# 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 $GCD->k2; 2 • fac (n) → Integer Computes the factorial n! = n·(n-1)·(n-2)·...·2·1. ##### Parameters  Int n >=0 ##### Returns  Integer n! • floor (a) → Rational The floor function. Returns the smallest integral number not larger than a. ##### Parameters  Rational a ##### Returns  Rational Example: • > print floor(1.8); 1 • gcd (a, b) → Int Computes the greatest common divisor of two integers. ##### Parameters  Int a Int b ##### Returns  Int Example: • > print gcd(6,9); 3 • gcd (v) → Element Compute the greatest common divisor of the elements of the given vector. ##### Parameters  Vector v ##### Returns  Element Example: • >$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 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 a list of rational numbers (other coefficents are not implemented).
##### Returns
 Map 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

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 p
##### Returns
 Array>

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 p
##### Returns
 Matrix

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 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

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

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

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 v
##### Returns
 Vector
•
dense <Element> (m) → Matrix<Element>

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
•
dense (s, dim) → Vector<Int>

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

##### Parameters
 Set s Int dim
##### Returns
 Vector
•
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

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

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

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

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

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 > , 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 > 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 >
•
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:
•

### 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 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 elem_labels optional labels for elements; if data is an IncidenceMatrix, Array, 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 M the matrix whose rows are to be written
##### Options
 Array coord_labels changes the labels of the coordinates Array 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 row_labels labels for the rows Array elem_labels optional labels for elements; if data is an IncidenceMatrix, Array, or similar, each element will be replaced by its label.
##### Returns
 Array 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 elem_labels labels for the elements
##### Returns
 Array 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 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.

•

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

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

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

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

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

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

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

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

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

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> 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> 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

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($M)); 1 0 0 2 0 0 0 1 0 0 0 1 • basis_rows (A) → Set<Int> Computes a subset of the rows of A that form a basis for the linear space spanned by A. ##### Parameters  Matrix A ##### Returns  Set Example: • Here we have a nice matrix:>$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
•

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,$b); 1 -2 -2 • moore_penrose_inverse (M) → Matrix<Float> Moore-Penrose Inverse of a Matrix ##### Parameters  Matrix M ##### Returns  Matrix • normalized (A) → Matrix<Float> Normalize a matrix by dividing each row by its length (l2-norm). ##### Parameters  Matrix A ##### Returns  Matrix Example: • >$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($b); -2 1 0 -3 0 1 • ones_vector <Element> (d) → Vector<Element> Creates a vector with all elements equal to 1. ##### Type Parameters  Element default: Rational. ##### Parameters  Int d vector dimension. If omitted, a vector of dimension 0 is created, which can adjust itself when involved in a block matrix operation. ##### Returns  Vector Example: • To create the all-ones Int vector of dimension 3, do this:>$v = ones_vector<Int>(3); You can print the result using the print statement:> print $v; 1 1 1 • pluecker (V) → Vector Compute the vector of maximal minors of a matrix. WARNING: interpretation different in tropical::lifted_pluecker ##### Parameters  Matrix V ##### Returns  Vector • project_to_orthogonal_complement (points, orthogonal) Projects points into the orthogonal complement of a subspace given via an orthogonal basis. The given points will be overwitten. ##### Parameters  Matrix points will be changed to orthogonal ones Matrix orthogonal basis of the subspace • qr_decomp (M) → Pair<Matrix,Matrix> QR decomposition of a Matrix M with rows > cols ##### Parameters  Matrix M ##### Returns  Pair Example: • >$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 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

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 AT of an incidence matrix A, i.e., (aT)ij = aji.

##### Parameters
 IncidenceMatrix A
##### Returns
 IncidenceMatrix
•
transpose (A) → Matrix

Computes the transpose AT of a matrix A, i.e., (aT)ij = aji.

##### 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

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 # @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

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

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 ab.

##### Parameters
 Int a minimal element of the set Int b maximal element of the set
##### Returns
 Set

Example:
• > print range(23,27); {23 24 25 26 27}
•
scalar2set (s) → Set<SCALAR>

Returns the singleton set {s}.

##### Parameters
 SCALAR s
##### Returns
 Set

Example:
• > print scalar2set(23); {23}
•
select_subset (s, indices) → Set

Returns the subset of s given by the indices.

##### Parameters
 Set s Set 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

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

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> array
##### Returns
 Map>, 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 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" 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 .html suffix if needed. 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" 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 .tikz suffix if needed. 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
•
col (i) → Vector<Element>

Returns the i-th column.

##### Parameters
 Int i
##### Returns
 Vector
•
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
•
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 • 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 • Polynomial <Coefficient, Exponent> UNDOCUMENTED ##### Type Parameters  Coefficient default: Rational Exponent default: Int #### User Methods of Polynomial • 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 • 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 • 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 • 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 • 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) ... (pk vk), where l is the dimension of the vector and each pair (pi vi) denotes an entry with value vi at position pi. 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.:

or

#### 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 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
 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 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
•
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.

$A->minor(All, ~[0]); will give you the minor of a matrix containing all rows and all but the 0-th column. ##### Parameters  Set r the rows Set c the columns ##### Returns  IncidenceMatrix • row (i) → SparseVector<Int> Returns the i-th row. ##### Parameters  Int i ##### Returns  SparseVector • rows () → Int Returns the number of rows. ##### Returns  Int • squeeze () Removes empty rows and columns. The remaining rows and columns are renumbered without gaps. • squeeze_cols () Removes empty columns. The remaining columns are renumbered without gaps. • squeeze_rows () Removes empty rows. The remaining rows are renumbered without gaps. • Map <Key, Value> Maps are sorted associative containers that contain unique key/value pairs. Maps are sorted by their keys. Accessing or inserting a value needs logarithmic time O(log n), where n is the size of the map. You can create a new Map mapping Ints to Strings by$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 PointColor color of the spheres or rectangles representing the points Flexible VertexColor alias for PointColor Flexible PointThickness scaling factor for the size of the spheres or rectangles representing the points Flexible VertexThickness alias for PointThickness Flexible PointBorderColor color of the border line of rectangles representing the points Flexible VertexBorderColor alias for PointBorderColor Flexible PointBorderThickness scaling factor for the thickness of the border line of rectangles representing the points Flexible VertexBorderThickness alias for PointBorderThickness Flexible PointStyle if set to "hidden", neither point nor its label is rendered Flexible VertexStyle alias for PointStyle Vector ViewPoint ViewPoint for Sketch visualization Vector ViewDirection ViewDirection for Sketch visualization Vector ViewUp ViewUp for Sketch visualization Float Scale scale for Sketch visualization Flexible 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 FacetColor filling color of the polygons Flexible FacetTransparency transparency factor of the polygons between 0 (opaque) and 1 (completely translucent) Flexible 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 EdgeColor color of the lines representing the edges Flexible EdgeThickness scaling factor for the thickness of the lines representing the edges EdgeMap EdgeLabels textual labels to be placed along the edges Flexible EdgeStyle if set to "hidden", neither the edge nor its label is rendered