playground:playground

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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

  • PermBase:
    Base class for permutations of `big' objects
  • Visual::Container:
    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.
  • Visual::Object:
    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().

Functions

db_print_searchable_fields

  • UNDOCUMENTED

db_get_type_information

  • 20171204: Made type_information_key undef by default to allow users to pass the key “default” explicitly, otherwise default keys have the form “default.<collection>” as there may be more than one collection in the db with a default type information entry

Arithmetic

These are functions that perform arithmetic computations.

fac(Int n)

  • Parameters:
  • Returns: Integer
  • Computes the factorial n! = n&middot;(n-1)&middot;(n-2)&middot;…&middot;2&middot;1.


is_zero(SCALAR s)

  • Parameters:
    • SCALAR s :
  • Returns: Bool
  • Compare with the zero (0) value of the corresponding data type.

div(Int a, Int b)

  • Parameters:
  • Returns: Div
  • Compute the quotient and remainder of a and b in one operation.
  • Example:

     > $d = div(10,3);
 > print $d->quot;
 3



     > print $d->rem;
 1

isinf(SCALAR a)

  • Parameters:
    • SCALAR a :
  • Returns: Int
  • Check whether the given number has an infinite value. Return -1/+1 for infinity and 0 for all finite values.
  • Example:

     > print isinf('inf');
 1



     > print isinf(23);
 0

div_exact(Integer a, Integer b)

  • Parameters:
  • Returns: Integer
  • Computes the ratio of two given integral numbers under the assumption that the dividend is a multiple of the divisor.
  • Example:

     > print div_exact(10,5);
 2

ceil(Rational a)

  • Parameters:
  • Returns: Rational
  • The ceiling function. Returns the smallest integral number not smaller than a.

set_var_names(String names …)

  • Parameters:
    • String names … : variable names; may also be bundled in an array
      an empty list resets to the default naming scheme
  • Set the list of variable names used for pretty printing and string parsing of the given polynomial class
    When the number of variables in a polynomial is greater than the size of the name list, the excess variable names
    are produced from a template “${last_var_name}_{EXCESS}”, where EXCESS starts at 0 for the variable corresponding
    to the last name in the list. If the last name already has a form “{Name}_{Number}”, the following variables are enumerated
    starting from that Number plus 1.
    The default naming scheme consists of a single letter “x”, “y”, “z”, “u”, “v”, or “w” chosen according to the nesting depth
    of polynomial types in the coefficient type. That is, variables of simple polynomials (those with pure numerical coefficients)
    are named x_0, x_1, …, variables of polynomials with simple polynomial coefficients are named y_0, y_1, etc.

monomials<Coefficient, Exponent>(Int n)

  • Template Parameters:
    • Coefficient : The polynomial coefficient type. Rational by default.
  • Exponent : The exponent type. Int by default.
  • Parameters:
  • Int n : The number of variables
  • Returns: UniPolynomial<monomials
  • Create degree one monomials of the desired polynomial type.

isfinite(SCALAR a)

  • Parameters:
    • SCALAR a :
  • Returns: Bool
  • Check whether the given number has a finite value.
  • Example:

     > print isfinite('inf');
 false



     > print isfinite(23);
 true

local_var_names(String names …)

  • Parameters:
  • Set the list of variable names for given polynomial class temporarily.
    The existing name list or the default scheme is restored at the end of the current user cycle,
    similarly to prefer_now.

numerator

get_var_names

  • Returns: Array<String>
  • Get the current list of variable names used for pretty printing and string parsing of the given polynomial class

sum_of_square_roots_naive(Array<Rational> input_array)

  • Parameters:
    • Array<Rational> input_array : a list of rational numbers (other coefficents are not implemented).
  • Returns: Map<Rational,Rational>
  • Make a naive attempt to sum the square roots of the entries of the input array.
  • Example:
    To obtain sqrt{3/4} + sqrt{245}, type 
     > print sum_of_square_roots_naive(new Array<Rational>([3/4, 245]));
 {(3 1/2) (5 7)}


    This output represents sqrt{3}/2 + 7 sqrt{5}.
    If you are not satisfied with the result, please use a symbolic algebra package.

gcd

denominator

ext_gcd(Int a, Int b)

  • Parameters:
  • Returns: ExtGCD
  • Compute the greatest common divisor of two numbers (a,b) and accompanying co-factors.
  • 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; \\ \\ ...and the quotients k1 of a and k2 of b by g. \\ <code> > print $GCD→k1;

5 </code>

 > print $GCD->k2;
 2

floor(Rational a)

  • Parameters:
  • Returns: Rational
  • The floor function. Returns the smallest integral number not larger than a.
  • Example:

     > print floor(1.8);
 1

is_one(SCALAR s)

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

lcm

Combinatorics

This category contains combinatorial functions.

permutation_order(Array<Int> p)

  • Parameters:
  • Returns: Int
  • Returns the order of a permutation
  • Example:

     > print permutation_order(new Array<Int>([1,2,0,4,3]));
 6

binomial(Int n, Int k)

  • Parameters:
  • Returns: Int
  • Computes the binomial coefficient n choose k.
    Negative values of n (and k) are supported.
  • Example:
    Print 6 choose 4 like this: 
     > print binomial(6,4);
 15

permutation_sign(Array<Int> p)

  • Parameters:
  • Returns: Int
  • Returns the sign of the permutation given by p.
  • Example:

     > print permutation_sign([1,0,3,2]);
 1

permutation_matrix<Scalar>(Array<Int> p)

  • Template Parameters:
    • Scalar : default: Int
  • Parameters:
  • Array<Int> p :
  • Returns: Matrix<permutation
  • Returns the permutation matrix of the permutation given by p.
  • 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)

find_permutation(Array a, Array b)

  • Parameters:
  • Returns: Array<Int>
  • Returns the permutation that maps a to b.
  • Example:

     > $p = find_permutation([1,8,3,4],[3,8,4,1]);
 > print $p;
 2 1 3 0

all_permutations(Int n)

  • Parameters:
  • Returns: ARRAY
  • Returns a list of all permutations of the set {0…n-1} as a perl-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 using foreach. The print statement produces the string obtained by dereferencing
    the current entry concatenated with the string “ ”. 

     > foreach( @a ){
 > print @{ $_ }, " ";
 > }
 012 102 201 021 120 210

n_fixed_points(Array<Int> p)

  • Parameters:
  • Returns: Int
  • Returns the number of fixed points of the permutation given by p.
  • Example:

     > print n_fixed_points([1,0,2,4,3]);
 1

are_permuted(Array a, Array b)

  • Parameters:
  • Returns: Bool
  • Determine whether two arrays a and b are permuted copies of each other.
  • Example:

     > print are_permuted([1,8,3,4],[3,8,4,1]);
 true

permutation_cycles(Array<Int> p)

  • Parameters:
  • Returns: ARRAY
  • Returns the cycles of a permutation given by p.
  • Example:

     > print permutation_cycles([1,0,3,2]);
 {0 1}{2 3}

permutation_cycle_lengths(Array<Int> p)

  • Parameters:
  • Returns: Array<Int>
  • Returns the sorted cycle lengths of a permutation
  • Example:

     > print permutation_cycle_lengths(new Array<Int>([1,2,0,4,3]));
 2 3

Data Conversion

This contains functions for data conversions and type casts.

rows

repeat_col(Vector v, Int i)

  • Parameters:
  • Create a Matrix by repeating the given Vector as cols.
  • Example:

     > $v = new Vector(23,42,666);
 > $M = repeat_col($v,3);
 > print $M;
 23 23 23
 42 42 42
 666 666 666

cast<Target>(Core::Object object)

  • Template Parameters:
    • Target : the desired new type
  • Parameters:
  • Core::Object object : to be modified
  • Returns: Core::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.

concat_rows(Matrix A)

  • Parameters:
  • Returns: Vector
  • Concatenates the rows of A.
  • Example:
    Make a vector out of the rows of the vertex matrix of a cube: 
     > $v = concat_rows(polytope::cube(2)->VERTICES);
 > print $v;
 1 -1 -1 1 1 -1 1 -1 1 1 1 1

  • Example:
    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)

vector2col(Vector v)

  • Parameters:
  • Returns: Matrix
  • Convert a Vector to a Matrix with a single column.
  • 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>

toTropicalPolynomial(String s, String vars)

  • Parameters:
    • String s : The string to be parsed
    • String vars : Optional 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»
  • 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.

toMatrix<Scalar>(IncidenceMatrix A)

  • Template Parameters:
    • Scalar :
  • Parameters:
  • Returns: SparseMatrix<toMatrix
  • Convert an IncidenceMatrix to a SparseMatrix.
  • Example:

     > $M = toMatrix<Int>(polytope::cube(2)->VERTICES_IN_FACETS);
 > print $M->type->full_name;
 SparseMatrix<Int, NonSymmetric>

indices(SparseVector v)

  • Parameters:
  • Returns: Set<Int>
  • Get the positions of non-zero entries of a sparse vector.
  • Example:

     > $v = new SparseVector(0,1,1,0,0,0,2,0,3);
 > print indices($v);
 {1 2 6 8}

repeat_row(Vector v, Int i)

  • Parameters:
  • Create a Matrix by repeating the given Vector as rows.
  • Example:

     > $v = new Vector(23,42,666);
 > $M = repeat_row($v,3);
 > print $M;
 23 42 666
 23 42 666
 23 42 666

convert_to

support(Vector v)

  • Parameters:
  • Returns: Set<Int>
  • Get the positions of non-zero entries of a vector.
  • Example:

     > print support(new Vector(0,23,0,0,23,0,23,0,0,23));
 {1 4 6 9}

toVector<Scalar>(Set S, Int d)

  • Template Parameters:
    • Scalar : type of apparent 1's
  • Parameters:
  • Set S :
  • Int d : dimension of the result
  • Returns: SparseVector<toVector
  • Create a sparse vector having 1's at positions contained in the given set

vector2row(Vector v)

  • Parameters:
  • Returns: Matrix
  • Convert a Vector to a Matrix with a single row.
  • 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>

cols

index_matrix(SparseMatrix m)

  • Parameters:
  • Returns: IncidenceMatrix
  • Get the positions of non-zero entries of a sparse matrix.
  • Example:

     > $S = new SparseMatrix([1,2,0,0,0,0],[0,0,5,0,0,32]);
 > print index_matrix($S);
 {0 1}
 {2 5}

dense

lex_ordered(FacetList f)

  • Parameters:
  • Returns: PowerSet<Int>
  • Visit the facets of f sorted lexicographically.
  • Example:

     > $f = new FacetList(polytope::cube(2)->VERTICES_IN_FACETS);
 > print lex_ordered($f);
 {{0 1} {0 2} {1 3} {2 3}}

Database

Here you can find the functions to access the polymake database.

db_write_db_info(String db)

  • Parameters:
    • String db : name of the database the description applies to
  • Add a db documentation.
    You need write access for this.

db_ids(HASH query)

  • Parameters:
    • HASH query :
  • Returns: Array<String>
  • Returns the IDs of all objects in the database db in collection that match the query. This is only recommended for a reasonably small number of matching objects. If you expect many such objects you should instead construct a DBCursor.

db_query(HASH query)

  • Parameters:
    • HASH query :
  • Returns: Array<Core::Object>
  • Returns all objects in the database db in collection that match the query. This is only recommended for a reasonably small number of matching objects. If you expect many such objects you should instead use a database cursor.

db_metadata(Core::Object p)

  • Parameters:
  • Returns: HASH
  • Retrieve the metadata of an object

db_get_list_col_for_db

  • Returns: Array
  • Returns a list of available collections in a database.

db_info

  • Print information about available databases and collections.

db_searchable_fields(String db, String collection)

  • Parameters:
  • Returns: Array<String>
  • Return the list of property names that can be searched in the database for a given database db,
    collection col and optional template key.

db_get_list_db_col

  • Returns: Array
  • Returns a list of available databases and collections (in the form db.collection).

db_get_list_db

  • Returns: Array
  • Returns a list of available databases.

db_count(HASH query)

  • Parameters:
    • HASH query :
  • Returns: Int
  • Returns the number of objects in the database db in collection that match the query.

db_cursor(HASH query)

  • Parameters:
    • HASH query : database query
  • Returns: DBCursor
  • Returns a cursor on the entries for the database db in collection that match the query.

db_print_metadata(Core::Object p)

  • Parameters:
  • print the metadata of an object

db_write_collection_info

  • Add documentation for a collection
    You need write access for this.

Database Admin

These are administrative functions. You need admin access to the database for these. This category also contains functions that I want to hide from the public because they are not yet completely presentable.

db_set_type_information

  • Set or update type (and template) information for collection col in the database db.
    Note that you need write access to the type database for this.

Database Write Access

These are the functions to insert and update objects. You need write access to the database for these.

db_insert(Core::Object obj)

  • Parameters:
  • Returns: String
  • Adds an object obj to the collection col in the database db.
    Note that you need write access to the database for this.

db_remove(String id)

  • Parameters:
  • Returns: String
  • Removes the object with a given id from the collection col in the database db.
    Note that you need write access to the database for this.

Formatting

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

latex(Matrix data, Array<String> elem_labels)

  • Parameters:
  • Returns: String
  • LaTeX output of a matrix.

numbered(Vector data)

  • Parameters:
  • Returns: String
  • Equivalent to labeled with omitted elem_labels argument.
  • Example:

     > $data = new Vector(23,42,666);
 > print numbered($data);
 0:23 1:42 2:666

rows_numbered(Matrix data)

  • Parameters:
  • Returns: Array<String>
  • Equivalent to rows_labeled with omitted row_labels argument.
    Formerly called “numbered”.
  • Example:

     > print rows_numbered(polytope::cube(2)->VERTICES);
 0:1 -1 -1
 1:1 1 -1
 2:1 -1 1
 3:1 1 1

labeled(Vector data, Array<String> elem_labels)

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

     > $v = new Vector(0,1,2);
 > print labeled($v,["zeroth","first","second"]);
 zeroth:0 first:1 second:2

rows_labeled

print_constraints(Matrix<print M)

  • Parameters:
    • Matrix<print M : the matrix whose rows are to be written
  • 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.
  • 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

Graph Operations

Operations on graphs.

node_edge_incidences<Coord>(Graph graph)

  • Template Parameters:
    • Coord : coordinate type for the resulting matrix, default: Int
  • Parameters:
  • Graph graph :
  • Returns: SparseMatrix<node
  • Returns the node-edge incidence matrix of a graph.
  • Example:

     > print node_edge_incidences(graph::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)

adjacency_matrix(Graph graph)

  • Parameters:
  • Returns: 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 playground, with entries encoding the number of parallel edges between two nodes.

induced_subgraph(Graph graph, Set set)

  • Parameters:
    • Graph graph :
    • Set set : indices of selected nodes
  • Returns: Graph
  • Creates an induced subgraph for the given subset of nodes.
  • Example:

     > $g = new props::Graph(graph::cycle_graph(5)->ADJACENCY);
 > $s1 = new Set(1,2,3);
 > print induced_subgraph($g,$s1);
 (5)
 (1 {2})
 (2 {1 3})
 (3 {2})

edges(Graph graph)

  • Parameters:
  • Returns: 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.

nodes(Graph graph)

  • Parameters:
  • Returns: Set<Int>
  • Returns the sequence of all valid nodes of a graph.
  • Example:

     > print nodes(graph::cycle_graph(5)->ADJACENCY);
 {0 1 2 3 4}

Lattice Tools

Functions for lattice related computations.

eliminate_denominators_entire(Matrix v)

  • Parameters:
  • Returns: Matrix<Integer>
  • Scales entire matrix with the least common multiple of the denominators of its coordinates.
  • 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(Vector v)

  • Parameters:
  • Returns: Vector<Integer>
  • Scale a vector with the least common multiple of the denominators of its coordinates.
  • 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_affine(Matrix v)

  • Parameters:
  • Returns: Matrix<Integer>
  • Scales entire matrix with the least common multiple of the denominators of its coordinates (ignore first column).
  • 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(Matrix M)

  • Parameters:
  • Returns: Matrix<Integer>
  • Scale a matrix row-wise with the least common multiple of the denominators of its coordinates.
  • 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

primitive

primitive_affine

is_integral

Linear Algebra

These functions are for algebraic computations and constructions of special matrices.

null_space

unit_vector<Element>(Int d, Int pos)

  • Template Parameters:
  • Parameters:
  • Int d : the dimension of the vector
  • Int pos : the position of the 1
  • Returns: SparseVector<unit
  • Creates a SparseVector of given length d with a one entry at position pos and zeroes elsewhere.
  • Example:
    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>
  • Example:
    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)

null_space_integer(Matrix A)

  • Parameters:
  • Returns: SparseMatrix
  • Computes the lattice null space of the integer matrix A.

inv(Matrix A)

  • Parameters:
  • Returns: Matrix
  • Computes the inverse A-1 of an invertible matrix A using Gauss elimination.
  • Example:
    We save the inverse of a small matrix M in the variable $iM: \\ <code> > $M = new Matrix([1,2],[3,4]);

> $iM = inv($M); </code>
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

rank(Matrix A)

  • Parameters:
  • Returns: Int
  • Computes the rank of a matrix.

det(Matrix< A)

  • Parameters:
  • Returns: SCALAR
  • Computes the determinant of a matrix using Gaussian elimination.
    If Scalar is not of field type, but element of a Euclidean ring R,
    type upgrade to element of the quotient field is performed.
    The result is recast as a Scalar, which is possible without roundoff
    since the so-computed determinant is an element of the (embedded) ring R.
  • Example:

     > print det(unit_matrix(3));
 1

  • Example:

     > $p = new UniPolynomial<Rational,Int>("x2+3x");
 > $M = new Matrix<UniPolynomial<Rational,Int>>([[$p, $p+1],[$p+1,$p]]);
 > print det($M);
 -2*x^2 -6*x - 1

qr_decomp(Matrix<Float> M)

  • Parameters:
  • Returns: Pair<Matrix,Matrix>
  • QR decomposition of a Matrix M with rows > cols
  • 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

lin_solve(Matrix A, Vector b)

  • Parameters:
  • Returns: Vector
  • Computes the vector x that solves the system Ax = b
  • 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

zero_matrix<Element>(Int i, Int j)

  • Template Parameters:
  • Parameters:
  • Int i : number of rows
  • Int j : number of columns
  • Returns: SparseMatrix<zero
  • Creates a zero matrix of given dimensions
  • Example:
    The following stores a 2×3 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
  • Example:
    The following stores a 2×3 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>

basis_affine(Matrix A)

smith_normal_form(Matrix M, Bool inv)

  • Parameters:
    • Matrix M : must be of integer type
    • Bool inv : optional, if true, compute the inverse of the companion matrices
  • Returns: SmithNormalForm<Integer>
  • Compute the Smith normal form of a given matrix M.
    M = LSR in normal case, or S = LMR in inverted case.
  • 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

ones_matrix<Element>(Int m, Int n)

  • Template Parameters:
  • Parameters:
  • Int m : number of rows
  • Int n : number of columns
  • Returns: Matrix<ones
  • Creates a matrix with all elements equal to 1.
  • Example:
    The following creates an all-ones matrix with Rational coefficients. 
     > $M = ones_matrix<Rational>(2,3);
 > print $M;
 1 1 1
 1 1 1

eigenvalues(Matrix<Float> A)

barycenter(Matrix A)

  • Parameters:
  • Returns: Vector
  • Calculate the average over the rows of a matrix.
  • Example:

     > $A = new Matrix([3,0,0],[0,3,0],[0,0,3]);
 > print barycenter($A);
 1 1 1

basis(Matrix A)

  • Parameters:
  • Returns: 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.
  • 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

lineality_space(Matrix A)

  • Parameters:
  • Returns: Matrix
  • Compute the lineality space of a matrix A.
  • Example:

     > $M = new Matrix([1,1,0,0],[1,0,1,0]);
 > print lineality_space($M);
 0 0 0 1

singular_value_decomposition(Matrix<Float> M)

  • Parameters:
  • Returns: 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.
  • 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

diag

reduce(Matrix A, Vector b)

  • Parameters:
  • Returns: Vector
  • Reduce a vector with a given matrix using Gauss elimination.

hadamard_product(Matrix M1, Matrix M2)

  • Parameters:
  • Returns: Matrix
  • Compute the Hadamard product of two matrices with same dimensions.

basis_cols(Matrix A)

  • Parameters:
  • Returns: Set<Int>
  • Computes a subset of the columns of A that form a basis for the linear space spanned by A.
  • 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

equal_bases(Matrix M1, Matrix M2)

  • Parameters:
  • Returns: Bool
  • Check whether both matrices are bases of the same linear subspace.
    Note: It is assumed that they are *bases* of the row space.
  • 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);
 true

normalized(Matrix<Float> A)

  • Parameters:
  • Returns: Matrix<Float>
  • Normalize a matrix by dividing each row by its length (l2-norm).
  • Example:

     > $A = new Matrix<Float>([1.5,2],[2.5,2.5]);
 > print normalized($A);
 0.6 0.8
 0.7071067812 0.7071067812

cramer(Matrix A, Vector b)

  • Parameters:
  • Returns: Vector
  • Computes the solution of the system Ax = b by applying Cramer's rule
  • Example:
    from the Wikipedia: 
     > $A = new Matrix([3,2,-1],[2,-2,4],[-1,1/2,-1]);
 > $b = new Vector(1,-2,0);
 > print cramer($A,$b);
 1 -2 -2

unit_matrix<Element>(Int d)

  • Template Parameters:
  • Parameters:
  • Int d : dimension of the matrix
  • Returns: SparseMatrix<unit
  • Creates a unit matrix of given dimension
  • Example:
    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)
  • Example:
    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>

trace(Matrix A)

  • Parameters:
  • Returns: Int
  • Computes the trace of a matrix.
  • Example:

     > $M = new Matrix([1,2,3],[23,24,25],[0,0,1]);
 > print trace($M);
 26

sqr(Vector< v)

  • Parameters:
  • Returns: SCALAR
  • Return the sum of the squared entries of a vector v.

remove_zero_rows(Matrix m)

  • Parameters:
  • Returns: Matrix
  • Remove all zero rows from a matrix.

pluecker(Matrix V)

  • Parameters:
  • Returns: Vector
  • Compute the vector of maximal minors of a matrix.
    WARNING: interpretation different in lifted_pluecker

totally_unimodular(Matrix A)

  • Parameters:
  • Returns: 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.
  • 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);
 true

basis_rows_integer(Matrix A)

  • Parameters:
  • Returns: Set<Int>
  • Computes a lattice basis of the span of the rows of A.

moore_penrose_inverse(Matrix M)

basis_rows(Matrix A)

  • Parameters:
  • Returns: Set<Int>
  • Computes a subset of the rows of A that form a basis for the linear space spanned by A.
  • 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

householder_trafo(Vector<Float> b)

  • Parameters:
  • Returns: Matrix<Float>
  • Householder transformation of Vector b. Only the orthogonal matrix reflection H is returned.

transpose

ones_vector<Element>(Int d)

  • Template Parameters:
  • 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<ones
  • Creates a vector with all elements equal to 1.
  • 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

solve_left(Matrix A, Matrix B)

  • Parameters:
  • Returns: Matrix
  • Computes a matrix X that solves the system XA = B. This is useful, for instance, for computing the coordinates of some vectors
    with respect to a basis. The rows of the matrix solve_left(B,V) are the coordinates of the rows of V with respect to the rows of B.
  • Example:
    Define the matrices 
     > $V = new Matrix([[-4,2,2],[3,-2,-1]]);
 > $B = new Matrix([[-1,1,0],[0,-1,1]]);


    so that the rows of B are a basis of the subspace of vectors with zero coordinate sum. Then the rows of 

     > print solve_left($B, $V);
 4 2
 -3 -1


    contain the coordinates of the rows of V with respect to the rows of B.

zero_vector<Element>(Int d)

  • Template Parameters:
  • 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<zero
  • Creates a vector with all elements equal to zero.
  • Example:
    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
  • Example:
    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>
  • Example:
    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

hermite_normal_form(Matrix M)

  • Parameters:
    • Matrix M : Matrix to be transformed.
  • Returns: HermiteNormalForm
  • 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.
  • Example:
    The following stores the result for a small matrix M in H and then prints both hnf and companion: 
     > $M = new Matrix<Integer>([1,2],[2,3]);
 > $H = hermite_normal_form($M);
 > print $H->hnf;
 1 0
 0 1



     > print $H->companion;
 -3 2
 2 -1

anti_diag

project_to_orthogonal_complement(Matrix points, Matrix orthogonal)

  • Parameters:
    • Matrix points : will be changed to orthogonal ones
    • Matrix orthogonal : basis of the subspace
  • Projects points into the orthogonal complement of a subspace given via an orthogonal basis.
    The given points will be overwitten.

solve_right(Matrix A, Matrix B)

  • Parameters:
  • Returns: Matrix
  • Computes a matrix X that solves the system AX = B
  • Example:
    A non-degenerate example: 
     > $A = new Matrix([[1,0,0],[1,1,0],[1,0,1],[1,1,1]]);
 > $B = new Matrix([[1,0,0],[1,0,1],[1,1,0],[1,1,1]]);
 > print solve_right($A,$B);
 1 0 0
 0 0 1
 0 1 0

  • Example:
    A degenerate example: 

     > $A = new Matrix([[1,0,0,0,0],[0,1,0,0,0],[1,0,1,0,0],[0,1,1,0,0]]);
 > $B = new Matrix([[0,1,0,0,0],[1,0,0,0,0],[0,1,1,0,0],[1,0,1,0,0]]);
 > print solve_right($A,$B);
 0 1 0 0 0
 1 0 0 0 0
 0 0 1 0 0
 0 0 0 0 0
 0 0 0 0 0

Set Operations

This category contains functions performing operations on Sets.

scalar2set(SCALAR s)

  • Parameters:
    • SCALAR s :
  • Returns: Set<
  • Returns the singleton set {s}.
  • Example:

     > print scalar2set(23);
 {23}

range(Int a, Int b)

  • Parameters:
    • Int a : minimal element of the set
    • Int b : maximal element of the set
  • Returns: Set<Int>
  • Create the Set {a, a+1, …, b-1, b} for a &le; b. See also: sequence
  • Example:

     > print range(23,27);
 {23 24 25 26 27}

range_from(Int a)

  • Parameters:
    • Int a : start index
  • Returns: Set<Int>
  • Create an index range starting at a, the last index is implied by the context.
    To be used with minor, slice, and similar methods selecting a subset of elements.
  • Example:

     > $v=new Vector<Int>(10,20,30,40);
 > print $v->slice(range_from(2));
 30 40

sequence(Int a, Int c)

  • Parameters:
    • Int a : the smallest element
    • Int c : the cardinality
  • Returns: Set<Int>
  • Create the Set {a, a+1, …, a+c-1}. See also: range
  • Example:

     > print sequence(23,6);
 {23 24 25 26 27 28}

select_subset(Set s, Set<Int> indices)

  • Parameters:
  • Returns: Set
  • Returns the subset of s given by the indices.
  • Example:

     > $s = new Set<Int>(23,42,666,789);
 > $ind = new Set<Int>(0,2);
 > $su = select_subset($s,$ind);
 > print $su;
 {23 666}

incl(Set s1, Set s2)

  • Parameters:
  • Returns: Int
  • Analyze the inclusion relation of two sets.
  • Example:

     > $s1 = new Set(1,2,3);
 > $s2 = $s1 - 1;
 > print incl($s1, $s2);
 1



     > print incl($s2, $s1);
 -1



     > print incl($s1, $s1);
 0



     > print incl($s2, $s1-$s2);
 2

Utilities

Miscellaneous functions.

fibonacci(Int m)

  • Parameters:
  • Returns: ARRAY
  • Returns the first m Fibonacci numbers.

average(ARRAY array)

  • Parameters:
    • ARRAY array :
  • Returns the average value of the array elements.
  • Example:

     > print average([1,2,3]);
 2

histogram(ARRAY data)

  • Parameters:
    • ARRAY data :
  • Returns: Map<
  • Produce a histogram of an array: each different element value is mapped on the number of its occurences.
  • 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<Set<Int» array)

  • Parameters:
  • Returns: HashMap<Array<Set<Int»,Int>
  • Return a map indexing an array of sets
  • Example:

     > $s1 = new Set(1,2,3);
 > $s2 = $s1 - 1;
 > $a = new Array<Set>($s1,$s2,$s1);
 > print index_of($a);
 {({1 2 3} 2) ({2 3} 1)}

minimum(ARRAY array)

  • Parameters:
    • ARRAY array :
  • Returns the minimal element of an array.
  • Example:

     > print minimum([23,42,666]);
 23

is_unimodal(ARRAY data)

  • Parameters:
    • ARRAY data :
  • Returns: Bool
  • Checks if a given sequence is unimodal
  • Example:

     > print is_unimodal([1,1,2,3,3,2,2,1]);
 1                     



     > print is_unimodal([3,3,2,-1]);
 1



     > print is_unimodal([1,3,2,3]);
 0

median(ARRAY array)

  • Parameters:
    • ARRAY array :
  • Returns the median value of the array elements.
  • Example:

     > print median([1,2,3,9]);
 2.5

bounding_box(Matrix m)

  • Parameters:
  • Returns: Matrix
  • Compute a column-wise bounding box for the given Matrix m.

rand_perm(Int n)

  • Parameters:
  • Returns: Array<Int>
  • gives a random permutation

maximum(ARRAY array)

  • Parameters:
    • ARRAY array :
  • Returns the maximal element of an array.
  • Example:

     > print maximum([1,2,3,4,5,6,7,23]);
 23

distance_matrix(Matrix S, CODE d)

  • Parameters:
  • Returns: Matrix
  • Given a metric, return a triangle matrix whose (i,j)-entry contains the distance between point i and j of the point set S for i<j. All entrys below the diagonal are zero. The metric is passed as a perl subroutine mapping two input vectors to a real value.
  • Example:
    The following defines the perl subroutine dist as the euclidean metric and then saves the distance matrix of the 3-cubes vertices in the variable $M: \\ <code> > sub dist($$) {

> my $v = $_[0] - $_[1]; > return sqrt(new Float($v*$v)); } > $M = distance_matrix(polytope::cube(3)→VERTICES, \&dist); </code>

is_nonnegative(ARRAY data)

  • Parameters:
    • ARRAY data :
  • Returns: Bool
  • Checks if a given sequence is nonnegative
  • Example:

     > print is_nonnegative([1,0,3]);
 1

perturb_matrix(Matrix M, Float eps, Bool not_hom)

  • 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)
  • Returns: 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.

Visualization

These functions are for visualization.

threejs(Visual::Object vis_obj)

  • Parameters:
  • Produce an html file with given visual objects.

postscript(Visual::Object vis_obj …)

  • Parameters:
  • Create a Postscript ™ drawing with the given visual objects.

povray(Visual::Object vis_obj …)

  • Parameters:
  • Run POVRAY to display given visual objects.

static(Visual::Object vis_obj)

  • Parameters:
    • Visual::Object vis_obj : drawing, e.g. created by VISUAL_GRAPH or SCHLEGEL.
  • Returns: Visual::Object
  • Suppress creation of dynamic (interactive) scenes.

jreality(Visual::Object vis_obj …)

compose

sketch(Visual::Object vis_obj …)

  • Parameters:
  • Produce a Sketch input file with given visual objects.

x3d(Visual::Object vis_obj …)

  • Parameters:
  • Create an X3D drawing with the given visual objects.

tikz(Visual::Object vis_obj)

  • Parameters:
  • Produce a TikZ file with given visual objects.

Small Object Types

Algebraic Types

This category contains all “algebraic” types, such as matrices, vectors, polynomials, rings, …

RationalFunction<Coefficient, Exponent>

  • Template Parameters:
  • Exponent : default: Int
  • the same with type deduction

Matrix<Element, Sym>

  • Template Parameters:
  • Methods of Matrix:
    • diagonal(Int i)
      • 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<Matrix
      • Returns the diagonal of the matrix.
    • row(Int i)
    • resize(Int r, Int c)
      • Parameters:
        • Int r : new number of rows
        • Int c : new number of columns
      • Change the dimensions; when growing, set added elements to 0.
    • rows
      • Returns: Int
      • Returns the number of rows.
    • clear(Int r, Int c)
      • Parameters:
        • Int r : new number of rows
        • Int c : new number of columns
      • Change the dimensions setting all elements to 0.
    • anti_diagonal(Int i)
      • 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<Matrix
      • Returns the anti-diagonal of the matrix.
    • minor(Set r, Set c)
      • Parameters:
        • Set r : the rows
        • Set c : the columns
      • Returns: 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.
    • col(Int i)
    • cols
      • Returns: Int
      • Returns the number of columns.
    • elem(Int r, Int c)
      • Parameters:
        • Int r : the row index
        • Int c : the column index
      • Returns: Matrix
      • Returns an element of the matrix.
        The return value is an `lvalue', that is, it can be modified if the matrix object is mutable.
    • div_exact(Int a)
      • Parameters:
      • Returns: Matrix
      • Divides every entry by a (assuming that every entry is divisible by a).

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)
    .
  • Methods of SparseMatrix:
    • squeeze_rows
      • Removes empty rows.
        The remaining rows are renumbered without gaps.
    • squeeze_cols
      • Removes empty columns.
        The remaining columns are renumbered without gaps.
    • squeeze
      • Removes empty rows and columns.
        The remaining rows and columns are renumbered without gaps.
    • resize(Int r, Int c)
      • Parameters:
        • Int r : new number of rows
        • Int c : new number of columns
      • Resize the matrix
        All elements in added rows and/or columns are implicit zeros.

Vector<Element>

  • Template Parameters:
    • 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]);
  • Methods of Vector:
    • slice(Set s)
      • Parameters:
        • Set s : indices to select from the vector
      • Returns: Vector
      • Returns a Vector containing all entries whose index is in a Set s.
    • div_exact(Int a)
      • Parameters:
      • Returns: Vector
      • Divides every entry by a (assuming that every entry is divisible by a).
    • dim
      • Returns: Int
      • The length of the vector

SparseVector<Element>

  • Template Parameters:
    • 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.:
    $v = new SparseVector<Int>(« '.');
    (6) (1 1) (2 2)
    .
  • Methods of SparseVector:
    • size
      • Returns: Int
      • The number of non-zero entries.

Polynomial<Coefficient, Exponent>

  • Template Parameters:
  • Exponent : default: Int
  • Methods of Polynomial:
    • mapvars(Array<Int> indices, Int nvars)
      • Parameters:
        • Array<Int> indices : indices of the target variables
        • Int nvars : new number of variables, default: maximal index
      • Returns: Polynomial
      • Map the variables of the Polynomial to the given indices.
        The same index may bei given multiple times and also some may be omitted
        for embedding with a higher number of variables.
        The length of the given Array must be the same as the number of variables
        of the original polynomial.
  • Example:

     > $pm = new Polynomial("1+x_0^2*x_3+x_1+3*x_3");
   > print $pm->mapvars([0,1,1,4],6);
   x_0^2*x_4 + x_1 + 3*x_4 + 1
    • project
      • Returns: Polynomial
      • Project the Polynomial to the given list of variables.
        The number of variables will be reduced to the number of indices,
        i.e. the variables will be renamed.
        Keeping the names of the variables is possible by using substitute
        with a Map.
  • Example:

     > $pm = new Polynomial("1+x_0^2*x_3+x_1+3*x_3");
   > print $pm->project([1,3]);
   x_0 + 4*x_1 + 1
    • print_ordered(Matrix m)
      • Parameters:
      • Print a polynomial with terms sorted according to a given Matrix m.
    • coefficients_as_vector
    • lm
      • Returns: Int
      • The exponent of the leading monomial.
    • lc
      • Returns: Int
      • The leading coefficient.
    • get_var_names
      • set the variable names
    • deg
      • Returns: Int
      • The degree of the polynomial
    • n_vars
      • Returns: Int
      • Number of variables
    • trivial
      • Returns: Bool
      • The polynomial is zero.
    • monomial(Int var_index, Int n_vars)
      • Parameters:
        • Int var_index : index of the variable
        • Int n_vars : number of variables
      • construct a monomial of degree 1 with a given variable
    • embed(Int nvars)
      • Parameters:
        • Int nvars : new number of variables
      • Returns: Polynomial
      • Embed the Polynomial in a polynomial ring with the given
        number of variables.
        The old variables will be put at the beginning, for more control use mapvars.
  • Example:

     > $pm = new Polynomial("1+x_0^2*x_3+x_1+3*x_3");
   > print $pm->project([1,3]);
   x_0 + 4*x_1 + 1
    • set_var_names
      • set the variable names
    • initial_form(Vector<Polynomial v)
    • monomials_as_matrix
    • constant_coefficient
      • Returns: Int
      • The constant coefficient.
    • substitute
      • Substitute a list of values in a Polynomial with Int exponents.
        Either with an array of values, or with a Map mapping variable indices
        to values.
  • Example:

     > $pm = new Polynomial("1+x_0^2*x_3+x_1+3*x_3");
   > print $pm->substitute([0,11,2,3]);
   21
  



     > $map = new Map<Int,Rational>([0,5/2],[1,1/5],[2,new Rational(7/3)]);
   > print $pm->substitute($map);
   37/4*x_3 + 6/5
    • reset_var_names
      • reset the variable names according to the default naming scheme

UniPolynomial<Coefficient, Exponent>

  • Template Parameters:
  • Exponent : default: Int
  • A class for univariate polynomials.
  • Methods of UniPolynomial:
    • lower_deg
      • Returns: UniPolynomial
      • The lowest degree occuring in the polynomial.
    • set_var_names
      • set the variable name
    • monomials_as_vector
    • constant_coefficient
      • Returns: Int
      • The constant coefficient.
    • reset_var_names
      • reset the variable name according to the default naming scheme
    • substitute
      • Substitute some value in a UniPolynomial with Int exponents.
        When all exponents are positive the argument can be any scalar, matrix or polynomial type.
        With negative exponents, polynomials are not supported (use RationalFunction instead)
        and any given matrix must be invertible
    • evaluate(UniPolynomial x, UniPolynomial exp)
      • Parameters:
      • Returns: UniPolynomial
      • Evaluate a UniPolynomial at a number (x^exp).
        Note that for non-integral exponents this may require intermediate floating point
        computations depending on the input:
        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.
    • lc
      • Returns: Int
      • The leading coefficient.
    • deg
      • Returns: UniPolynomial
      • The highest degree occuring in the polynomial.
    • get_var_names
      • get the variable name
    • print_ordered(UniPolynomial x)
      • Parameters:
      • Print a polynomial with terms sorted according to exponent*x.
    • coefficients_as_vector
    • monomial
      • create a monomial of degree 1
    • n_vars
      • Number of variables
    • evaluate_float(Float x)
      • Parameters:
      • Returns: Float
      • Approximate evaluation of the polynomial at a Float x.

Plucker<Scalar>

  • Template Parameters:
  • Methods of Plucker:
    • permuted
      • UNDOCUMENTED
    • coordinates
      • UNDOCUMENTED

all_rows_or_cols

  • Use the keyword “All” for all rows or columns, e.g. when constructing a minor.

PuiseuxFraction<MinMax, Coefficient, Exponent>

  • Template Parameters:
    • MinMax : type of tropical addition: either Min or Max

Arithmetic

These types are needed as return types of arithmetic computations.

Div<Scalar>

  • Template Parameters:
    • Scalar :
  • The complete result of an integral division, quotient and remainder.
  • Methods of Div:
    • rem
      • Returns: Div
      • The remainder.
    • quot
      • Returns: Div
      • The quotient.

Max

  • tropical addition: max
  • Methods of Max:
    • apply
      • UNDOCUMENTED
    • orientation
      • UNDOCUMENTED

Min

  • tropical addition: min
  • Methods of Min:
    • orientation
      • UNDOCUMENTED
    • apply
      • UNDOCUMENTED

TropicalNumber<Addition, Scalar>

  • Template Parameters:
  • Methods of TropicalNumber:
    • zero
      • Returns: TropicalNumber
      • The zero element of the tropical semiring of this element.
    • orientation
      • Returns: Int
      • The orientation of the associated addition, i.e.
        +1 if the corresponding 0 is +inf
        -1 if the corresponding 0 is -inf

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
  • Methods of ExtGCD:
    • k1
      • Returns: Int
      • The factor of a.
    • p
      • Returns: Int
      • The co-factor of a.
    • g
      • Returns: Int
      • The greatest common divisor of a and b.
    • q
      • Returns: Int
      • The co-factor of b.
    • k2
      • Returns: Int
      • The factor of b.

Addition

  • Type parameter of many functions and data types from the domain of tropical geometry.
    Specifies the mode of tropical addition, must be Min or Max.
    There is on purpose no default value for it.

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.

Symmetric

DirectedMulti

  • Type tag for a directed multigraph.

LocalFloatEpsilon

Serialized<X>

  • Template Parameters:
    • X :

UndirectedMulti

  • Type tag for an undirected multigraph.

CachedObjectPointer

EdgeIterator

NonSymmetric

EdgeList

Directed

  • Type tag for a directed Graph.

Container

  • One-dimensional collection of objects, aka container in STL
    It can be accessed like a perl array. Sequential iteration via foreach() is always supported;
    random access might be supported depending on the underlying C++ object.

Iterator<Element>

  • Template Parameters:
    • Element :
  • An iterator over a sequence of objects.
    The objects may be stored in a container or be generated on the fly.

Undirected

  • Type tag for an undirected Graph.

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, …

Bool

  • Corresponds to the C++ type bool.

AccurateFloat

  • A wrapper for AccurateFloat.

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.

String

  • Corresponds to the C++ type std::string.

Float

  • Corresponds to the C++ type double.
  • Methods of Float:
    • minus_inf
      • produce an infinitely large negative value
    • inf
      • produce an infinitely large positive value

Pair<First, Second>

  • Template Parameters:
    • First :
    • Second :
  • Corresponds to the C++ type std::pair.

Int

  • Corresponds to the C++ type int.

QuadraticExtension<Field>

  • Template Parameters:
  • 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).

Integer

  • An integer of arbitrary size.
  • Methods of Integer:
    • inf
      • produce an infinitely large positive value
    • minus_inf
      • produce an infinitely large negative value

Rational

  • A class for rational numbers.
    You can easily create a Rational like that: $x = 1/2;
  • Methods of Rational:
    • inf
      • Produce an infinitely large positive value.
    • minus_inf
      • Produce an infinitely large negative value.

Text

  • Plain text without any imposed structure.

List<Element>

  • Template Parameters:
    • Element :
  • Corresponds to the C++ type std::list.

Array<Element>

  • Template Parameters:
    • Element :
  • An array with elements of type Element.
    Corresponds to the C++ type Array.
  • Methods of Array:
    • size
      • Returns: Int
      • Returns the size.

Database

Here you can find the functions to access the polymake database.

DBCursor

  • A database cursor. Initialize it with a database name, a collection name and a query.
    You can then iterate over the objects matching the query with next.
    It lazily fetches more objects from the database server.
    (Note that you have to create a new cursor if you want to start from the beginning.)

MongoClient

  • A handler for the polyDB database internally controlling the connection to the MongoDB database

Graph Types

This contains all property types that are related to graphs.

EdgeHashMap<Dir, Element>

  • Element : data associated with edges
  • Sparse mapping of edges to data items.
  • Methods of EdgeHashMap:
    • erase(Int from, Int to)
      • Parameters:
        • Int from : source node
        • Int to : target node
      • Delete the data associated with an edge between two given nodes.
    • edge(Int from, Int to)
      • Parameters:
        • Int from : source node
        • Int to : target node
      • Access the data associated with an edge between two given nodes.
        The new data element is created on demand.
    • find(Int from, Int to)
      • Parameters:
        • Int from : source node
        • Int to : target node
      • Returns: Iterator
      • Access the data associated with an edge between two given nodes.

NodeHashMap<Dir, Element>

  • Element : data associated with nodes
  • Sparse mapping of nodes to data items.

NodeMap<Dir, Element>

  • Element : data associated with nodes
  • Dense mapping of nodes to data items.

EdgeMap<Dir, Element>

  • Element : data associated with edges
  • Dense mapping of edges to data items.
  • Methods of EdgeMap:
    • edge(Int from, Int to)
      • Parameters:
        • Int from : source node
        • Int to : target node
      • Access the data associated with an edge between two given nodes.

Graph<Dir>

  • Methods of Graph:
    • out_adjacent_nodes(Int node)
      • Parameters:
      • Returns: Set
      • Returns the set of indices of the nodes with an edge arriving from node.
    • contract_edge(Int node1, Int node2)
      • Parameters:
      • 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.
    • node_out_of_range(Int node)
      • Parameters:
      • Returns: Bool
      • Returns true if the given node index is out of valid range.
    • invalid_node(Int node)
      • Parameters:
      • Returns: Bool
      • Returns true if the given node index is either out of valid range or points to a formerly deleted node.
    • add_edge(Int tail_node, Int head_node)
      • Parameters:
        • Int tail_node :
        • Int head_node :
      • Returns: 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.
    • squeeze
      • Renumbers the valid nodes as to eliminate all gaps left after deleting.
    • degree(Int node)
      • Parameters:
      • Returns: Int
      • Returns the number of edges incident to node.
    • adjacent_nodes(Int node)
      • Parameters:
      • Returns: Set
      • Returns the set of indices of nodes adjacent to node.
    • out_degree(Int node)
      • Parameters:
      • Returns: Int
      • Returns the number of edges leaving node.
    • edge(Int tail_node, Int head_node)
      • Parameters:
        • Int tail_node :
        • Int head_node :
      • Returns: 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.
    • all_edges(Int tail_node, Int head_node)
      • Parameters:
        • Int tail_node :
        • Int head_node :
      • Returns: Iterator
      • Returns an iterator visiting all (parallel) edges connecting two given nodes.
    • in_degree(Int node)
      • Parameters:
      • Returns: Int
      • Returns the number of edges heading to node.
    • dim
      • Returns: Int
      • Returns the maximal node index + 1.
        If the graph does not have gaps caused by node deletion, the result is equivalent to nodes().
    • add_node
      • Returns: Int
      • Add a new node without incident edes.
    • out_edges(Int node)
      • Parameters:
      • Returns: EdgeList
      • Returns a sequence of edges leaving (in Directed case) or incident to (in Undirected case) node.
    • edge_exists(Int tail_node, Int head_node)
      • Parameters:
        • Int tail_node :
        • Int head_node :
      • Returns: Bool
      • Checks whether two given nodes are connected by (at least) one edge.
    • edges
      • Returns: Int
      • Get the total number of edges.
    • squeeze_isolated
      • Deletes all nodes of degree 0,
        then renumbers the remaining nodes without gaps.
    • delete_all_edges(Int tail_node, Int head_node)
      • Parameters:
        • Int tail_node :
        • Int head_node :
      • Deletes all edges in a multigraph connecting two given nodes.
    • in_edges(Int node)
      • Parameters:
      • Returns: EdgeList
      • Returns a sequence of edges heading to (in Directed case) or incident to (in Undirected case) node.
    • nodes
      • Returns: Int
      • Get the total number of nodes.
    • permute_nodes
      • permute the nodes
        param perm permutation of node indexes
    • node_exists(Int node)
      • Parameters:
      • Returns: Bool
      • Check whether the node with given index exists.
    • permute_inv_nodes
      • permute the nodes
        param perm inverse permutation of node indexes
    • delete_edge
    • in_adjacent_nodes(Int node)
      • Parameters:
      • Returns: Set
      • Returns the set of indices of the nodes that have an edge heading to node.
    • has_gaps
      • Returns: Bool
      • Returns true if some nodes have been deleted since the last squeeze operation.
    • delete_node(Int node)
      • Parameters:
      • Deletes all edges incident to the given node and marks it as invalid.
        The numeration of other nodes stays unchanged.

GraphMap<Dir, Element>

  • Element : data associated with nodes or edges
  • The common abstract base class for all kinds of associative containers that can be attached to a Graph.

Linear Algebra

These types are needed as return types of algebraic computations.

HermiteNormalForm

  • Complete result of the Hermite normal form computation.
    Contains the following fields:
    Matrix<Scalar> hnf: the Hermite normal form N of M
    SparseMatrix<Scalar> companion: unimodular Matrix R such that M*R = N.
    Int rank: rank of M

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, …

IncidenceMatrix<Sym>

  • A 0/1 incidence matrix.
  • Methods of IncidenceMatrix:
    • rows
      • Returns: Int
      • Returns the number of rows.
    • squeeze_rows
      • Removes empty rows.
        The remaining rows are renumbered without gaps.
    • row(Int i)
    • squeeze
      • Removes empty rows and columns.
        The remaining rows and columns are renumbered without gaps.
    • elem(Int r, Int c)
      • Parameters:
        • Int r : the row index
        • Int c : the column index
      • Returns: 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.
    • cols
      • Returns: Int
      • Returns the number of columns.
    • minor(Set r, Set c)
      • Parameters:
        • Set r : the rows
        • Set c : the columns
      • Returns: 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.
        For example,
        $A→minor(All, ~[0]);
        will give you the minor of a matrix containing all rows and all but the 0-th column.
    • col(Int i)
    • squeeze_cols
      • Removes empty columns.
        The remaining columns are renumbered without gaps.

Bitset

  • Container class for dense sets of integers.
    A special class optimized for representation of a constrained range of
    non-negative integer numbers.

HashMap<Key, Value>

  • Template Parameters:
    • Key : type of the key values
  • Value : type of the mapped value
  • An unordered map based on a hash table. Its interface is similar to that of Map,
    but the order of elements is not stable across polymake sessions and does not correlate with key values.
    Accessing and inserting 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).
  • Methods of HashMap:
    • size
      • Returns: Int
      • Size of the map

PowerSet<Element>

  • Template Parameters:
    • Element : default: Int
  • A Set whose elements are of type Set<Element>.

Map<Key, Value>

  • Template Parameters:
    • Key : type of the key values
  • Value : type of the mapped 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};

Set<Element>

  • Template Parameters:
    • Element : default: Int
  • A type for ordered 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
  • Methods of Set:
    • size
      • Returns: Int
      • The cardinality of the set.
    • back
      • Returns: Int
      • The last element of the set, that is, the largest element
    • contains(Set e)
      • Parameters:
        • Set e : element check for
      • Returns: Bool
      • Check if e is contained in the set.
    • collect(Set e)
      • Parameters:
        • Set e : element to insert into the set
      • Returns: Bool
      • Add to the set, report true if existed formerly.
    • front
      • Returns: Int
      • The first element of the set, that is, the smallest element

ApproximateSet<Element>

  • Template Parameters:
    • Element : default: Float
  • A specialization of Sets containing elements of type Element,
    but where equality is enforced only up to a global epsilon.
    You can for example create a new ApproximateSet with two elements by:
    $s = new ApproximateSet(1.1, 1.2, 1.200000001);

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 playground 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.
  • Methods of FacetList:
    • eraseSupersets(Set s)
      • Parameters:
        • Set s : filter for removal
      • Returns: Int
      • Remove all supersets of a given set
    • eraseSubsets(Set s)
      • Parameters:
        • Set s : filter for removal
      • Returns: Int
      • Remove all subsets of a given set
    • findSubsets(Set s)
      • Parameters:
      • Returns: Iterator
      • Find all subsets of a given set.
    • n_vertices
      • Returns: Int
      • The number of vertices
    • insertMin(Set f)
      • Parameters:
        • Set f : facet to add
      • Returns: 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.
    • size
      • Returns: Int
      • The number of facets in the list.
    • find(Set f)
      • Parameters:
        • Set f : facet to find
      • Returns: Iterator
      • Look up a facet.
    • insert(Set f)
      • Parameters:
        • Set f : facet to add.
      • 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.
    • findSupersets(Set s)
      • Parameters:
      • Returns: Iterator
      • Find all supersets of a given set.
    • erase(Set f)
      • Parameters:
        • Set f : facet to remove
      • Returns: Bool
      • Remove a facet.
    • insertMax(Set f)
      • Parameters:
        • Set f : facet to add
      • Returns: 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.

HashSet<Element>

  • Template Parameters:
    • Element :
  • An unordered set of elements based on a hash table.
    Can be traversed as a normal container, but the order of elements is not stable across polymake sessions,
    even if the set is restored from the same data file every time.
  • Methods of HashSet:
    • size
      • Returns: Int
      • The cardinality of the set.

Visualization

These property_types are for visualization.

HSV

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

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.

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 hex triplet with a leading # symbol,
    or a list of three floating-point values from the range 0..1,
    or a symbolic name from the X11 color names.
  • playground/playground.1553273008.txt.gz
  • Last modified: 2019/03/22 16:43
  • by oroehrig