int
Example:
> print gcd(6,9); 3
gcd (v) → Element
Compute the greatest common divisor of the elements of the given vector.
Parameters
Vector<Element> v
Returns
Element
Example:
> $v = new Vector<Int>(3,6,9);> print gcd($v); 3
gcd (p, q) → UniPolynomial
Returns the greatest common divisor of two univariate polynomials.
Parameters
UniPolynomial p
UniPolynomial q
Returns
UniPolynomial
Example:
We first create the polynomial ring with one variable, Rational coefficients and Int exponents. $r=new Ring<Rational, Int>(1); Then we create two UniPolynomials with said coefficient and exponent type:> $p = new UniPolynomial<Rational,Int>([2,2],[3,2],$r);> $q = new UniPolynomial<Rational,Int>([6,4],[4,2],$r); Printing them reveals what the constructor does:> print $p; 2*x^3 + 2*x^2> print $q; 4*x^4 + 6*x^2 Now we can calculate their gcd:> print gcd($p,$q); x^2
isfinite (a) → Bool
Check whether the given number has a finite value.
Parameters
Integer a
Returns
Bool
Example:
> print isfinite('inf'); 0> print isfinite(23); 1
isinf (a) → Bool
Check whether the given number has an infinite value. Return -1/+1 for infinity and 0 for all finite values.
Parameters
Integer a
Returns
Bool
Example:
> print isinf('inf'); 1> print isinf(23); 0
is_one (s) → Bool

Compare with the one (1) value of the corresponding data type.
Parameters
SCALAR s
Returns
Bool
is_zero (s) → Bool

Compare with the zero (0) value of the corresponding data type.
Parameters
SCALAR s
Returns
Bool
lcm (a, b) → Int
Computes the least common multiple of two integers.
Parameters
Int a
Int b
Returns
Int
Example:
> print lcm(6,9); 18
lcm (v) → Element
Compute the least common multiple of the elements of the given vector.
Parameters
Vector<Element> v
Returns
Element
Example:
> $v = new Vector<Integer>(1,3,6,9);> print lcm($v); 18
numerator (a) → Integer

Returns the numerator of a in a reduced representation.
Parameters
Rational a
Returns
Integer
numerator (f) → Polynomial

Returns the numerator of a RationalFunction f.
Parameters
RationalFunction f
Returns
Polynomial
numerator (f) → Polynomial

Returns the numerator of a PuiseuxFraction f.
Parameters
PuiseuxFraction f
Returns
Polynomial

sum_of_square_roots_naive (a) → Map<Rational, Rational>

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

Parameters

Array<Rational>

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.

Combinatorics
This category contains combinatorial functions.
- all_permutations (n) → ARRAY
  
  Returns a list of all permutations of the set {0...n-1} as a perl-array
  Parameters
  Int n
  Returns
  ARRAY
  
  Example:
  To store the result in the perl array @a, type this:> @a = all_permutations(3); The array contains pointers to arrays. To access the 0-th pointer, do this:> $a0 = $a[0]; To print the 0-th array itself, you have to dereference it as follows:> print @{ $a0 }; 012 You can loop through @a by using foreach. The print statement gets the string obtained by dereferencing the current entry concatenated with the string " ".> foreach( @a ){> print @{ $_ }," ";> } 012 102 201 021 120 210
- binomial (n, k) → Int
  
  Computes the binomial coefficient n choose k. Negative values of n (and k) are supported.
  Parameters
  Int n
  Int k
  Returns
  Int
  n choose k
- find_permutation (a, b) → Array<Int>
  
  Returns the permutation that maps a to b.
  Parameters
  Array a
  Array b
  Returns
  Array<Int>
  
  Example:
  > $p = find_permutation([1,8,3,4],[3,8,4,1]);> print $p; 2 1 3 0
- permutation_cycles (p) → Array<List<Int>>
  
  Returns the cycles of a permutation given by p.
  Parameters
  Array<Int> p
  Returns
  Array<List<Int>>
  
  Example:
  > print permutation_cycles([1,0,3,2]); {0 1}{2 3}
- permutation_matrix <Scalar> (p) → Matrix<Scalar>
  
  Returns the permuation matrix of the permutation given by p.
  Type Parameters
  Scalar
  default: Int
  Parameters
  Array<Int> p
  Returns
  Matrix<Scalar>
  
  Example:
  The following prints the permutation matrix in sparse representation.> print permutation_matrix([1,0,3,2]); (4) (1 1) (4) (0 1) (4) (3 1) (4) (2 1)
- permutation_sign (p) → Int
  
  Returns the sign of the permutation given by p.
  Parameters
  Array<Int> p
  Returns
  Int
  +1 or -1
  
  Example:
  > print permutation_sign([1,0,3,2]); 1

Data Conversion

This contains functions for data conversions and type casts.

cast <Target> (object) → Object

Change the type of the polymake object to one of its base types (aka ancestor in the inheritance hierarchy). The object loses all properties that are unknown in the target type.
Type Parameters
Target
the desired new type
Parameters
Object object
to be modified
Returns
Object
the same object, but with modified type
cols ()

UNDOCUMENTED
cols (A) → ARRAY<Vector>
Returns an array containing the columns of A.
Parameters
Matrix A
Returns
ARRAY<Vector>
Example:
The following saves the columns of the vertex matrix of a square in the variable $w and then prints its contents using a foreach loop and concatenating each entry with the string " ".> $w = rows(cube(2)->VERTICES);> foreach( @$w ){> print @{$_}," ";> } 1111 -11-11 -1-111
concat_rows (A) → Vector
Concatenates the rows of A.
Parameters
Matrix A
Returns
Vector
Examples:
Make a vector out of the rows of the vertex matrix of a cube:> $v = concat_rows(cube(2)->VERTICES);> print $v; 1 -1 -1 1 1 -1 1 -1 1 1 1 1
For a sparse matrix, the resulting vector is sparse, too.> $vs = concat_rows(unit_matrix(3));> print $vs; (9) (0 1) (4 1) (8 1)
convert_to <Target> (s) → Target

Explicit conversion to different scalar type.
Type Parameters
Target
Parameters
SCALAR s
Returns
Target
convert_to <Target> (v) → Vector<Target>
Explicit conversion to a different element type.
Type Parameters
Target
Parameters
Vector v
Returns
Vector<Target>
Example:
> $v = new Vector<Rational>(1/2,2/3,3/4);> $vf = convert_to<Float>($v);> print $vf; 0.5 0.6666666667 0.75
convert_to <Target> (m) → Matrix<Target>
Explicit conversion to a different element type.
Type Parameters
Target
Parameters
Matrix m
Returns
Matrix<Target>
Example:
> $M = new Matrix<Rational>([1/2,2],[3,2/3]);> $Mf = convert_to<Float>($M);> print $Mf; 0.5 2 3 0.6666666667
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<Element>

Convert to an equivalent dense vector of the same element type.
Type Parameters
Element
Parameters
SparseVector<Element> v
Returns
Vector<Element>
dense <Element> (m) → Matrix<Element>

Convert to an equivalent dense matrix of the same element type.
Type Parameters
Element
Parameters
SparseMatrix<Element> m
Returns
Matrix<Element>
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
Example:
> $S = new SparseMatrix([1,2,0,0,0,0],[0,0,5,0,0,32]);> print index_matrix($S); {0 1} {2 5}
indices (v) → Set<Int>
Get the positions of non-zero entries of a sparse vector.
Parameters
SparseVector v
Returns
Set<Int>
Example:
> $v = new SparseVector(0,1,1,0,0,0,2,0,3);> print indices($v); {1 2 6 8}
lex_ordered (f) → PowerSet<Int>
Visit the facets of f sorted lexicographically.
Parameters
FacetList f
Returns
PowerSet<Int>
Example:
> $f = new FacetList(cube(2)->VERTICES_IN_FACETS);> print lex_ordered($f); {{0 1} {0 2} {1 3} {2 3}}
repeat_col (v, i)
Create a Matrix by repeating the given Vector as cols.
Parameters
Vector v
Int i
Example:
> $v = new Vector(23,42,666);> $M = repeat_col($v,3);> print $M; 23 23 23 42 42 42 666 666 666
repeat_row (v, i)
Create a Matrix by repeating the given Vector as rows.
Parameters
Vector v
Int i
Example:
> $v = new Vector(23,42,666);> $M = repeat_row($v,3);> print $M; 23 42 666 23 42 666 23 42 666
rows ()

UNDOCUMENTED
rows (A) → ARRAY<Vector>
Returns an array containing the rows of A.
Parameters
Matrix A
Returns
ARRAY<Vector>
Example:
The following saves the rows of the vertex matrix of a square in the variable $w and then prints its contents using a foreach loop and concatenating each entry with the string " ".> $w = rows(cube(2)->VERTICES);> foreach( @$w ){> print @{$_}," ";> } 1-1-1 11-1 1-11 111
support (v) → Set<Int>
Get the positions of non-zero entries of a vector.
Parameters
Vector v
Returns
Set<Int>
Example:
> print support(new Vector(0,23,0,0,23,0,23,0,0,23)); {1 4 6 9}
toMatrix <Scalar> (A) → SparseMatrix<Scalar>
Convert an IncidenceMatrix to a SparseMatrix.
Type Parameters
Scalar
Parameters
IncidenceMatrix A
Returns
SparseMatrix<Scalar>
Example:
> $M = toMatrix<Int>(cube(2)->VERTICES_IN_FACETS);> print $M->type->full_name; SparseMatrix<Int, NonSymmetric>
toPolynomial (s, vars)

Read a Polynomial from a String.
Parameters
String s
ARRAY vars
toTropicalPolynomial (s, vars) → Polynomial<TropicalNumber<Addition,Rational> >

This converts a string into a tropical polynomial. The syntax for the string is as follows: It is of the form "min(...)" or "max(...)" or "min{...}" or "max{...}", where ... is a comma-separated list of sums of the form "a + bx + c + dy + ...", where a,c are rational numbers, b,d are Ints and x,y are variables. Such a sum can contain several such terms for the same variable and they need not be in any order. Any text that starts with a letter and does not contain any of +-*,(){} or whitespace can be a variable. A term in a sum can be of the form "3x", "3*x", but "x3"will be interpreted as 1 * "x3". Coefficients should not contain letters and there is no evaluation of arithmetic, i.e. "(2+4)*x" does not work (though "2x+4x" would be fine). In fact, further brackets should only be used (but are not necessary!) for single coefficienst, e.g. "(-3)*x". Warning: The parser will remove all brackets before parsing the individual sums. If no further arguments are given, the function will take the number of occuring variables as total number of variables and create a ring for the result. The variables will be sorted alphabetically.
Parameters
String s
The string to be parsed
String vars
Optional. A list of variables. If this is given, all variables used in s must match one of the variables in this list.
Returns
Polynomial<TropicalNumber<Addition,Rational> >
, where Addition depends on whether min or max was used in the string.

toTropicalPolynomial (s, r, match_variables_by_order) → Polynomial<TropicalNumber<Addition,Rational> >

Same as toTropicalPolynomial(String,...), except that the result will live in the specified ring.

Parameters

String	s	The string to be parsed
Ring<TropicalNumber<Addition,Rational> >	r	The polynomial ring in which the result will live. By default, all variables in s must match variables of the ring (this can be changed with the next argument) and the ring must be over the TropicalNumbers.
Bool	match_variables_by_order	Optional and false by default. If true, variables in s can be arbitrary, though their total number has to match the total number of ring variables. The variables of the string will be assigned to ring variables in alphabetical order.

Returns

Polynomial<TropicalNumber<Addition,Rational> >

toVector <Scalar> (S)

UNDOCUMENTED
Type Parameters
Scalar
Parameters
Set S
vector2col (v) → Matrix
Convert a Vector to a Matrix with a single column.
Parameters
Vector v
Returns
Matrix
Example:
This converts a vector into a column and prints it and its type:> $v = new Vector([1,2,3,4]);> $V = vector2col($v);> print $V; 1 2 3 4> print $V->type->full_name; Matrix<Rational, NonSymmetric>
vector2row (v) → Matrix
Convert a Vector to a Matrix with a single row.
Parameters
Vector v
Returns
Matrix
Example:
This converts a vector into a row and prints it and its type:> $v = new Vector([1,2,3,4]);> $V = vector2row($v);> print $V; 1 2 3 4> print $V->type->full_name; Matrix<Rational, NonSymmetric>

Formatting

Functions for pretty printing, labels or latex output of polymake types.

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
Example:
> $v = new Vector(0,1,2);> print labeled($v,["zeroth","first","second"]); zeroth:0 first:1 second:2
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
Example:
> $data = new Vector(23,42,666);> print numbered($data); 0:23 1:42 2:666

print_constraints (M)

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

Parameters

Matrix<Scalar>

the matrix whose rows are to be written

Options

Array<String>	coord_labels	changes the labels of the coordinates
Array<String>	row_labels	changes the labels of the rows
Bool	homogeneous	false if the first coordinate should be interpreted as right hand side
Bool	equations	true if the rows represent equations instead of inequalities

Example:

> $M = new Matrix([1,2,3],[4,5,23]);> print_constraints($M,equations=>1); 0: 2 x1 + 3 x2 = -1 1: 5 x1 + 23 x2 = -4

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

Example:

> print rows_labeled(cube(2)->VERTICES,['a','b','c','d']); a:1 -1 -1 b:1 1 -1 c:1 -1 1 d:1 1 1

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
Example:
> print rows_labeled(cycle_graph(4)->ADJACENCY,['a','b','c','d']); a:b d b:a c c:b d d:a c
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
Example:
> print rows_numbered(cube(2)->VERTICES); 0:1 -1 -1 1:1 1 -1 2:1 -1 1 3:1 1 1

Graph Operations
Operations on graphs.
- adjacency_matrix (graph) → IncidenceMatrix
  
  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.
  Parameters
  Graph graph
  Returns
  IncidenceMatrix
  both rows and columns correspond to the nodes
- edges (graph) → EdgeList
  
  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.
  Parameters
  Graph graph
  Returns
  EdgeList
- induced_subgraph (graph, set) → Graph
  
  Creates an induced subgraph for the given subset of nodes.
  Parameters
  Graph graph
  Set set
  indices of selected nodes
  Returns
  Graph
  
  Example:
  $g = new props::Graph(cycle_graph(5)->ADJACENCY);> $s1 = new Set(1,2,3);> print induced_subgraph($g,$s1); {2} {1 3} {2}
- nodes (graph) → NodeSet
  
  Returns the sequence of all valid nodes of a graph.
  Parameters
  Graph graph
  Returns
  NodeSet
  
  Example:
  > print nodes(cycle_graph(5)->ADJACENCY); {0 1 2 3 4}
- node_edge_incidences <Coord> (graph) → SparseMatrix<Coord>
  
  Returns the node-edge incidence matrix of a graph.
  Type Parameters
  Coord
  coordinate type for the resulting matrix, default: Int
  Parameters
  Graph graph
  Returns
  SparseMatrix<Coord>
  
  Example:
  > print node_edge_incidences(cycle_graph(5)->ADJACENCY); (5) (0 1) (3 1) (5) (0 1) (1 1) (5) (1 1) (2 1) (5) (2 1) (4 1) (5) (3 1) (4 1)
Lattice Tools
Functions for lattice related computations.
- 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>
  
  Example:
  > $v = new Vector(1/2,1/3,1/4,1/5);> $ve = eliminate_denominators($v);> print $ve; 30 20 15 12
- eliminate_denominators_entire (v) → Matrix<Integer>
  
  Scales entire matrix with the least common multiple of the denominators of its coordinates.
  Parameters
  Matrix v
  Returns
  Matrix<Integer>
  
  Example:
  > $M = new Matrix([1/2,1/3],[1/5,7],[1/4,4/3]);> $Me = eliminate_denominators_entire($M);> print $Me; 30 20 12 420 15 80
- eliminate_denominators_entire_affine (v) → Matrix<Integer>
  
  Scales entire matrix with the least common multiple of the denominators of its coordinates (ignore first column).
  Parameters
  Matrix v
  Returns
  Matrix<Integer>
  
  Example:
  > $M = new Matrix([1,1/2,1/3],[1,1/5,7],[1,1/4,4/3]);> $Me = eliminate_denominators_entire_affine($M);> print $Me; 1 30 20 1 12 420 1 15 80
- eliminate_denominators_in_rows (M) → Matrix<Integer>
  
  Scale a matrix row-wise with the least common multiple of the denominators of its coordinates.
  Parameters
  Matrix M
  Returns
  Matrix<Integer>
  
  Example:
  > $M = new Matrix([1/2,1/3],[1/5,7],[1/4,4/3]);> $Me = eliminate_denominators_in_rows($M);> print $Me; 3 2 1 35 3 16
- is_integral (v) → Bool
  
  Checks whether all coordinates of a rational vector are integral.
  Parameters
  Vector v
  Returns
  Bool
  
  Example:
  This rational vector has only integral entries:> $v = new Vector<Rational>(1,2,3,4); polytope > print is_integral($v); 1 But if we append 1/2, it hasn't anymore:> print is_integral($v|1/2);
- is_integral (m) → Bool
  
  Checks whether all coordinates of a rational matrix are integral.
  Parameters
  Matrix m
  Returns
  Bool
  
  Example:
  This rational matrix has only integral entries:> $m = new Matrix<Rational>([1,2],[3,4]);> print is_integral($m); 1 But if we multiply it with 1/2, that is not the case anymore.> print is_integral(1/2 * $m);
- primitive (v) → Vector<Integer>
  
  Scales the vector to a primitive integral vector.
  Parameters
  Vector v
  Returns
  Vector<Integer>
  
  Example:
  > print primitive(new Vector(3,3/2,3,3)); 2 1 2 2
- primitive (M) → Matrix<Integer>
  
  Scales each row of the matrix to a primitive integral vector.
  Parameters
  Matrix M
  Returns
  Matrix<Integer>
  
  Example:
  > print primitive(new Matrix([1,3/2],[3,1])); 2 3 3 1
- primitive_affine (v) → Vector<Integer>
  
  Scales the affine part of a vector to a primitive integral vector.
  Parameters
  Vector v
  Returns
  Vector<Integer>
  
  Example:
  > print primitive_affine(new Vector(1,3/2,1,1)); 1 3 2 2
- 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>
  
  Example:
  > print primitive_affine(new Matrix([1,1,3/2],[1,3,1])); 1 2 3 1 3 1
Linear Algebra
These functions are for algebraic computations and constructions of special matrices.
- anti_diag (d) → SparseMatrix
  
  Produces a SparseMatrix from its anti-diagonal.
  Parameters
  Vector d
  the anti-diagonal entries
  Returns
  SparseMatrix
- anti_diag (m1, m2) → SparseMatrix
  
  Returns a block anti-diagonal matrix with blocks m1 and m2.
  Parameters
  Matrix m1
  Matrix m2
  Returns
  SparseMatrix
- 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.
  
  Example:
  Here we have a nice matrix:> $M = new Matrix([[1,0,0,0],[2,0,0,0],[0,1,0,0],[0,0,1,0]]); Let's print bases for the row and column space:> ($row,$col) = basis($M);> print $M->minor($row,All); 1 0 0 0 0 1 0 0 0 0 1 0> print $M->minor(All,$col); 1 0 0 2 0 0 0 1 0 0 0 1
- 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>
  
  Example:
  Here we have a nice matrix:> $M = new Matrix([[1,0,0,0],[2,0,0,0],[0,1,0,0],[0,0,1,0]]); Let's print a basis of its column space:> print $M->minor(All,basis_cols($M)); 1 0 0 2 0 0 0 1 0 0 0 1
- 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>
  
  Example:
  Here we have a nice matrix:> $M = new Matrix([[1,0,0,0],[2,0,0,0],[0,1,0,0],[0,0,1,0]]); Let's print a basis of its row space:> print $M->minor(basis_rows($M),All); 1 0 0 0 0 1 0 0 0 0 1 0
- det (A) → Int
  
  Computes the determinant of a matrix using Gauss elimination.
  Parameters
  Matrix A
  Returns
  Int
  det(A)
  
  Example:
  > print det(unit_matrix(3)); 1
- diag (d) → SparseMatrix
  
  Produces a SparseMatrix from its diagonal.
  Parameters
  Vector d
  the diagonal entries
  Returns
  SparseMatrix
  
  Example:
  > $v = new Vector(1,2,3,4);> $D = diag($v);> print $D; (4) (0 1) (4) (1 2) (4) (2 3) (4) (3 4)
- diag (m1, m2) → SparseMatrix
  
  Returns a block diagonal matrix with blocks m1 and m2.
  Parameters
  Matrix m1
  Matrix m2
  Returns
  SparseMatrix
  
  Example:
  > $m1 = new Matrix([1,2],[3,4]);> $m2 = new Matrix([1,0,2],[3,4,0]);> $D = diag($m1,$m2);> print $D; (5) (0 1) (1 2) (5) (0 3) (1 4) 0 0 1 0 2 0 0 3 4 0
- equal_bases (M1, M2) → Bool
  
  Check whether both matrices are bases of the same linear subspace. Note: It is assumed that they are *bases* of the row space.
  Parameters
  Matrix M1
  Matrix M2
  Returns
  Bool
  
  Example:
  > $M1 = new Matrix([1,1,0],[1,0,1],[0,0,1]);> $M2 = new Matrix([1,0,0],[0,1,0],[0,0,1]);> print equal_bases($M1,$M2); 1
- hadamard_product (M1, M2) → Matrix
  
  Compute the Hadamard product of two matrices with same dimensions.
  Parameters
  Matrix M1
  Matrix M2
  Returns
  Matrix
- hermite_normal_form (M) → List
  
  Computes the (column) Hermite normal form of an integer matrix. Pivot entries are positive, entries to the left of a pivot are non-negative and strictly smaller than the pivot.
  Parameters
  Matrix M
  Matrix to be transformed.
  Options
  Bool reduced
  If this is false, entries to the left of a pivot are left untouched. True by default
  Returns
  List
  (Matrix N, SparseMatrix R) such that M*R=N, R quadratic unimodular.
  
  Example:
  The following stores the result for a small matrix M in H and then prints both N and R:> $M = new Matrix<Integer>([1,2],[2,3]);> @H = hermite_normal_form($M);> print $H[0]; 1 0 0 1> print $H[1]; -3 2 2 -1
- householder_trafo (b) → Vector
  
  Householder tranformation of Vector b. Only the orthogonal matrix reflection H is returned.
  Parameters
  Vector b
  Returns
  Vector
- inv (A) → Matrix
  
  Computes the inverse A^-1 of an invertible matrix A using Gauss elimination.
  Parameters
  Matrix A
  Returns
  Matrix
  
  Example:
  We save the inverse of a small matrix M in the variable $iM:> $M = new Matrix([1,2],[3,4]);> $iM = inv($M); To print the result, type this:> print $iM; -2 1 3/2 -1/2 As we can see, that is in fact the inverse of M.> print $M * $iM; 1 0 0 1
- lineality_space (A) → Matrix
  
  Compute the lineality space of a matrix A.
  Parameters
  Matrix A
  Returns
  Matrix
  
  Example:
  > $M = new Matrix([1,1,0,0],[1,0,1,0]);> print lineality_space($M); 0 0 0 1
- lin_solve (A, b) → Vector
  
  Computes the solution of the system Ax = b
  Parameters
  Matrix A
  must be invertible
  Vector b
  Returns
  Vector
  
  Example:
  from the Wikipedia:> $A = new Matrix([3,2,-1],[2,-2,4],[-1,1/2,-1]);> $b = new Vector(1,-2,0);> print lin_solve($A,$b); 1 -2 -2
- moore_penrose_inverse (M) → Matrix<Float>
  
  Moore-Penrose Inverse of a Matrix
  Parameters
  Matrix M
  Returns
  Matrix<Float>
- normalized (A) → Matrix<Float>
  
  Normalize a matrix by dividing each row by its length (l2-norm).
  Parameters
  Matrix<Float> A
  Returns
  Matrix<Float>
  
  Example:
  > $A = new Matrix<Float>([1.5,2],[2.5,2.5]);> print normalized($A); 0.6 0.8 0.7071067812 0.7071067812
- null_space (A) → Matrix
  
  Compute the null space of a matrix A.
  Parameters
  Matrix A
  Returns
  Matrix
  
  Example:
  > $A = new Matrix([1,2,0],[2,0,2]);> print null_space($A); -1 1/2 1
- null_space (b) → Matrix
  
  Compute the null space of a vector b.
  Parameters
  Vector b
  Returns
  Matrix
  
  Example:
  > $b = new Vector(1,2,3); polytope > print null_space($b); -2 1 0 -3 0 1
- ones_vector <Element> (d) → Vector<Element>
  
  Creates a vector with all elements equal to 1.
  Type Parameters
  Element
  default: 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.
  Returns
  Vector<Element>
  
  Example:
  To create the all-ones Int vector of dimension 3, do this:> $v = ones_vector<Int>(3); You can print the result using the print statement:> print $v; 1 1 1
- pluecker (V) → Vector
  
  Compute the vector of maximal minors of a matrix. WARNING: interpretation different in tropical::lifted_pluecker
  Parameters
  Matrix V
  Returns
  Vector
- project_to_orthogonal_complement (points, orthogonal)
  
  Projects points into the orthogonal complement of a subspace given via an orthogonal basis. The given points will be overwitten.
  Parameters
  Matrix points
  will be changed to orthogonal ones
  Matrix orthogonal
  basis of the subspace
- qr_decomp (M) → Pair<Matrix,Matrix>
  
  QR decomposition of a Matrix M with rows > cols
  Parameters
  Matrix<Float> M
  Returns
  Pair<Matrix,Matrix>
  
  Example:
  > $M = new Matrix<Float>([23,4],[6,42]);> $qr = qr_decomp($M);> print $qr->first; 0.9676172724 0.2524218971 0.2524218971 -0.9676172724> print $qr->second; 23.76972865 14.47218877 0 -39.63023785> print $qr->first * $qr->second ; 23 4 6 42
- rank (A) → Int
  
  Computes the rank of a matrix.
  Parameters
  Matrix A
  Returns
  Int
- reduce (A, b) → Vector
  
  Reduce a vector with a given matrix using Gauss elimination.
  Parameters
  Matrix A
  Vector b
  Returns
  Vector
- singular_value_decomposition (M) → SingularValueDecomposition
  
  SVD decomposition of a Matrix. Computes the SVD of a matrix into a diagonal Marix (S), orthogonal square Matrix (U), orthogonal square Matrix (V), such that U*S*V^T=M The first element of the output array is S, the second U and the thrid V.
  Parameters
  Matrix<Float> M
  Returns
  SingularValueDecomposition
  
  Example:
  > $M = new Matrix<Float>([1,2],[23,24]);> $SVD = singular_value_decomposition($M); The following prints the three matrices, seperated by newline characters.> print $SVD->left_companion ,"\n", $SVD->sigma ,"\n", $SVD->right_companion; 0.06414638608 0.9979404998 0.9979404998 -0.06414638608 33.31011547 0 0 0.6604600341 0.6909846321 -0.7228694476 0.7228694476 0.6909846321
- smith_normal_form (M, inv) → SmithNormalForm<Integer>
  
  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
  Returns
  SmithNormalForm<Integer>
  
  Example:
  > $M = new Matrix<Integer>([1,2],[23,24]);> $SNF = smith_normal_form($M); The following line prints the three matrices seperated by newline characters.> print $SNF->left_companion ,"\n", $SNF->form ,"\n", $SNF->right_companion; 1 0 23 1 1 0 0 -22 1 2 0 1
- totally_unimodular (A) → Bool
  
  The matrix A is totally unimodular if the determinant of each square submatrix equals 0, 1, or -1. This is the naive test (exponential in the size of the matrix).
  For a better implementation try Matthias Walter's polymake extension at https://github.com/xammy/unimodularity-test/wiki/Polymake-Extension.
  Parameters
  Matrix A
  Returns
  Bool
  
  Example:
  > $M = new Matrix<Int>([-1,-1,0,0,0,1],[1,0,-1,-1,0,0],[0,1,1,0,-1,0],[0,0,0,1,1,-1]);> print totally_unimodular($M); 1
- trace (A) → Int
  
  Computes the trace of a matrix.
  Parameters
  Matrix A
  Returns
  Int
  trace(A)
  
  Example:
  > $M = new Matrix([1,2,3],[23,24,25],[0,0,1]);> print trace($M); 26
- transpose (A) → IncidenceMatrix
  
  Computes the transpose A^T of an incidence matrix A, i.e., (a^T)_ij = a_ji.
  Parameters
  IncidenceMatrix A
  Returns
  IncidenceMatrix
- transpose (A) → Matrix
  
  Computes the transpose A^T of a matrix A, i.e., (a^T)_ij = a_ji.
  Parameters
  Matrix A
  Returns
  Matrix
  
  Example:
  > $M = new Matrix([1,2,23],[23,22,21]);> $Mt = transpose($M);> print $Mt; 1 23 2 22 23 21
- unit_matrix <Element> (d) → SparseMatrix<Element>
  
  Creates a unit matrix of given dimension
  Type Parameters
  Element
  default: Rational
  Parameters
  Int d
  dimension of the matrix
  Returns
  SparseMatrix<Element>
  
  Examples:
  The following stores the 3-dimensional unit matrix (ones on the diagonal and zeros otherwise) in a variable and prints it:> $M = unit_matrix(3);> print $M; (3) (0 1) (3) (1 1) (3) (2 1)
  The following stores the 3-dimensional unit matrix (ones on the diagonal and zeros otherwise) from type Int in a variable and prints it:> $M = unit_matrix<Int>(3);> print $M->type->full_name; SparseMatrix<Int, Symmetric>
- 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
  default: Rational
  Parameters
  Int d
  the dimension of the vector
  Int pos
  the position of the 1
  Returns
  SparseVector<Element>
  # @example The following stores a vector of dimension 5 with a single 1 (as a Rational) at position 2: > $v = unit_vector(5,2); > print $v; | (5) (2 1)
  
  Examples:
  The following stores a vector of dimension 5 with a single 1 (as a Int) at position 2:> $v = unit_vector<Int>(5,2);> print $v->type->full_name; SparseVector<Int>
  The following concatenates a unit vector of dimension 3 with a 1 at position 2 and a unit vector of dimension 2 with a 1 at position 1:> $v = unit_vector(3,2) | unit_vector(2,1);> print $v; (5) (2 1) (4 1)
- zero_matrix <Element> (i, j) → SparseMatrix<Element>
  
  Creates a zero matrix of given dimensions
  Type Parameters
  Element
  default: Rational
  Parameters
  Int i
  number of rows
  Int j
  number of columns
  Returns
  SparseMatrix<Element>
  
  Examples:
  The following stores a 2x3 matrix with 0 as entries (from type Rational) in a variable and prints it:> $M = zero_matrix(2,3);> print $M; 0 0 0 0 0 0
  The following stores a 2x3 matrix with 0 as entries from type Int in a variable and prints its type:> $M = zero_matrix<Int>(2,3);> print $M->type->full_name; Matrix<Int, NonSymmetric>
- zero_vector <Element> (d) → Vector<Element>
  
  Creates a vector with all elements equal to zero.
  Type Parameters
  Element
  default: 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.
  Returns
  Vector<Element>
  
  Examples:
  The following stores a vector of dimension 5 with 0 as entries (from type Rational) in a variable and prints it:> $v = zero_vector(5);> print $v; 0 0 0 0 0
  The following stores a vector of dimension 5 with 0 as entries from type Int in a variable and prints its type:> $v = zero_vector<Int>(5);> print $v->type->full_name; Vector<Int>
  The following concatenates a vector of dimension 2 of ones and a vector of length 2 of zeros:> $v = ones_vector(2) | zero_vector(2);> print $v; 1 1 0 0
Set Operations
This category contains functions performing operations on Sets.
- incl (s1, s2) → Int
  
  Analyze the inclusion relation of two sets.
  Parameters
  Set s1
  Set s2
  Returns
  Int
  0 if s1 = s2,
  -1 if s1 ⊂ s2,
  1 if s1 ⊃ s2,
  2 otherwise.
  
  Example:
  > $s1 = new Set(1,2,3); $s2 = $s2 - 1;> print incl($s1,$s2); 1> print incl($s2,$s1); -1> print incl($s1,$s1); 0> print incl($s2,$s1-$s2); 2
- range (a, b) → Set<Int>
  
  Creates the Set {a, a+1, ..., b-1, b} for a ≤ b.
  Parameters
  Int a
  minimal element of the set
  Int b
  maximal element of the set
  Returns
  Set<Int>
  
  Example:
  > print range(23,27); {23 24 25 26 27}
- scalar2set (s) → Set<SCALAR>
  
  Returns the singleton set {s}.
  Parameters
  SCALAR s
  Returns
  Set<SCALAR>
  
  Example:
  > print scalar2set(23); {23}
- select_subset (s, indices) → Set
  
  Returns the subset of s given by the indices.
  Parameters
  Set s
  Set<Int> indices
  Returns
  Set
  
  Example:
  > $s = new Set<Int>(23,42,666,789);> $ind = new Set<Int>(0,2);> $su = select_subset($s,$ind);> print $su; {23 666}
- sequence (a, c) → Set<Int>
  
  Creates the Set {a, a+1, ..., a+c-1}.
  Parameters
  Int a
  the smallest element
  Int c
  the cardinality
  Returns
  Set<Int>
  
  Example:
  > print sequence(23,6); {23 24 25 26 27 28}

Utilities

Miscellaneous functions.

average (array)
Returns the average value of the array elements.
Parameters
ARRAY array
Example:
> print average([1,2,3]); 2
bounding_box (m) → Matrix

Compute a column-wise bounding box for the given Matrix m.
Parameters
Matrix m
Returns
Matrix
a Matrix with two rows and m->cols columns; row(0) contains lower bounds, row(1) contains upper bounds.
fibonacci (m) → ARRAY

Returns the first m Fibonacci numbers.
Parameters
Int m
Returns
ARRAY
histogram (data) → Map<Element, Int>
Produce a histogram of an array: each different element value is mapped on the number of its occurences.
Parameters
ARRAY data
Returns
Map<Element, Int>
Example:
> $H = histogram([1,1,2,2,2,3,3,2,3,3,1,1,1,3,2,3]);> print $H; {(1 5) (2 5) (3 6)}
index_of (array) → Map<Array<Set<Int>>, Int>
Return a map indexing an array of sets
Parameters
Array<Set<Int>> array
Returns
Map<Array<Set<Int>>, Int>
Example:
> $s1 = new Set(1,2,3);> $s2 = $s2 - 1;> $a = new Array<Set>($s1,$s2,$s1);> print index_of($a); {({1 2 3} 2) ({2 3} 1)}
maximum (array)
Returns the maximal element of an array.
Parameters
ARRAY array
Example:
> print maximum([1,2,3,4,5,6,7,23]); 23
minimum (array)
Returns the minimal element of an array.
Parameters
ARRAY array
Example:
> print minimum([23,42,666]); 23

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

rand_perm (n) → Array<Int>

gives a random permutation
Parameters
Int n
Options
Int Seed
Returns
Array<Int>
random permutation

Visualization

These functions are for visualization.

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.
option list: Visual::Polygons::decorations
Returns
Visual::Container
if called in void context, immediately starts the preferred rendering program.
Example:
Draw a pretty 8-pointed star:> compose(cube(2)->VISUAL,cross(2,sqrt(2))->VISUAL,Title=>"A pretty star.",VertexLabels=>"hidden");
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.

javaview (vis_obj ...)

Run JavaView with the given visual objects.

Contained in extension bundled:javaview.

Parameters

Visual::Object

vis_obj ...

objects to display

Options

String

File

"filename" or "AUTO" Store the object description in a JVX file without starting the interactive GUI. The .jvx 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.

jreality (vis_obj ...)

Run jReality to display given visual objects.

Contained in extension bundled:jreality.

Parameters

Visual::Object

vis_obj ...

objects to display

Options

String

File

"filename" or "AUTO" Store the object description in a bean shell source file without starting the interactive GUI. The .bsh 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.

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.

threejs (vis_obj)

Produce an html file with given visual objects.

Parameters

Visual::Object

vis_obj

object to display

Options

String

File

"filename" or "AUTO"

An automatically generated file name is displayed in the verbose mode.

tikz (vis_obj)

Produce a TikZ file with given visual objects.

Parameters

Visual::Object

vis_obj

object to display

Options

String

File

"filename" or "AUTO"

An automatically generated file name is displayed in the verbose mode.

Property Types

Algebraic Types
This category contains all "algebraic" types, such as matrices, vectors, polynomials, rings, ...
- all_rows_or_cols
  
  Use the keyword "All" for all rows or columns, e.g. when constructing a minor.
- Matrix <Element, Sym>
  UNDOCUMENTED
  Type Parameters
  Element
  default: Rational
  Sym
  default: NonSymmetric
  User Methods of Matrix
  
  anti_diagonal (i) → Vector<Element>
  
  Returns the anti-diagonal of the matrix.
  Parameters
  Int i
  i=0: the main anti_diagonal (optional)
  i>0: the i-th anti_diagonal below the main anti_diagonal
  i<0: the i-th anti_diagonal above the main anti_diagonal
  Returns
  Vector<Element>
  
  col (i) → Vector<Element>
  
  Returns the i-th column.
  Parameters
  Int i
  Returns
  Vector<Element>
  
  cols () → Int
  
  Returns the number of columns.
  Returns
  Int
  
  diagonal (i) → Vector<Element>
  
  Returns the diagonal of the matrix.
  Parameters
  Int i
  i=0: the main diagonal (optional)
  i>0: the i-th diagonal below the main diagonal
  i<0: the i-th diagonal above the main diagonal
  Returns
  Vector<Element>
  
  div_exact (a) → Matrix
  
  Divides every entry by a (assuming that every entry is divisible by a).
  Parameters
  Int a
  Returns
  Matrix
  
  elem (r, c) → Element
  
  Returns an element of the matrix. The return value is an `lvalue', that is, it can be modified if the matrix object is mutable.
  Parameters
  Int r
  the row index
  Int c
  the column index
  Returns
  Element
  
  minor (r, c) → Matrix
  
  Returns a minor of the matrix containing the rows in r and the columns in c. You can pass All if you want all rows or columns and ~ for the complement of a set. E.g.
  $A->minor(All, ~[0]);
  will give you the minor of a matrix containing all rows and all but the 0-th column.
  Parameters
  Set r
  the rows
  Set c
  the columns
  Returns
  Matrix
  
  row (i) → Vector<Element>
  
  Returns the i-th row.
  Parameters
  Int i
  Returns
  Vector<Element>
  
  rows () → Int
  
  Returns the number of rows.
  Returns
  Int
- Monomial <Coefficient, Exponent>
  UNDOCUMENTED
  Type Parameters
  Coefficient
  default: Rational
  Exponent
  default: Int
- Plucker <Scalar>
  UNDOCUMENTED
  Type Parameters
  Scalar
  default: Rational
  User Methods of Plucker
  
  coordinates ()
  
  UNDOCUMENTED
  
  permuted ()
  
  UNDOCUMENTED
- 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
  
  print_ordered ()
  
  UNDOCUMENTED
  
  trivial ()
  
  UNDOCUMENTED
- PuiseuxFraction <MinMax, Coefficient, Exponent>
  UNDOCUMENTED
  Type Parameters
  MinMax
  type of tropical addition: either Min or Max
  Coefficient
  default: Rational
  Exponent
  default: Rational
  User Methods of PuiseuxFraction
  
  evaluate (m, x, exp) → Matrix<Coefficient>
  
  Evaluate all PuiseuxFractions in a Matrix at a Rational number (x^exp). Let explcm be the lcm of the denominators of all exponents. If there are no denominators or explcm divides exp, then the evaluation is computed exactly. Otherwise, some rational number close to the root (x^exp)^-explcm will be chosen via an intermediate floating point number.
  Parameters
  Matrix m
  Coefficient x
  Int exp
  (default: 1)
  Returns
  Matrix<Coefficient>
  
  evaluate (v, x, exp) → Vector<Coefficient>
  
  Evaluate all PuiseuxFractions in a Vector at a Rational number (x^exp). Let explcm be the lcm of the denominators of all exponents. If there are no denominators or explcm divides exp, then the evaluation is computed exactly. Otherwise, some rational number close to the root (x^exp)^-explcm will be chosen via an intermediate floating point number.
  Parameters
  Vector v
  Coefficient x
  Int exp
  (default: 1)
  Returns
  Vector<Coefficient>
  
  evaluate (x, exp) → Coefficient
  
  Evaluate a PuiseuxFraction at a Rational number (x^exp). Let explcm be the lcm of the denominators of all exponents. If there are no denominators or explcm divides exp, then the evaluation is computed exactly. Otherwise, some rational number close to the root (x^exp)^-explcm will be chosen via an intermediate floating point number.
  Parameters
  Coefficient x
  Int exp
  (default: 1)
  Returns
  Coefficient
  
  evaluate_float (x) → Float
  
  Approximate evaluation at x
  Parameters
  Float x
  Returns
  Float
  
  evaluate_float (m, x) → Float
  
  Approximate evaluation of a Matrix at x
  Parameters
  Matrix m
  Float x
  Returns
  Float
  
  evaluate_float (v, x) → Float
  
  Approximate evaluation of a Vector at x
  Parameters
  Vector v
  Float x
  Returns
  Float
  
  val () → TropicalNumber<MinMax>
  
  The valuation.
  Returns
  TropicalNumber<MinMax>
- RationalFunction <Coefficient, Exponent>
  UNDOCUMENTED
  Type Parameters
  Coefficient
  default: Rational
  Exponent
  default: Int
- 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
- SparseMatrix <Element, Sym>
  A SparseMatrix is a two-dimensional associative array with row and column indices as keys; elements equal to the default value (ElementType(), which is 0 for most numerical types) are not stored, but implicitly encoded by the gaps in the key set. Each row and column is organized as an AVL-tree.
  Use dense to convert this into its dense form.
  You can create a new SparseMatrix by entering its entries row by row, as a list of SparseVectors e.g.:
  $A = new SparseMatrix<Int>(<< '.');
  (5) (1 1)
  (5) (4 2)
  (5)
  (5) (0 3) (1 -1)
  .
  derived from: Matrix
  Type Parameters
  Element
  Sym
  one of Symmetric or NonSymmetric, default: NonSymmetric
  User Methods of SparseMatrix
  
  resize ()
  
  Resize the matrix
  
  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 an associative container with element indices (coordinates) as keys; elements equal to the default value (ElementType(), which is 0 for most numerical types) are not stored, but implicitly encoded by the gaps in the key set. It is based on an AVL tree.
  The printable representation of a SparseVector looks like a sequence (l) (p₁ v₁) ... (p_k v_k), where l is the dimension 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 printable encoding as described above, e.g.:
  $v = new SparseVector<Int>(<< '.');
  (6) (1 1) (2 2)
  .
  derived from: Vector
  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
  
  evaluate ()
  
  UNDOCUMENTED
  
  evaluate_float ()
  
  UNDOCUMENTED
  
  lc () → Int
  
  The leading coefficient.
  Returns
  Int
  
  lower_deg ()
  
  UNDOCUMENTED
  
  print_ordered ()
  
  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 elements, e.g.:
  $v = new Vector<Int>(1,2,3);
  or
  $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 (i) → Vector
  
  Returns a Vector containing only the entries after i.
  Parameters
  Int i
  Returns
  Vector
  
  slice (i, j) → Vector
  
  Returns a Vector containing only the entries from i to j-1.
  Parameters
  Int i
  Int j
  Returns
  Vector
Arithmetic
These types are needed as return types of arithmetic computations.
- Div <Scalar>
  The complete result of an integral division, quotient and remainder.
  Type Parameters
  Scalar
  User Methods of Div
  
  quot () → Scalar
  
  The quotient.
  Returns
  Scalar
  
  rem () → Scalar
  
  The remainder.
  Returns
  Scalar
- ExtGCD
  
  The complete result of the calculation of the greatest common divisor of two numbers a and b:
  
  g=gcd(a,b)
  
  g=a*p+b*q
  
  a=g*k1
  
  b=g*k2
  
  User Methods of ExtGCD
  
  g () → Int
  
  The greatest common divisor of a and b.
  Returns
  Int
  
  k1 () → Int
  
  The factor of a.
  Returns
  Int
  
  k2 () → Int
  
  The factor of b.
  Returns
  Int
  
  p () → Int
  
  The co-factor of a.
  Returns
  Int
  
  q () → Int
  
  The co-factor of b.
  Returns
  Int
- Max
  
  tropical addition: max
  
  User Methods of Max
  
  apply ()
  
  UNDOCUMENTED
  
  orientation ()
  
  UNDOCUMENTED
- Min
  
  tropical addition: min
  
  User Methods of Min
  
  apply ()
  
  UNDOCUMENTED
  
  orientation ()
  
  UNDOCUMENTED
- TropicalNumber <Addition, Scalar>
  UNDOCUMENTED
  Type Parameters
  Addition
  Scalar
  default: Rational
  User Methods of TropicalNumber
  
  orientation () → Int
  
  The orientation of the associated addition, i.e. +1 if the corresponding 0 is +inf -1 if the corresponding 0 is -inf
  Returns
  Int
  
  zero () → Scalar
  
  The zero element of the tropical semiring of this element.
  Returns
  Scalar
Artificial
These types are auxiliary artifacts helping to build other classes, primarily representing template parameters or enumeration constants. They should not be used alone as property types or function arguments. In the most cases they won't even have user-accessible constructors.
- ARRAY
  
  A perl array of whatever contents.
- Directed
  
  Type tag for a directed Graph.
- DirectedMulti
  
  Type tag for a directed multigraph.
- EdgeIterator
  
  UNDOCUMENTED
- EdgeList
  
  UNDOCUMENTED
- Iterator <Element>
  An iterator over a sequence of objects. The objects may be stored in a container or be generated on the fly.
  Type Parameters
  Element
- LocalFloatEpsilon
  
  UNDOCUMENTED
- NodeIterator
  
  UNDOCUMENTED
- NodeSet
  
  UNDOCUMENTED
- NonSymmetric
  
  Labels a Matrix or an IncidenceMatrix as non-symmetric.
- SCALAR
  
  A perl scalar of whatever contents.
- Serialized <X>
  UNDOCUMENTED
  Type Parameters
  X
- SparseIterator
  
  Iterator over a sequence of objects having indexes. Examples are non-zero elements of a SparseVector or of a row/column of a SparseMatrix.
- Symmetric
  
  Labels a Matrix or an IncidenceMatrix as symmetric.
- Undirected
  
  Type tag for an undirected Graph.
- UndirectedMulti
  
  Type tag for an undirected multigraph.
Basic Types
This category contains all basic types, in particular those that wrap C++, GMP or perl types such as Int, Integer, Rational, Long, Float, Array, String, ...
- Array <Element>
  An array with elements of type Element. Corresponds to the C++ type Array.
  Type Parameters
  Element
  User Methods of Array
  
  size () → Int
  
  Returns the size.
  Returns
  Int
- Bool
  
  Corresponds to the C++ type bool.
- Float
  
  Corresponds to the C++ type double.
  
  User Methods of Float
  
  inf ()
  
  produce an infinitely large positive value
  
  minus_inf ()
  
  produce an infinitely large negative value
- Int
  
  Corresponds to the C++ type int.
- Integer
  
  An integer of arbitrary size.
  
  User Methods of Integer
  
  inf ()
  
  produce an infinitely large positive value
  
  minus_inf ()
  
  produce an infinitely large negative value
- List <Element>
  Corresponds to the C++ type std::list.
  Type Parameters
  Element
- Long
  
  Corresponds to the C++ type long. This type is primarily needed for automatic generated function wrappers, because perl integral constants are always kept as a long.
  
  derived from: Int
- Pair <First, Second>
  Corresponds to the C++ type std::Pair.
  Type Parameters
  First
  Second
- QuadraticExtension <Field>
  Realizes quadratic extensions of fields.
  You can construct the value a+b$\sqrt r$ via QuadraticExtension(a, b, r) (where a, b, r are of type Field).
  Type Parameters
  Field
  default: Rational
- Rational
  
  A class for rational numbers. You can easily create a Rational like that: $x = 1/2;
  
  User Methods of Rational
  
  inf ()
  
  Produce an infinitely large positive value.
  
  minus_inf ()
  
  Produce an infinitely large negative value.
- String
  
  Corresponds to the C++ type std::string.
- Text
  
  Plain text without any imposed structure.
Graph Types
This contains all property types that are related to graphs.
- EdgeHashMap <Dir, Element>
  Sparse mapping of edges to data items.
  derived from: GraphMap
  Type Parameters
  Dir
  one of Undirected, Directed, UndirectedMulti or DirectedMulti
  Element
  data associated with edges
  User Methods of EdgeHashMap
  
  edge (from, to)
  
  Access the data associated with an edge between two given nodes. The new data element is created on demand.
  Parameters
  Int from
  source node
  Int to
  target node
  
  erase (from, to)
  
  Delete the data associated with an edge between two given nodes.
  Parameters
  Int from
  source node
  Int to
  target node
  
  find (from, to) → Iterator
  
  Access the data associated with an edge between two given nodes.
  Parameters
  Int from
  source node
  Int to
  target node
  Returns
  Iterator
  pointing to the data element (must be dereferenced as ${...}) or undef if the element does not exist.
- EdgeMap <Dir, Element>
  Dense mapping of edges to data items.
  derived from: GraphMap
  Type Parameters
  Dir
  kind of the host graph, Undirected, Directed, UndirectedMulti, or DirectedMulti
  Element
  data associated with edges
  User Methods of EdgeMap
  
  edge (from, to)
  
  Access the data associated with an edge between two given nodes.
  Parameters
  Int from
  source node
  Int to
  target node
- Graph <Dir>
  UNDOCUMENTED
  Type Parameters
  Dir
  one of Undirected, Directed, UndirectedMulti or DirectedMulti, default: Undirected
  User Methods of Graph
  
  add_edge (tail_node, head_node) → Int
  
  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 index of the (new) edge.
  Parameters
  Int tail_node
  Int head_node
  Returns
  Int
  
  add_node () → Int
  
  Adds a new node without incident edes, returns its index.
  Returns
  Int
  
  adjacent_nodes (node) → Set
  
  Returns the set of indices of nodes adjacent to node.
  Parameters
  Int node
  Returns
  Set
  
  all_edges (tail_node, head_node) → Iterator
  
  Returns an iterator visiting all (parallel) edges connecting two given nodes.
  Parameters
  Int tail_node
  Int head_node
  Returns
  Iterator
  
  contract_edge (node1, node2)
  
  Contract the edge(s) between node1 and 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) → Int
  
  Returns the number of edges incident to node.
  Parameters
  Int node
  Returns
  Int
  
  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 () → Int
  
  Returns the maximal node index + 1. If the graph does not have gaps caused by node deletion, the result is equivalent to nodes().
  Returns
  Int
  
  edge (tail_node, head_node) → Int
  
  Returns the index 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
  Returns
  Int
  
  edges () → Int
  
  Returns the total number of edges.
  Returns
  Int
  
  edge_exists (tail_node, head_node) → Bool
  
  Checks whether two given nodes are connected by (at least) one edge.
  Parameters
  Int tail_node
  Int head_node
  Returns
  Bool
  
  has_gaps () → Bool
  
  Returns true if some nodes have been deleted since the last squeeze operation.
  Returns
  Bool
  
  invalid_node (node) → Bool
  
  Returns true if the given node index is either out of valid range or points to a formerly deleted node.
  Parameters
  Int node
  Returns
  Bool
  
  in_adjacent_nodes (node) → Set
  
  Returns the set of indices of the nodes that have an edge heading to node.
  Parameters
  Int node
  Returns
  Set
  
  in_degree (node) → Int
  
  Returns the number of edges heading to node.
  Parameters
  Int node
  Returns
  Int
  
  in_edges (node) → EdgeList
  
  Returns a sequence of edges heading to (in Directed case) or incident to (in Undirected case) node.
  Parameters
  Int node
  Returns
  EdgeList
  
  nodes () → Int
  
  Returns the total number of nodes.
  Returns
  Int
  
  node_exists (node) → Bool
  
  Checks whether the node with given index exists.
  Parameters
  Int node
  Returns
  Bool
  
  node_out_of_range (node) → Bool
  
  Returns true if the given node index is out of valid range.
  Parameters
  Int node
  Returns
  Bool
  
  out_adjacent_nodes (node) → Set
  
  Returns the set of indices of the nodes with an edge arriving from node.
  Parameters
  Int node
  Returns
  Set
  
  out_degree (node) → Int
  
  Returns the number of edges leaving node.
  Parameters
  Int node
  Returns
  Int
  
  out_edges (node) → EdgeList
  
  Returns a sequence of edges leaving (in Directed case) or incident to (in Undirected case) node.
  Parameters
  Int node
  Returns
  EdgeList
  
  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.
- GraphMap <Dir, Element>
  The common abstract base class for all kinds of associative containers that can be attached to a Graph.
  Type Parameters
  Dir
  kind of the host graph: Undirected, Directed, UndirectedMulti, or DirectedMulti
  Element
  data associated with nodes or edges
- NodeHashMap <Dir, Element>
  Sparse mapping of nodes to data items.
  derived from: GraphMap
  Type Parameters
  Dir
  one of Undirected, Directed, UndirectedMulti or DirectedMulti
  Element
  data associated with nodes
- NodeMap <Dir, Element>
  Dense mapping of nodes to data items.
  derived from: GraphMap
  Type Parameters
  Dir
  kind of the host graph, Undirected, Directed, UndirectedMulti, or DirectedMulti
  Element
  data associated with nodes
Linear Algebra
These types are needed as return types of algebraic computations.
- SingularValueDecomposition
  
  Complete result of the singular value decomposition of a matrix M, such that left_companion * sigma * transpose(right_companion) = M Contains the following fields:
  
  Matrix<Float> sigma: the diagonalized matrix
  
  Matrix<Float> left_companion: matrix of left singular vectors
  
  Matrix<Float> right_companion: matrix of right singular vectors
- 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).
Set Types
In this category you find all property types related to sets, such as Set, Map, HashMap, IncidenceMatrix, ...
- boost_dynamic_bitset
  
  UNDOCUMENTED
  
  Contained in extension bundled:group.
  derived from: Set<Int>
  
  User Methods of boost_dynamic_bitset
  
  size () → Int
  
  The cardinality of the set.
  Returns
  Int
- FacetList
  
  A FacetList is a collection of sets of integral numbers from a closed contiguous range [0..n-1]. The contained sets usually encode facets of a simplicial complex, with set elements corresponding to vertices of a complex, therefore the name.
  
  From the structural perspective, FacetList is interchangeable with IncidenceMatrix, but they significantly differ in supported operations and their performance. IncidenceMatrix offers fast random access to elements, while FacetList is optimized for finding, inserting, and deleting facets fulfilling certain conditions like all subsets or supersets of a given vertex set.
  
  On perl side, FacetList behaves like a sequence of Set<Int> without random access to facets. Facets are visited in chronological order. Each facet has a unique integral ID generated at the moment of insertion. The IDs can be obtained via call to index() of iterators created by find() methods.
  
  User Methods of FacetList
  
  erase (f) → Bool
  
  Remove a facet.
  Parameters
  Set f
  facet to remove
  Returns
  Bool
  whether a facet existed before
  
  eraseSubsets (s) → Int
  
  Remove all subsets of a given set
  Parameters
  Set s
  filter for removal
  Returns
  Int
  number of removed facets
  
  eraseSupersets (s) → Int
  
  Remove all supersets of a given set
  Parameters
  Set s
  filter for removal
  Returns
  Int
  number of removed facets
  
  find (f) → Iterator
  
  Look up a facet.
  Parameters
  Set f
  facet to find
  Returns
  Iterator
  pointing to the facet or in an invalid state
  
  findSubsets (s) → Iterator
  
  Find all subsets of a given set.
  Parameters
  Set s
  Returns
  Iterator
  all facets equal to or included in s, visited in lexicographical order
  
  findSupersets (s) → Iterator
  
  Find all supersets of a given set.
  Parameters
  Set s
  Returns
  Iterator
  all facets equal to or including s, visited in reverse chronological order
  
  insert (f)
  
  Add a new facet. It may be a proper subset or a proper superset of existing facets. It must not be empty or coincide with any existing facet.
  Parameters
  Set f
  facet to add.
  
  insertMax (f) → Bool
  
  Add a new facet if and only if there are no facets including it. If this holds, remove all facets that are included in the new one.
  Parameters
  Set f
  facet to add
  Returns
  Bool
  whether the new facet was really included.
  
  insertMin (f) → Bool
  
  Add a new facet if and only if there are no facets included in it. If this holds, remove all facets including the new facet.
  Parameters
  Set f
  facet to add
  Returns
  Bool
  whether the new facet was really included.
  
  n_vertices () → Int
  
  The number of vertices
  Returns
  Int
  
  size () → Int
  
  The number of facets in the list.
  Returns
  Int
- HashMap <Key, Value>
  Similar to Map. HashMaps are associative containers that contain unique key/value pairs.
  The values are stored in a hash table. Accessing and interserting a value by its key works in constant time O(1).
  You can create a new HashMap mapping Ints to Strings by
  $myhashmap = new HashMap<Int, String>([1, "Monday"], [2, "Tuesday"]);
  On the perl side HashMaps are treated like hashrefs. You can work with a HashMap like you work with a Map (keeping in mind differences in performance and memory demand).
  Type Parameters
  Key
  type of the key values
  Value
  type of the mapped value
- HashSet <Element>
  Similar to Set. (But keep in mind differences in performance and memory demand.)
  Type Parameters
  Element
- IncidenceMatrix <Sym>
  A 0/1 incidence matrix.
  Type Parameters
  Sym
  one of Symmetric or NonSymmetric, default: NonSymmetric
  User Methods of IncidenceMatrix
  
  col (i) → SparseVector<Int>
  
  Returns the i-th column.
  Parameters
  Int i
  Returns
  SparseVector<Int>
  
  cols () → Int
  
  Returns the number of columns.
  Returns
  Int
  
  elem (r, c) → Bool
  
  Returns an element of the matrix as a boolean value. The return value is an `lvalue', that is, it can be assigned to, flipped, etc. if the matrix object is mutable.
  Parameters
  Int r
  the row index
  Int c
  the column index
  Returns
  Bool
  
  minor (r, c) → IncidenceMatrix
  
  Returns a minor of the matrix containing the rows in r and the columns in c. You can pass All if you want all rows or columns and ~ for the complement of a set. E.g.
  $A->minor(All, ~[0]);
  will give you the minor of a matrix containing all rows and all but the 0-th column.
  Parameters
  Set r
  the rows
  Set c
  the columns
  Returns
  IncidenceMatrix
  
  row (i) → SparseVector<Int>
  
  Returns the i-th row.
  Parameters
  Int i
  Returns
  SparseVector<Int>
  
  rows () → Int
  
  Returns the number of rows.
  Returns
  Int
  
  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.
- Map <Key, Value>
  Maps are sorted associative containers that contain unique key/value pairs. Maps are sorted by their keys.
  Accessing or inserting a value needs logarithmic time O(log n), where n is the size of the map.
  You can create a new Map mapping Ints to Strings by
  $mymap = new Map<Int, String>([1, "Monday"], [2, "Tuesday"]);
  On the perl side Maps are treated like hashrefs. You can add a new key/value pair by
  $mymap->{3} = "Wednesday";
  (If the key is already contained in the Map, the corresponding value is replaced by the new one.) or ask for the value of a key by
  print $mymap->{1};
  Type Parameters
  Key
  type of the key values
  Value
  type of the mapped value
- PowerSet <Element>
  A Set whose elements are of type Set<Element>.
  derived from: Set
  Type Parameters
  Element
  default: Int
- Set <Element>
  A type for sets containing elements of type Element.
  You can for example create a new Set by:
  $s = new Set(2, 3, 5, 7);
  You can perform set theoretic operations:
  $s1 + $s2 union
  $s1 * $s2 intersection
  $s1 - $s2 difference
  $s1 ^ $s2 symmetric difference
  Type Parameters
  Element
  default: Int
  User Methods of Set
  
  back () → Int
  
  The last element of the (implicitly sorted) set
  Returns
  Int
  
  front () → Int
  
  The first element of the (implicitly sorted) set
  Returns
  Int
  
  size () → Int
  
  The cardinality of the set.
  Returns
  Int
Visualization
These property_types are for visualization.
- Color
  
  This is a pseudo-type for documentation purposes only. A function expecting an argument or option of type Color can digest an object of type RGB or HSV as well as a string with an RGB value in hex notation "#RRGGBB" or a symbolic color name.
- 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.

Common Option Lists

Visualization

These options are for visualization.

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<Color>	PointColor	color of the spheres or rectangles representing the points
Flexible<Color>	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<Color>	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
Flexible<String>	LabelAlignment	Defines the alignment of the vertex labels: left, right or center

Visual::Polygon::decorations

Attributes modifying the appearance of filled polygons.

imports from: Visual::PointSet::decorations

Options

Color	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
Color	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<Color>	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)
Color	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<Color>	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

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.

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

Matrix	points	will be changed to orthogonal ones
Matrix	orthogonal	basis of the subspace

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

Int	a	minimal element of the set
Int	b	maximal element of the set

String	Title	name of the whole drawing; per default the name of the first Object is taken.
option list:	Visual::Polygons::decorations

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

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

MinMax	type of tropical addition: either Min or Max
Coefficient	default: Rational
Exponent	default: Rational

Element
Sym	one of Symmetric or NonSymmetric, default: NonSymmetric

Dir	one of Undirected, Directed, UndirectedMulti or DirectedMulti
Element	data associated with edges

Dir	kind of the host graph, Undirected, Directed, UndirectedMulti, or DirectedMulti
Element	data associated with edges

Dir	kind of the host graph: Undirected, Directed, UndirectedMulti, or DirectedMulti
Element	data associated with nodes or edges

Dir	one of Undirected, Directed, UndirectedMulti or DirectedMulti
Element	data associated with nodes

Navigation

application: common

Objects

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