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Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2
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' objectsPolyDB::Client
:
A live connection to a PolyDB server. This is a starting point for all operations on the database.PolyDB::Collection
:
Represents a collection in PolyDBPolyDB::Cursor
:
Database cursor over the results of a query operation. Objects can be retrieved in a loop, fetching one at a time, or all at once.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
Arithmetic
These are functions that perform arithmetic computations.
-
ceil(Rational a)
The ceiling function. Returns the smallest integral number not smaller than a.
- Parameters:
Rational
a
- Returns:
-
denominator(Rational a)
Returns the denominator of a in a reduced representation.
- Parameters:
Rational
a
- Returns:
-
denominator(RationalFunction f)
Returns the denominator of a
RationalFunction
f.- Parameters:
- Returns:
-
denominator(PuiseuxFraction f)
Returns the denominator of a
PuiseuxFraction
f.- Parameters:
- Returns:
-
div(Scalar a, Scalar b)
Compute the quotient and remainder of a and b in one operation.
-
div_exact(Integer a, Integer b)
Computes the ratio of two given integral numbers under the assumption that the dividend is a multiple of the divisor.
-
ext_gcd(Scalar a, Scalar b)
Compute the greatest common divisor of two numbers (a,b) and accompanying co-factors.
- Parameters:
Scalar
a
Scalar
b
- Returns:
- 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
…and the quotients k1 of a and k2 of b by g.
> print $GCD->k1; 5
> print $GCD->k2; 2
-
floor(Rational a)
The floor function. Returns the smallest integral number not larger than a.
-
gcd(Int a, Int b)
Computes the greatest common divisor of two integers.
-
gcd(Vector<Scalar> v)
Compute the greatest common divisor of the elements of the given vector.
-
gcd(UniPolynomial p, UniPolynomial q)
Returns the greatest common divisor of two univariate polynomials.
- Parameters:
- Returns:
- Example:
We create two UniPolynomials with said coefficient and exponent type:
> $p = new UniPolynomial<Rational,Int>([2,2],[3,2]); > $q = new UniPolynomial<Rational,Int>([6,4],[4,2]);
Printing them reveals what the constructor does:
> print $p; 2*x^3 + 2*x^2
> print $q; 6*x^4 + 4*x^2
Now we can calculate their gcd:
> print gcd($p,$q); x^2
-
get_var_names()
Get the current list of variable names used for pretty printing and string parsing of the given polynomial class
- Returns:
-
is_one(Scalar s)
Compare with the one (1) value of the corresponding data type.
- Parameters:
Scalar
s
- Returns:
-
is_zero(Scalar s)
Compare with the zero (0) value of the corresponding data type.
- Parameters:
Scalar
s
- Returns:
-
isfinite(Scalar a)
Check whether the given number has a finite value.
-
isinf(Scalar a)
Check whether the given number has an infinite value. Return -1/+1 for infinity and 0 for all finite values.
-
lcm(Int a, Int b)
Computes the least common multiple of two integers.
-
lcm(Vector<Scalar> v)
Compute the least common multiple of the elements of the given vector.
-
local_var_names(String names …)
Set the list of variable names for given polynomial class temporarily. The existing name list or the default scheme is restored at the end of the current user cycle, similarly to
prefer_now
.- Parameters:
String
names …
: variable names, seeset_var_names
.
-
monomials<Coefficient, Exponent>(Int n)
Create degree one monomials of the desired polynomial type.
- Type Parameters:
Coefficient
: The polynomial coefficient type. Rational by default.Exponent
: The exponent type. Int by default.- Parameters:
Int
n
: The number of variables- Returns:
UniPolynomial<Coefficient,Exponent>
-
numerator(Rational a)
Returns the numerator of a in a reduced representation.
- Parameters:
Rational
a
- Returns:
-
numerator(RationalFunction f)
Returns the numerator of a
RationalFunction
f.- Parameters:
- Returns:
-
numerator(PuiseuxFraction f)
Returns the numerator of a
PuiseuxFraction
f.- Parameters:
- Returns:
-
set_var_names(String names …)
Set the list of variable names used for pretty printing and string parsing of the given polynomial class When the number of variables in a polynomial is greater than the size of the name list, the excess variable names are produced from a template “${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.
- Parameters:
String
names …
: variable names; may also be bundled in an array an empty list resets to the default naming scheme
-
sum_of_square_roots_naive(Array<Rational> input_array)
Make a naive attempt to sum the square roots of the entries of the input array.
- Parameters:
- Returns:
- from extension:
- 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.
Combinatorics
Combinatorial functions.
-
all_permutations(Int n)
Returns a list of all permutations of the set {0…n-1} as a perl-array
-
are_permuted(Array a, Array b)
Determine whether two arrays a and b are permuted copies of each other.
-
binomial(Int n, Int k)
Compute the binomial coefficient n choose k. Negative values of n (and k) are supported.
-
fibonacci(Int n)
Compute the n-th Fibonacci number
-
n_fixed_points(Array<Int> p)
Returns the number of fixed points of the permutation given by p.
- Parameters:
- Returns:
- Example:
> print n_fixed_points([1,0,2,4,3]); 1
-
permutation_cycle_lengths(Array<Int> p)
Returns the sorted cycle lengths of a permutation
- Parameters:
- Returns:
- Example:
> print permutation_cycle_lengths(new Array<Int>([1,2,0,4,3])); 2 3
-
permutation_cycles(Array<Int> p)
Returns the cycles of a permutation given by p.
- Parameters:
- Returns:
ARRAY
- Example:
> print permutation_cycles([1,0,3,2]); {0 1}{2 3}
-
permutation_matrix<Scalar>(Array<Int> p)
Returns the permutation matrix of the permutation given by p.
-
permutation_order(Array<Int> p)
Returns the order of a permutation
- Parameters:
- Returns:
- Example:
> print permutation_order(new Array<Int>([1,2,0,4,3])); 6
-
permutation_sign(Array<Int> p)
Returns the sign of the permutation given by p.
- Parameters:
- Returns:
- Example:
> print permutation_sign([1,0,3,2]); 1
Data Conversion
This contains functions for data conversions and type casts.
-
cast<Target>(Core::BigObject 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:
Core::BigObject
object
: to be modified- Returns:
-
cols
UNDOCUMENTED
-
cols(Matrix A)
- Parameters:
Matrix
A
- Returns:
- 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 “ ”.
> $w = cols(polytope::cube(2)->VERTICES); > foreach( @$w ){ > print @{$_}, " "; > } 1111 -11-11 -1-111
-
concat_rows(Matrix M)
Concatenates the rows of the
Matrix
M. If M is aSparseMatrix
, then the resulting vector is sparse as well.- Parameters:
Matrix
M
- Returns:
- 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)
-
convert_to<Target>(Scalar s)
Explicit conversion to a different scalar type.
- Type Parameters:
Target
- Parameters:
Scalar
s
- Returns:
Target
-
convert_to<Target>(Vector v)
Explicit conversion to a different element type.
-
convert_to<Target>(Matrix m)
Explicit conversion to a different element type.
-
convert_to<Target>(Polynomial m)
Explicit conversion to a different coefficient type.
- Type Parameters:
Target
- Parameters:
- Returns:
Polynomial<Target>
-
convert_to<Target>(UniPolynomial m)
Explicit conversion to a different coefficient type.
- Type Parameters:
Target
- Parameters:
- Returns:
UniPolynomial<Target>
-
dense(Vector v)
Return the input
Vector
(which is already in the dense form).- Parameters:
Vector
v
- Returns:
-
dense(Matrix M)
Return the input
Matrix
(which is already in dense form).- Parameters:
Matrix
M
- Returns:
-
dense<Element>(SparseVector<Element> v)
Convert to an equivalent dense vector of the same element type. Returns the input
SparseVector
as a denseVector
of the same element type.- Type Parameters:
Element
- Parameters:
SparseVector<Element>
v
- Returns:
Vector<Element>
- Example:
> $v = new SparseVector<Integer>([0,1,2,0,0,0]); > print($v); (6) (1 1) (2 2)
> print(dense($v)); 0 1 2 0 0 0
-
dense<Element>(SparseMatrix<Element> M)
Returns the input
SparseMatrix
as a denseMatrix
of the same element type.- Type Parameters:
Element
- Parameters:
SparseMatrix<Element>
M
- Returns:
Matrix<Element>
- Example:
> $M = new SparseMatrix([[0,0,1],[0,0,0],[0,2,0]]); > print($M); (3) (2 1) (3) (3) (1 2)
> print(dense($M)); 0 0 1 0 0 0 0 2 0
-
dense(IncidenceMatrix M)
Converts an
IncidenceMatrix
to a dense 0/1Matrix
.- Parameters:
- Returns:
- Example:
> $M = polytope::cube(2)->VERTICES_IN_FACETS; > print(dense($M)); 1 0 1 0 0 1 0 1 1 1 0 0 0 0 1 1
-
dense(Set S, Int dim)
-
index_matrix(SparseMatrix M)
Gets the positions of non-zero entries of a
SparseMatrix
.- Parameters:
- Returns:
- 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(SparseVector v)
Get the positions of non-zero entries of a
SparseVector
.- Parameters:
- Returns:
- Example:
> $v = new SparseVector(0,1,1,0,0,0,2,0,3); > print indices($v); {1 2 6 8}
-
lex_ordered(FacetList f)
Visit the facets of f sorted lexicographically.
- Parameters:
- Returns:
- Example:
> $f = new FacetList(polytope::cube(2)->VERTICES_IN_FACETS); > print lex_ordered($f); {{0 1} {0 2} {1 3} {2 3}}
-
rows
UNDOCUMENTED
-
rows(Matrix A)
- Parameters:
Matrix
A
- Returns:
- 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 “ ”.
> $w = rows(polytope::cube(2)->VERTICES); > foreach( @$w ){ > print @{$_}, " "; > } 1-1-1 11-1 1-11 111
-
toMatrix<Scalar>(IncidenceMatrix M)
Converts an
IncidenceMatrix
to aSparseMatrix
.- Type Parameters:
Scalar
- Parameters:
- Returns:
SparseMatrix<Scalar>
- Example:
> $M = polytope::cube(2)->VERTICES_IN_FACETS; > print $M->type->full_name; IncidenceMatrix<NonSymmetric>
> print(toMatrix<Int>($M)->type->full_name); SparseMatrix<Int, NonSymmetric>
-
toTropicalPolynomial(String s, String vars)
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 occurring 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 parsedString
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>>
-
toVector<Scalar>(Set S, Int d)
Creates a
SparseVector
with dimension d and having 1's at positions contained in the givenSet
S.- Type Parameters:
Scalar
: type of apparent 1's- Parameters:
Set
S
Int
d
: dimension of the result- Returns:
SparseVector<Scalar>
- Example:
> $v = toVector<Float>(new Set([0,1]), 5); > print($v); (5) (0 1) (1 1)
-
vector2col(Vector v)
-
vector2row(Vector v)
Database Access
Core methods for connecting to the database and retrieving metadata.
-
polyDB(String host)
Connect to PolyDB server, create a session object
- Parameters:
String
host
: Host address of the PolyDB server in form “hostname” or “hostname:port”; default location is db.polymake.org, can be customized in $PolyDB::default::db_host- Options:
String
user
: user name for the database; default is “polymake” with read-only access to all public collections, can be customized in $PolyDB::default::db_userString
password
: password for the database; when omitted, will be prompted for in the polymake shell; can be customized in $PolyDB::default::db_pwdString
auth_db
: name of the authentication database where the user is defined; default is “admin”, can be customized in $PolyDB::default::db_auth_db- Returns:
- Example:
Connect to the public polymake server as a user “polymake” with read-only permissions
> $polyDB=polyDB();
Connect to a local PolyDB server for testing purposes without authentication
> $testDB=polyDB("localhost");
Connect to a custom server with authentication, prompting for a password input
> $otherDB=polyDB("otherdb.my.domain", user=>"myname");
Formatting
Functions for pretty printing, labels or latex output of polymake types.
-
labeled(Vector data, Array<String> elem_labels)
Prepares a vector for printing, prepends each element with a label and a colon.
-
latex(Matrix data, Array<String> elem_labels)
LaTeX output of a matrix.
- Parameters:
Matrix
data
: to be printedArray<String>
elem_labels
: optional labels for elements; if data is anIncidenceMatrix
, common, or similar, each element will be replaced by its label.- Returns:
-
print_constraints(Matrix<Scalar> M)
Write the rows of a matrix M as inequalities (equations=0) or equations (equations=1) in a readable way. It is possible to specify labels for the coordinates via an optional array coord_labels.
- Parameters:
Matrix<Scalar>
M
: the matrix whose rows are to be written- Options:
Bool
homogeneous
: false if the first coordinate should be interpreted as right hand sideBool
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(Matrix data, Array<String> row_labels, Array<String> elem_labels)
Prepares a matrix for printing, prepends each row with a label and a colon.
- Parameters:
Matrix
data
: to be printedArray<String>
elem_labels
: optional labels for elements; if data is anIncidenceMatrix
, common, or similar, each element will be replaced by its label.- Returns:
- Example:
> print rows_labeled(polytope::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(GraphAdjacency graph, Array<String> elem_labels)
Like above, but specialized for Graphs (defined for convenience: a PTL Graph is not a container)
- Parameters:
GraphAdjacency
graph
: to be printed- Returns:
- Example:
> print rows_labeled(graph::cycle_graph(4)->ADJACENCY, ['a','b','c','d']); a:b d b:a c c:b d d:a c
-
rows_numbered(Matrix data)
Equivalent to
rows_labeled
with omitted row_labels argument. Formerly called “numbered”.
Graph Operations
Operations on graphs.
-
adjacency_matrix(GraphAdjacency graph)
Returns the adjacency matrix of graph nodes. For a normal graph, it will be a kind of
IncidenceMatrix
, for multigraph, it will be a common, with entries encoding the number of parallel edges between two nodes.- Parameters:
GraphAdjacency
graph
- Returns:
-
edges(GraphAdjacency graph)
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:
GraphAdjacency
graph
- Returns:
-
induced_subgraph(GraphAdjacency graph, Set set)
Creates an induced subgraph for the given subset of nodes.
- Parameters:
GraphAdjacency
graph
Set
set
: indices of selected nodes- Returns:
- Example:
> $g = new GraphAdjacency(graph::cycle_graph(5)->ADJACENCY); > $s1 = new Set(1,2,3); > print induced_subgraph($g,$s1); (5) (1 {2}) (2 {1 3}) (3 {2})
-
node_edge_incidences<Coord>(GraphAdjacency graph)
Returns the node-edge incidence matrix of a graph.
- Type Parameters:
Coord
: coordinate type for the resulting matrix, default:Int
- Parameters:
GraphAdjacency
graph
- Returns:
SparseMatrix<Coord>
- 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)
-
nodes(GraphAdjacency graph)
Returns the sequence of all valid nodes of a graph.
- Parameters:
GraphAdjacency
graph
- Returns:
- Example:
> print nodes(graph::cycle_graph(5)->ADJACENCY); {0 1 2 3 4}
Lattice Tools
Functions for lattice related computations.
-
eliminate_denominators(Vector v)
Scale a vector with the least common multiple of the denominators of its coordinates.
-
eliminate_denominators_entire(Matrix v)
Scales entire matrix with the least common multiple of the denominators of its coordinates.
-
eliminate_denominators_entire_affine(Matrix v)
Scales entire matrix with the least common multiple of the denominators of its coordinates (ignore first column).
-
eliminate_denominators_in_rows(Matrix M)
Scale a matrix row-wise with the least common multiple of the denominators of its coordinates.
-
is_integral(Vector v)
Checks whether all coordinates of a rational vector are integral.
-
is_integral(Matrix m)
Checks whether all coordinates of a rational matrix are integral.
-
primitive(Vector v)
Scales the vector to a primitive integral vector.
-
primitive(Matrix M)
Scales each row of the matrix to a primitive integral vector.
-
primitive_affine(Vector v)
Scales the affine part of a vector to a primitive integral vector.
-
primitive_affine(Matrix M)
Scales the affine part of each row of the matrix to a primitive integral vector.
Linear Algebra
These functions are for algebraic computations and constructions of special matrices.
-
anti_diag(Vector d)
Produces a
SparseMatrix
with the givenVector
d as anti-diagonal.-
anti_diag(Matrix M1, Matrix M2)
Returns a block anti-diagonal matrix with blocks M1 and M2.
- Parameters:
Matrix
M1
Matrix
M2
- Returns:
- Example:
> $M = anti_diag(unit_matrix(2),unit_matrix(3)); > print $M; (5) (2 1) (5) (3 1) (5) (4 1) (5) (0 1) (5) (1 1)
To print a more human-readable representation, use the dense() function:
> print dense($M); 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0
-
barycenter(Matrix<Scalar> M)
Calculate the average over the rows of a
Matrix
, i.e., where i-th entry corrsponds to the mean of i-th column vector.
-
basis(Matrix A)
Computes subsets of the rows and columns of the
Matrix
A that form a basis for the linear space spanned by A.- Parameters:
Matrix
A
- Returns:
- 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(Matrix A)
Does the same as
basis
ignoring the first column of the matrix.- Parameters:
Matrix
A
- Returns:
- Example:
Let us illustrate this using the same matrix in the example for
basis
.> $M = new Matrix ([[1,0,0,0],[2,0,0,0],[0,1,0,0],[0,0,1,0]]); > ($row,$col) = basis_affine($M); > print"$row \n$col"; {2 3} {1 2}
> print $M->minor($row,All); 0 1 0 0 0 0 1 0
> print $M->minor(All,$col); 0 0 0 0 1 0 0 1
-
basis_cols(Matrix M)
Computes a subset of the columns of the
Matrix
M that form a basis for the linear space spanned by M.
-
basis_rows(Matrix M)
Computes a subset of the rows of the
Matrix
M that form a basis for the linear space spanned by the rows of M.
-
det(Matrix<Scalar> A)
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.
-
diag(Vector d)
Produces a
SparseMatrix
with givenVector
d as diagonal.-
diag(Matrix m1, Matrix m2)
Returns a block diagonal matrix with blocks m1 and m2.
- Parameters:
Matrix
m1
Matrix
m2
- Returns:
- Example:
> $m1 = new Matrix([1,2],[3,4]); > $m2 = new Matrix([1,0,2],[3,4,0]); > $D = diag($m1,$m2); > print $D; (5) (0 1) (1 2) (5) (0 3) (1 4) 0 0 1 0 2 0 0 3 4 0
To print a more human-readable representation, use the dense() function:
> print dense($D); 1 2 0 0 0 3 4 0 0 0 0 0 1 0 2 0 0 3 4 0
-
eigenvalues(Matrix<Float> M)
Returns the eigenvalues of the given
Matrix
M.- Parameters:
- Returns:
- Example:
> $M = new Matrix<Float>([[2,0,0],[0,3,4],[0,4,9]]); > print(eigenvalues($M)); 2 11 1
-
equal_bases(Matrix M1, Matrix M2)
Check whether both matrices are bases of the same linear subspace. Note: It is assumed that they are *bases* of the row space.
-
hadamard_product(Matrix M1, Matrix M2)
Compute the Hadamard product of two matrices with same dimensions.
-
hermite_normal_form(Matrix M)
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:
- 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
-
householder_trafo(Vector<Float> b)
Householder transformation of
Vector
b. Only the orthogonal matrix reflection H is returned.- Parameters:
- Returns:
-
lattice_basis(Matrix<Integer> A)
Computes a lattice basis of the span of the rows of A.
- Parameters:
- Returns:
- Example:
No two of the rows of the following matrix form a basis.
> $A = new Matrix<Integer>([[2,3],[1,3],[2,4]]); > print lattice_basis($A); 2 3 -1 -1
-
moore_penrose_inverse(Matrix<Float> M)
Computes the Moore-Penrose Inverse of a
Matrix
M.- Parameters:
- Returns:
- Example:
> $M = new Matrix<Float>([1,0],[0,1],[0,1]); > print(moore_penrose_inverse($M)); 1 0 0 0 0.5 0.5
-
normalized(Matrix<Float> M)
Normalize a
Matrix
by dividing each row by its length (l2-norm).- Parameters:
- Returns:
- Example:
> $M = new Matrix<Float>([1.5,2],[2.5,2.5]); > print normalized($M); 0.6 0.8 0.7071067812 0.7071067812
-
null_space(Matrix M)
Computes the null space of a
Matrix
M.-
null_space(Vector v)
Computes the null space of a
Vector
v.
-
ones_matrix<Element>(Int m, Int n)
Creates a
Matrix
with \\n
rows and \\m
columns such that all elements equal to 1.
-
ones_vector<Element>(Int d)
Creates a
Vector
with all elements equal to 1.- Type Parameters:
Element
: default:Rational
.- Parameters:
Int
d
: vector dimension. If omitted, a vector of dimension 0 is created, which can adjust itself when involved in a block matrix operation.- Returns:
Vector<Element>
- Example:
The following stores and prints an integer vector of dimension 3 with all entries equal to 1:
> $v = ones_vector<Int>(3); > print $v; 1 1 1
-
pluecker(Matrix M)
Compute the vector of maximal minors of the matrix M. See also
tpluecker
which is related.
-
project_to_orthogonal_complement(Matrix M, Matrix N)
Projects the row
Matrix
M of points into the orthogonal complement of a subspace given by the rows of theMatrix
N. The points of M will be overwitten.
-
rank_mod_p(Matrix M, Integer p)
Computes the rank of a given matrix M in GF(p), where p is a prime number.
-
singular_value_decomposition(Matrix<Float> M)
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 third V.
- Parameters:
- Returns:
- Example:
> $M = new Matrix<Float>([1,2],[23,24]); > $SVD = singular_value_decomposition($M);
The following prints the three matrices, separated 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
> print $SVD->left_companion * $SVD->sigma * transpose($SVD->right_companion); 1 2 23 24
-
smith_normal_form(Matrix M, Bool inv)
Computes the Smith normal form of a given
Matrix
M. M = L*S*R in normal case, or S = L*M*R in inverted case.- Parameters:
Matrix
M
: must be of integer typeBool
inv
: if true, the companion matrices in the result will be inverted- Returns:
- Example:
> $M = new Matrix<Integer>([1,2],[23,24]); > $SNF = smith_normal_form($M);
The following line prints the three matrices separated 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
-
smith_normal_form_flint(Matrix M)
Compute the Smith normal form of a given matrix M via flint.
-
solve_left(Matrix A, Matrix B)
Computes a
Matrix
X that solves the system XA = B.- Parameters:
Matrix
A
Matrix
B
- Returns:
- Example:
This is useful, for instance, for computing the coordinates of some vectors with respect to a basis. The rows of the matrix solve_left(A,B) are the coordinates of the rows of B with respect to the rows of A. Define the matrices
> $A = new Matrix([[-1,1,0],[0,-1,1]]); > $B = new Matrix([[-4,2,2],[3,-2,-1]]);
so that the rows of A are a basis of the subspace of vectors with zero coordinate sum. Then the rows of
> print solve_left($A, $B); 4 2 -3 -1
contain the coordinates of the rows of B with respect to the rows of A.
-
solve_right(Matrix A, Matrix B)
Computes a
Matrix
X that solves the system AX = B- Parameters:
Matrix
A
Matrix
B
- Returns:
- 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
-
totally_unimodular(Matrix M)
A naive test(exponential in the size of matrix) to check if a
Matrix
M is totally unimodular. The matrix M is totally unimodular if the determinant of each square submatrix equals 0, 1, or -1. For a better implementation try Matthias Walter's polymake extension at https://github.com/xammy/unimodularity-test/wiki/Polymake-Extension.
-
transpose(IncidenceMatrix A)
Computes the transpose AT of an incidence matrix A, i.e., (aT)ij = aji.
- Parameters:
- Returns:
-
transpose(Matrix M)
Computes the transpose MT of a
Matrix
M, i.e., (MT)ij = Mji.
-
unit_matrix<Element>(Int d)
Creates a unit matrix of given dimension.
- Type Parameters:
Element
: default:Rational
- Parameters:
Int
d
: dimension of the matrix- Returns:
SparseMatrix<Element>
- Example:
The following stores the 3-dimensional unit matrix (ones on the diagonal and zeros otherwise) and prints it:
> $M = unit_matrix(3); > print $M; (3) (0 1) (3) (1 1) (3) (2 1)
> print $M->type->full_name; SparseMatrix<Rational, Symmetric>
- Example:
The following stores the 3-dimensional unit matrix (ones on the diagonal and zeros otherwise) of type Int in a variable and prints it:
> $M = unit_matrix<Int>(3); > print $M->type->full_name; SparseMatrix<Int, Symmetric>
-
unit_vector<Element>(Int d, Int pos)
Creates a
SparseVector
of given length d with an entry 1 at position pos and zeroes elsewhere.- Type Parameters:
Element
: default:Rational
- Parameters:
Int
d
: the dimension of the vectorInt
pos
: the position of the 1- Returns:
SparseVector<Element>
- Example:
The following stores a vector of dimension 5 with a single 1 (as a Rational) at position 2:
> $v = unit_vector(5,2); > print $v; (5) (2 1)
> print(dense($v)); 0 0 1 0 0
- 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)
-
zero_matrix<Element>(Int i, Int j)
Creates a zero matrix of given dimensions.
- Type Parameters:
Element
: default:Rational
- Parameters:
Int
i
: number of rowsInt
j
: number of columns- Returns:
Matrix<Element>
- 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
> print $M->type->full_name; Matrix<Rational, NonSymmetric>
- 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>
-
zero_vector<Element>(Int d)
Creates a
Vector
with all elements equal to zero.- Type Parameters:
Element
: default:Rational
- Parameters:
Int
d
: vector dimension. If omitted, a vector of dimension 0 is created, which can adjust itself when involved in a block matrix operation.- Returns:
Vector<Element>
- 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
Set Operations
This category contains functions performing operations on Sets.
-
all_subsets(Any c)
Returns all subsets of size k of a given container as perl-array.
- Parameters:
- Returns:
-
all_subsets_of_k(Any c, Int k)
Returns all subsets of size k of a given container as perl-array.
- Parameters:
Int
k
: size of the subsets- Returns:
-
range_from(Int a)
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.
-
scalar2set(Scalar s)
Returns the singleton set {s}.
Utilities
Miscellaneous functions.
-
average(ARRAY array)
Returns the average value of the array elements.
- Parameters:
ARRAY
array
- Example:
> print average([1,2,3]); 2
-
bounding_box(Matrix m)
Compute the column-wise bounds for the given Matrix m.
- Parameters:
Matrix
m
- Returns:
-
distance_matrix(Matrix S, CODE d)
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.
- Parameters:
Matrix
S
CODE
d
- Returns:
- 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:
-
expand(Map<Integer,Int> factorization)
Use flint to expand the prime factorization of an Integer This is the inverse operation of
factor
- Parameters:
- Returns:
- from extension:
-
factor(Integer n)
Use flint to compute the prime factorization of an Integer
- Parameters:
Integer
n
- Returns:
- from extension:
-
histogram(ARRAY data)
Produce a histogram of an array: each different element value is mapped on the number of its occurences.
- Parameters:
ARRAY
data
- Returns:
- 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)
Return a map indexing an array of sets
- Parameters:
- Returns:
- 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)}
-
is_nonnegative(ARRAY data)
Checks if a given sequence is nonnegative
- Parameters:
ARRAY
data
- Returns:
- Example:
> print is_nonnegative([1,0,3]); 1
-
is_unimodal(ARRAY data)
Checks if a given sequence is unimodal
-
maximum(ARRAY array)
Returns the maximal element of an array.
- Parameters:
ARRAY
array
- Example:
> print maximum([1,2,3,4,5,6,7,23]); 23
-
median(ARRAY array)
Returns the median value of the array elements. Running time O(n log(n)) if n is the length of the input.
- Parameters:
ARRAY
array
- Example:
> print median([1,2,3,9]); 2.5
-
minimum(ARRAY array)
Returns the minimal element of an array.
- Parameters:
ARRAY
array
- Example:
> print minimum([23,42,666]); 23
-
perturb_matrix(Matrix M, Float eps, Bool not_hom)
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: 1Bool
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:
-
sum(ARRAY array)
Returns the sum of the array elements.
- Parameters:
ARRAY
array
- Example:
> print sum([1,2,3]); 6
-
valuation(Rational x, Integer p)
Use flint's Integer factorization to compute the p-adic valuation of a Rational x
Visualization
These functions are for visualization.
-
compose(Visual::Object vis_obj …)
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:
- Example:
Draw a pretty 8-pointed star:
> compose(cube(2)->VISUAL, cross(2,sqrt(2))->VISUAL, Title=>"A pretty star.", VertexLabels=>"hidden");
-
compose(Visual::Container vis_container, Visual::Object vis_obj …)
Add new objects to a composite drawing.
- Parameters:
Visual::Container
vis_container
: drawing produced by some visualization functionVisual::Object
vis_obj …
: objects to be added- Options:
String
Title
: new name for the drawingAny
decorations
: to be applied to all components as default values.- Returns:
-
geomview(Visual::Object vis_obj …)
Run geomview to display given visual objects.
- Parameters:
Visual::Object
vis_obj …
: objects to display- Options:
String
File
: “filename” or “AUTO” Store the objects in agcl
(geomview control language) file instead of starting the interactive GUI. The geometric data inOFF
format is embedded in the Lisp-style commands, but can be easily extracted using any text editor, if needed. The.gcl
suffix is automatically added to the file name. Specify AUTO if you want the filename be automatically derived from the drawing title. You can also use any expression allowed for theopen
function, including “-” for terminal output, “&HANDLE” for an already opened file handle, or “| program” for a pipe.
-
javaview(Visual::Object vis_obj …)
Run JavaView with the given visual objects.
- 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 theopen
function, including “-” for terminal output, “&HANDLE” for an already opened file handle, or “| program” for a pipe.- from extension:
-
jreality(Visual::Object vis_obj …)
Run jReality to display given visual objects.
- Parameters:
Visual::Object
vis_obj …
: objects to display- Options:
String
File
: “filename” or “AUTO” Store the object description in a 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 theopen
function, including “-” for terminal output, “&HANDLE” for an already opened file handle, or “| program” for a pipe.- from extension:
-
postscript(Visual::Object vis_obj …)
Create a Postscript ™ 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 theopen
function, including “-” for terminal output, “&HANDLE” for an already opened file handle, or “| program” for a pipe.
-
povray(Visual::Object vis_obj …)
Run POVRAY to display given visual objects.
- Parameters:
Visual::Object
vis_obj …
: objects to display- Options:
String
File
: “filename” or “AUTO” Store the object description in a POVRAY source file without actual rendering. The.pov
suffix is automatically added to the file name. Specify AUTO if you want the filename be automatically derived from the drawing title. You can also use any expression allowed for theopen
function, including “-” for terminal output, “&HANDLE” for an already opened file handle, or “| program” for a pipe.
-
sketch(Visual::Object 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 theopen
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(Visual::Object vis_obj)
Suppress creation of dynamic (interactive) scenes.
- Parameters:
Visual::Object
vis_obj
: drawing, e.g. created byVISUAL_GRAPH
orSCHLEGEL
.- Returns:
-
svg(Visual::Object vis_obj …)
Create a Svg 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.svg
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 theopen
function, including “-” for terminal output, “&HANDLE” for an already opened file handle, or “| program” for a pipe.
-
threejs(Visual::Object 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 theopen
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(Visual::Object 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 theopen
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.
-
x3d(Visual::Object vis_obj …)
Create an X3D 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.x3d
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 theopen
function, including “-” for terminal output, “&HANDLE” for an already opened file handle, or “| program” for a pipe.
no category
-
evaluate(PuiseuxFraction f, Coefficient x, Int exp)
Evaluate a
PuiseuxFraction
at aRational
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(Matrix m, Coefficient x, Int exp)
Evaluate all
PuiseuxFraction
s in aMatrix
at aRational
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.-
evaluate(Vector v, Coefficient x, Int exp)
Evaluate all
PuiseuxFraction
s in aVector
at aRational
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.
-
evaluate_float(PuiseuxFraction f, Float x)
Approximate evaluation at x
- Parameters:
Float
x
- Returns:
-
evaluate_float(Matrix m, Float x)
Approximate evaluation of a
Matrix
at x-
evaluate_float(Vector v, Float x)
Approximate evaluation of a
Vector
at x
Small Object Types
Algebraic Types
This category contains all “algebraic” types, such as matrices, vectors, polynomials, rings, …
-
Matrix<Element, Sym>
A type for matrices with entries of type Element. You can create a new Matrix by entering its entries as an array of row vectors:
> 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: :''[[.:common#Vector |Vector]]<Element>'' ? Example: :: <code perl> > $M = new Matrix([1,2,3],[4,5,6],[7,8,9]); > print $M->anti_diagonal; 3 5 7 </code> :: <code perl> > print $M->anti_diagonal(-1); 2 4 </code> ? **''clear([[.:common#Int |Int]] r, [[.:common#Int |Int]] c)''** :: Change the dimensions setting all elements to 0. ? Parameters: :: ''[[.:common#Int |Int]]'' ''r'': new number of rows :: ''[[.:common#Int |Int]]'' ''c'': new number of columns ? **''col([[.:common#Int |Int]] i)''** :: Returns the //i//-th column. ? Parameters: :: ''[[.:common#Int |Int]]'' ''i'' ? Returns: :''[[.:common#Vector |Vector]]<Element>'' ? Example: :: <code perl> > $M = new Matrix([1,2,3],[4,5,6]); > print $M->col(1); 2 5 </code> ? **''cols()''** :: Returns the number of columns. ? Returns: :''[[.:common#Int |Int]]'' ? Example: :: <code perl> > $M = new Matrix([1,2,3],[4,5,6]); > print $M->cols; 3 </code> ? **''diagonal([[.:common#Int |Int]] i)''** :: Returns the __diagonal__ of the matrix. ? Parameters: :: ''[[.:common#Int |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: :''[[.:common#Vector |Vector]]<Element>'' ? Example: :: <code perl> > $M = new Matrix([1,2,3],[4,5,6],[7,8,9]); > print $M->diagonal; 1 5 9 </code> :: <code perl> > print $M->diagonal(1); 4 8 </code> ? **''div_exact([[.:common#Int |Int]] a)''** :: Divides every entry of ''[[.:common#Matrix |Matrix]]'' //M// by ''[[.:common#Int |Int]]'' //a//. If type of //M// is ''[[.:common#Int |Int]]'', then only the integer part of the each entry is taken into account. ? Parameters: :: ''[[.:common#Int |Int]]'' ''a'' ? Returns: :''[[.:common#Matrix |Matrix]]'' ? Example: :: <code perl> > $M = new Matrix([1,2],[2,1]); > print $M->div_exact(2); 1/2 1 1 1/2 </code> ? **''elem([[.:common#Int |Int]] r, [[.:common#Int |Int]] c)''** :: 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: :: ''[[.:common#Int |Int]]'' ''r'': the row index :: ''[[.:common#Int |Int]]'' ''c'': the column index ? Returns: :''Element'' ? Example: :: <code perl> > $M = new Matrix([1,2,3],[4,5,6]); > print $M->elem(1,1); 5 </code> ? **''minor([[.:common#Set |Set]] r, [[.:common#Set |Set]] c)''** :: Returns a __minor__ of the matrix containing the rows in //r// and the columns in //c//. You can pass [[.:common#all_rows_or_cols |All]] if you want all rows or columns and ~ for the complement of a set. ? Parameters: :: ''[[.:common#Set |Set]]'' ''r'': the rows :: ''[[.:common#Set |Set]]'' ''c'': the columns ? Returns: :''[[.:common#Matrix |Matrix]]'' ? Example: :: <code perl> > $M = new Matrix([1,2,3],[4,5,6]); > print $M->minor([0,1],[0,1]); 1 2 4 5 </code> :: <code perl> > print $M->minor(All,~[0]); 2 3 5 6 </code> ? **''resize([[.:common#Int |Int]] r, [[.:common#Int |Int]] c)''** :: Resizes the dimension of ''[[.:common#Matrix |Matrix]]'' //M// to //r// x //c//; missing entries are filled with zero and for all i > //r// or j > //c// the (i,j)-th entry of //M// is forgotten. ? Parameters: :: ''[[.:common#Int |Int]]'' ''r'': new number of rows :: ''[[.:common#Int |Int]]'' ''c'': new number of columns ? Example: :: <code perl> > $M = new Matrix([1,2,3],[4,5,6]); > $M->resize(3,2); > print $M; 1 2 4 5 0 0 </code> ? **''row([[.:common#Int |Int]] i)''** :: Returns the //i//-th row. ? Parameters: :: ''[[.:common#Int |Int]]'' ''i'' ? Returns: :''[[.:common#Vector |Vector]]<Element>'' ? Example: :: <code perl> > $M = new Matrix([1,2,3],[4,5,6]); > print $M->row(1); 4 5 6 </code> ? **''rows()''** :: Returns the number of rows. ? Returns: :''[[.:common#Int |Int]]'' ? Example: :: <code perl> > $M = new Matrix([1,2,3],[4,5,6]); > print $M->rows; 2 </code>
-
Plucker<Scalar>
A class to hold and process Plücker coordinates of a subspace.
- Type Parameters:
Scalar
: default:Rational
- Methods of Plucker:
-
coordinates
UNDOCUMENTED
-
permuted
UNDOCUMENTED
-
-
Polynomial<Coefficient, Exponent>
- Type Parameters:
Coefficient
: default:Rational
Exponent
: default:Int
- Methods of Polynomial:
-
coefficients_as_vector()
The vector of all coefficients. The sorting agrees with
monomials_as_matrix
.- Returns:
Vector<Coefficient>
-
constant_coefficient()
The constant coefficient.
- Returns:
-
deg()
The degree of the polynomial
- Returns:
-
embed(Int nvars)
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 usemapvars
.-
get_var_names
set the variable names
-
initial_form(Vector<Exponent> v)
The initial form with respect to a weight-vector v
- Parameters:
Vector<Exponent>
v
: weights- Returns:
-
lc()
The leading coefficient.
- Returns:
-
lm()
The exponent of the leading monomial.
- Returns:
-
mapvars(Array<Int> indices, Int nvars)
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 givenArray
must be the same as the number of variables of the original polynomial.-
monomial(Int var_index, Int n_vars)
construct a monomial of degree 1 with a given variable
-
monomials_as_matrix()
The matrix of all exponent vectors (row-wise). The sorting agrees with
coefficients_as_vector
.- Returns:
SparseMatrix<Exponent>
-
n_vars()
Number of variables
- Returns:
-
print_ordered(Matrix m)
Print a polynomial with terms sorted according to a given
Matrix
m.- Parameters:
Matrix
m
-
project()
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 usingsubstitute
with aMap
.- Returns:
- 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
-
reset_var_names
reset the variable names according to the default naming scheme
-
set_var_names
set the variable names
-
substitute
Substitute a list of values in a
Polynomial
withInt
exponents. Either with an array of values, or with aMap
mapping variable indices to values.-
trivial()
The polynomial is zero.
- Returns:
-
-
PuiseuxFraction<MinMax, Coefficient, Exponent>
- Type Parameters:
Coefficient
: default:Rational
Exponent
: default:Rational
- Methods of PuiseuxFraction:
-
val()
The valuation.
- Returns:
TropicalNumber<MinMax>
- Example:
> $x = monomials<Rational,Rational>(1); > print new PuiseuxFraction<Max>(2*$x*$x-$x+2)->val; 2
- Example:
The valuation can also be applied to Containers via convert_to
> $m = long_and_winding(3)->FACETS; > print convert_to<TropicalNumber<Max>>($m); (7) (0 2) (1 0) (7) (0 1) (2 0) (7) (1 1) (3 0) (7) (2 1) (3 0) (7) (1 1/2) (2 1/2) (4 0) (7) (3 1) (5 0) (7) (4 1) (5 0) (7) (3 3/4) (4 3/4) (6 0) (7) (5 0) (7) (6 0)
-
-
RationalFunction<Coefficient, Exponent>
-
SparseMatrix<Element, Sym>
A SparseMatrix is a special form of a
Matrix
optimized for large matrices with few non-zero elements. Physically it is stored as a two-dimensional grid of balanced binary trees with row and column indexes as search keys. Usedense
to convert a SparseMatrix into the dense form.- Type Parameters:
Element
Sym
: one ofSymmetric
orNonSymmetric
, default:NonSymmetric
- derived from:
- Example:
construct a SparseMatrix from a list of row vectors. Each row can be specified in a dense or sparse form. Number of columns is determined by the size of dense rows or by the maximal index value in sparse rows.
> $M = new SparseMatrix<Integer>( > [0, 1, 0, 0, 0], > {4 => 2}, > {}, > [3, -1, 0, 0, 0]); > print $M; (5) (1 1) (5) (4 2) (5) (5) (0 3) (1 -1)
- Example:
construct a SparseMatrix with explicitly given number of columns:
> $M = new SparseMatrix<Integer>( > {1 => 10}, > {2 => 20}, > {cols => 5}); > print $M; (5) (1 10) (5) (2 20)
- Methods of SparseMatrix:
-
resize(Int r, Int c)
Resizes the
Matrix
according to given number of rows and columns. All elements in added rows and/or columns are implicit zeros. If the number of rows and/or columns of the original matrix is less than the given number or rows and/or columns, the extra entries are deleted.-
squeeze
Adjusts the given
SparseMatrix
by removing empty rows and columns. The remaining rows and columns are renumbered without gaps.- Example:
> $M = new SparseMatrix<Integer>({1 =>2, _dim => 6},{5 => 3, _dim => 6}, {});
print(dense($M));
> $M->squeeze(); > print(dense($M)); 2 0 0 3
-
squeeze_cols
Adjusts the given
SparseMatrix
by removing empty columns. The remaining columns are renumbered without gaps.-
squeeze_rows
Adjusts the given
SparseMatrix
by removing empty rows. The remaining rows are renumbered without gaps.- Example:
> $M = new SparseMatrix<Integer>({1 =>2, _dim => 6},{5 => 3, _dim => 6}, {});
print(dense($M));
> $M->squeeze_rows(); > print(dense($M)); 0 2 0 0 0 0 0 0 0 0 0 3
-
-
SparseVector<Element>
A SparseVector is a special form of a
Vector
optimized for large vectors with few non-zero elements. Physically it is stored as a balanced binary tree with element index as a search key. 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. Usedense
to convert a sparse vector into the dense form.- Type Parameters:
Element
- derived from:
- Example:
construct a SparseVector from a contiguous list of scalars; zero values are filtered out automatically:
> $v = new SparseVector<Integer>(0, 1, 2, 0, 0, 0); > print $v, "\n"; (6) (1 1) (2 2)
- Example:
construct a SparseVector from a hash map with non-zero values; dimension is provided explicitly:
> $v = new SparseVector<Integer>({ 1 => 1, 2 => 2, _dim => 6 }); > print $v, "\n"; (6) (1 1) (2 2)
- Methods of SparseVector:
-
size()
The number of non-zero entries of a sparse vector.
- Returns:
- Example:
> $v = new SparseVector<Integer>(0, 1, 2, 0, 0, 0); > print($v->size()); 2
-
-
UniPolynomial<Coefficient, Exponent>
A class for univariate polynomials.
- Type Parameters:
Coefficient
: default:Rational
Exponent
: default:Int
- Methods of UniPolynomial:
-
coefficients_as_vector()
The vector of all coefficients. The sorting agrees with
monomials_as_vector
.- Returns:
Vector<Coefficient>
-
constant_coefficient()
The constant coefficient.
- Returns:
-
deg()
The highest degree occurring in the polynomial.
- Returns:
Exponent
-
evaluate(Coefficient x, Exponent exp)
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.- Parameters:
Coefficient
x
Exponent
exp
: (default: 1)- Returns:
Coefficient
-
evaluate_float(Float x)
Approximate evaluation of the polynomial at a
Float
x.- Parameters:
Float
x
- Returns:
-
get_var_names
get the variable name
-
lc()
The leading coefficient.
- Returns:
-
lower_deg()
The lowest degree occurring in the polynomial.
- Returns:
Exponent
-
monomial
create a monomial of degree 1
-
monomials_as_vector()
The vector of all exponents. The order agrees with
coefficients_as_vector
.- Returns:
Vector<Exponent>
-
n_vars
Number of variables
-
print_ordered(Exponent x)
Print a polynomial with terms sorted according to exponent*x.
- Parameters:
Exponent
x
-
reset_var_names
reset the variable name according to the default naming scheme
-
set_var_names
set the variable name
-
substitute
Substitute some value in a
UniPolynomial
withInt
exponents. When all exponents are positive the argument can be any scalar, matrix or polynomial type. With negative exponents, polynomials are not supported (useRationalFunction
instead) and any given matrix must be invertible
-
-
Vector<Element>
A type for vectors with entries of type Element. You can perform algebraic operations such as addition or scalar multiplication.
- Type Parameters:
Element
- Example:
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:
-
all_rows_or_cols
Allows the use of the keyword “All” for all rows or columns, e.g. when constructing a
minor
.
Arithmetic
These types are needed as return types of arithmetic computations.
-
Div<Scalar>
The complete result of an integral division
- Type Parameters:
Scalar
: type supporting modulo division, e.g.Integer
orUniPolynomial
- Methods of Div:
-
quot()
the quotient
- Returns:
Scalar
-
rem()
the remainder
- Returns:
Scalar
-
-
ExtGCD<Scalar>
The complete result of the calculation of the greatest common divisor of two numbers a and b:
-
Max
tropical addition: max
- Methods of Max:
-
apply
UNDOCUMENTED
-
orientation
UNDOCUMENTED
-
-
Min
tropical addition: min
- Methods of Min:
-
apply
UNDOCUMENTED
-
orientation
UNDOCUMENTED
-
-
TropicalNumber<Addition, Scalar>
- Type Parameters:
Addition
Scalar
: default:Rational
- Methods of TropicalNumber:
-
orientation()
The orientation of the associated addition, i.e. +1 if the corresponding 0 is +inf -1 if the corresponding 0 is -inf
- Returns:
-
zero()
The zero element of the tropical semiring of this element.
- Returns:
Scalar
-
Artificial
These types are auxiliary artifacts helping to build other classes, primarily representing template parameters or enumeration constants. They should not be used alone as property types or function arguments. In the most cases they won't even have user-accessible constructors.
-
CachedObjectPointer
-
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.
-
Directed
Type tag for a directed graph.
-
DirectedMulti
Type tag for a directed multigraph.
-
EdgeIterator
-
EdgeList
-
Iterator<Element>
An iterator over a sequence of objects. The objects may be stored in a container or be generated on the fly.
- Type Parameters:
Element
-
LocalFloatEpsilon
-
NonSymmetric
Labels a
Matrix
or anIncidenceMatrix
as non-symmetric.
-
Serialized<X>
- Type Parameters:
X
-
Symmetric
Labels a
Matrix
or anIncidenceMatrix
as symmetric.
-
Undirected
Type tag for an undirected graph.
-
UndirectedMulti
Type tag for an undirected multigraph.
Basic Types
This category contains all basic types, in particular those that wrap C++, GMP or perl types such as Int
, Integer
, Rational
, Float
, Array
, String
, …
-
AccurateFloat
A wrapper for AccurateFloat.
-
Array<Element>
An array with elements of type Element. Corresponds to the C++ type
Array
.- Type Parameters:
Element
- Methods of Array:
-
size()
Returns the size.
- Returns:
-
-
Bool
Corresponds to the C++ type bool.
-
Float
Corresponds to the C++ type double.
- Methods of Float:
-
inf
produce an infinitely large positive value
-
minus_inf
produce an infinitely large negative value
-
-
GF2
The Galois field of order 2
-
Int
Corresponds to the C++ type Int.
- Methods of Int:
-
max
produce the highest possible value
-
min
produce the lowest possible value
-
-
Integer
An integer of arbitrary size.
- Methods of Integer:
-
inf
produce an infinitely large positive value
-
minus_inf
produce an infinitely large negative value
-
-
List<Element>
Corresponds to the C++ type std::list.
- Type Parameters:
Element
-
Pair<First, Second>
Corresponds to the C++ type std::pair.
- Type Parameters:
First
Second
-
QuadraticExtension<Field>
Realizes quadratic extensions of fields. You can construct the value a+b\(\sqrt r\) via QuadraticExtension(a, b, r) (where a, b, r are of type Field). Caution: QuadraticExtension requires flint for normalization. For example, the input (1,1,4) will not be reduced to 3 without flint.
- Type Parameters:
Field
: default:Rational
-
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.
-
-
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.
- Type Parameters:
Dir
: one ofUndirected
,Directed
,UndirectedMulti
orDirectedMulti
Element
: data associated with edges- derived from:
- Methods of EdgeHashMap:
-
edge(Int from, Int to)
Access the data associated with an edge between two given nodes. The new data element is created on demand.
-
erase(Int from, Int to)
Delete the data associated with an edge between two given nodes.
-
find(Int from, Int to)
Access the data associated with an edge between two given nodes.
-
-
EdgeMap<Dir, Element>
Dense mapping of edges to data items.
- Type Parameters:
Dir
: kind of the host graph,Undirected
,Directed
,UndirectedMulti
, orDirectedMulti
Element
: data associated with edges- derived from:
- Methods of EdgeMap:
-
GraphAdjacency<Dir>
- Type Parameters:
Dir
: one ofUndirected
,Directed
,UndirectedMulti
orDirectedMulti
, default:Undirected
- Methods of GraphAdjacency:
-
add_edge(Int tail_node, Int head_node)
In a multigraph, creates a new edge connecting two given nodes. In a normal graph, creates a new edge only if the nodes were not connected yet. Returns the index of the (new) edge.
-
add_node()
Add a new node without incident edes.
- Returns:
-
adjacent_nodes(Int node)
Returns the set of indices of nodes adjacent to node.
- Parameters:
Int
node
- Returns:
-
all_edges(Int tail_node, Int head_node)
Returns an iterator visiting all (parallel) edges connecting two given nodes.
-
contract_edge(Int node1, Int 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.
-
degree(Int node)
Returns the number of edges incident to node.
- Parameters:
Int
node
- Returns:
-
delete_all_edges(Int tail_node, Int head_node)
Deletes all edges in a multigraph connecting two given nodes.
-
delete_edge(Int tail_node, Int head_node)
Deletes the edge connecting two given nodes, if there was one. In a multigraph, deletes one arbitrary edge from the parallel bundle.
-
delete_edge(Iterator iterator)
Delete the edge in a multigraph pointed to by the given iterator
- Parameters:
-
delete_node(Int node)
Deletes all edges incident to the given node and marks it as invalid. The numeration of other nodes stays unchanged.
- Parameters:
Int
node
-
dim()
Returns the maximal node index + 1. If the graph does not have gaps caused by node deletion, the result is equivalent to
nodes
().- Returns:
-
edge(Int tail_node, Int head_node)
Returns the index of the edge connecting two given nodes. The edge is created if was not there and if the graph is not read-only. In a multigraph, an arbitrary edge from the parallel bundle will be picked. Warning: If there is no
EdgeMap
attached to the graph, the returned number will always be zero.- Parameters:
Int
tail_node
Int
head_node
- Returns:
- Example:
Get edge numbers for the
HASSE_DIAGRAM
of a square.> $c = polytope::cube(2); > $g = $c->HASSE_DIAGRAM->ADJACENCY; > print $g; {1 2 3 4} {5 7} {6 7} {5 8} {6 8} {9} {9} {9} {9} {}
> $E = new EdgeMap<Directed,Int>($g); > print $g->edge(0,1); 0
> print $g->edge(1,5); 4
> print $g->edge(5,9); 12
- Example:
If there is no
EdgeMap
the returned number is always zero.> $c = polytope::cube(2); > $g = $c->HASSE_DIAGRAM->ADJACENCY; > print $g->edge(0,1); 0
> print $g->edge(1,5); 0
> print $g->edge(5,9); 0
- Example:
If the graph is read-only it cannot add an edge.
> $c = polytope::cube(2); > $g = $c->HASSE_DIAGRAM->ADJACENCY; > $E = new EdgeMap<Directed,Int>($g); > print $g->edge(0,1); 0
> print $g->edge(0,5); polymake: ERROR: non-existing edge
> $gg = new GraphAdjacency<Directed>($g); > $EE = new EdgeMap<Directed,Int>($gg); > print $gg->edge(0,5); 4
-
edge_exists(Int tail_node, Int head_node)
Checks whether two given nodes are connected by (at least) one edge.
-
edges()
Get the total number of edges.
- Returns:
-
has_gaps()
Returns true if some nodes have been deleted since the last
squeeze
operation.- Returns:
-
in_adjacent_nodes(Int node)
Returns the set of indices of the nodes that have an edge heading to node.
- Parameters:
Int
node
- Returns:
-
in_degree(Int node)
Returns the number of edges heading to node.
- Parameters:
Int
node
- Returns:
-
in_edges(Int node)
Returns a sequence of edges heading to (in Directed case) or incident to (in Undirected case) node.
- Parameters:
Int
node
- Returns:
-
invalid_node(Int node)
Returns true if the given node index is either out of valid range or points to a formerly deleted node.
- Parameters:
Int
node
- Returns:
-
node_exists(Int node)
Check whether the node with given index exists.
- Parameters:
Int
node
- Returns:
-
node_out_of_range(Int node)
Returns true if the given node index is out of valid range.
- Parameters:
Int
node
- Returns:
-
nodes()
Get the total number of nodes.
- Returns:
-
out_adjacent_nodes(Int node)
Returns the set of indices of the nodes with an edge arriving from node.
- Parameters:
Int
node
- Returns:
-
out_degree(Int node)
Returns the number of edges leaving node.
- Parameters:
Int
node
- Returns:
-
out_edges(Int node)
Returns a sequence of edges leaving (in Directed case) or incident to (in Undirected case) node.
- Parameters:
Int
node
- Returns:
-
permute_inv_nodes
permute the nodes param perm inverse permutation of node indexes
-
permute_nodes
permute the nodes param perm permutation of node indexes
-
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
GraphAdjacency
.- Type Parameters:
Dir
: kind of the host graph:Undirected
,Directed
,UndirectedMulti
, orDirectedMulti
Element
: data associated with nodes or edges
-
NodeHashMap<Dir, Element>
Sparse mapping of nodes to data items.
- Type Parameters:
Dir
: one ofUndirected
,Directed
,UndirectedMulti
orDirectedMulti
Element
: data associated with nodes- derived from:
-
NodeMap<Dir, Element>
Dense mapping of nodes to data items.
- Type Parameters:
Dir
: kind of the host graph,Undirected
,Directed
,UndirectedMulti
, orDirectedMulti
Element
: data associated with nodes- derived from:
Linear Algebra
These types are needed as return types of algebraic computations.
-
HermiteNormalForm<Scalar>
Complete result of the Hermite normal form computation of the input matrix M.
- Type Parameters:
Scalar
: matrix element type- Methods of HermiteNormalForm:
-
companion()
unimodular matrix R such that M*R = H
- Returns:
SparseMatrix<Scalar>
-
hnf()
the Hermite normal form
- Returns:
Matrix<Scalar>
-
rank()
rank of M
- Returns:
-
-
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:- Methods of SingularValueDecomposition:
-
left_companion()
matrix of left singular vectors
- Returns:
-
right_companion()
matrix of right singular vectors
- Returns:
-
sigma()
the diagonalized matrix
- Returns:
-
-
SmithNormalForm<Scalar>
Complete result of the Smith normal form computation of the input matrix M.
- Type Parameters:
Scalar
: matrix element type- Methods of SmithNormalForm:
-
form()
the Smith normal form
- Returns:
SparseMatrix<Scalar>
-
left_companion()
unimodular matrix L such that M = L*S*R
- Returns:
SparseMatrix<Scalar>
-
rank()
rank of M
- Returns:
-
right_companion()
unimodular matrix R such that M = L*S*R
- Returns:
SparseMatrix<Scalar>
-
torsion()
absolute values of the entries greater than 1 of the diagonal together with their multiplicity
- Returns:
-
Set Types
In this category you find all property types related to sets, such as Set, Map, HashMap, IncidenceMatrix, …
-
ApproximateSet<Element>
A specialization of Sets containing elements of type Element, but where equality is enforced only up to a global epsilon.
- Type Parameters:
Element
: default:Float
- derived from:
- Example:
You can for example create a new ApproximateSet with two elements by:
> $s = new ApproximateSet(1.1, 1.2, 1.200000001);
-
Bitset
Container class for dense sets of integers. A special class optimized for representation of a constrained range of non-negative integer numbers.
- derived from:
-
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 common 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:
-
erase(Set f)
Remove a facet.
- Parameters:
Set
f
: facet to remove- Returns:
-
eraseSubsets(Set s)
Remove all subsets of a given set
- Parameters:
Set
s
: filter for removal- Returns:
-
eraseSupersets(Set s)
Remove all supersets of a given set
- Parameters:
Set
s
: filter for removal- Returns:
-
find(Set f)
Look up a facet.
- Parameters:
Set
f
: facet to find- Returns:
-
findSubsets(Set s)
Find all subsets of a given set.
- Parameters:
Set
s
- Returns:
-
findSupersets(Set s)
Find all supersets of a given set.
- Parameters:
Set
s
- Returns:
-
insert(Set 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(Set f)
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:
-
insertMin(Set f)
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:
-
n_vertices()
The number of vertices
- Returns:
-
size()
The number of facets in the list.
- Returns:
-
-
HashMap<Key, 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). On the perl side HashMaps are treated like hashrefs. You can work with a HashMap like you work with aMap
(keeping in mind differences in performance and memory demand).- Type Parameters:
Key
: type of the key valuesValue
: type of the mapped value- Example:
You can create a new HashMap mapping Ints to Strings by
> $myhashmap = new HashMap<Int, String>([1, "Monday"], [2, "Tuesday"]);
- Methods of HashMap:
-
size()
Size of the map
- Returns:
-
-
HashSet<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.
- Type Parameters:
Element
- Methods of HashSet:
-
size()
The cardinality of the set.
- Returns:
-
-
IncidenceMatrix<Sym>
A 0/1 incidence matrix.
- Type Parameters:
Sym
: one ofSymmetric
orNonSymmetric
, default:NonSymmetric
- Methods of IncidenceMatrix:
-
col(Int i)
Returns the i-th column.
- Parameters:
Int
i
- Returns:
-
cols()
Returns the number of columns.
- Returns:
-
elem(Int r, Int c)
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.
-
minor(Set r, Set c)
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.
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row(Int i)
Returns the i-th row.
- Parameters:
Int
i
- Returns:
-
rows()
Returns the number of rows.
- Returns:
-
squeeze
Removes empty rows and columns. The remaining rows and columns are renumbered without gaps.
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squeeze_cols
Removes empty columns. The remaining columns are renumbered without gaps.
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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.
- Type Parameters:
Key
: type of the key valuesValue
: type of the mapped value- Example:
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}; Monday
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Set<Element>
A type for ordered sets containing elements of type Element.
- Type Parameters:
Element
: default:Int
- Example:
You can for example create a new Set by:
> $s1 = new Set(2, 3, 5, 7); > $s2 = new Set(4, 5);
You can perform set theoretic operations: union
> print $s1 + $s2; {2 3 4 5 7}
intersection
> print $s1 * $s2; {5}
difference
> print $s1 - $s2; {2 3 7}
symmetric difference
> print $s1 ^ $s2; {2 3 4 7}
- Methods of Set:
-
back()
The last element of the set, that is, the largest element
- Returns:
-
collect(Element e)
Add to the set, report true if existed formerly.
- Parameters:
Element
e
: element to insert into the set- Returns:
-
contains(Element e)
Check if e is contained in the set.
- Parameters:
Element
e
: element check for- Returns:
-
front()
The first element of the set, that is, the smallest element
- Returns:
-
size()
The cardinality of the set.
- Returns:
-
Visualization
These property_types are for visualization.
-
Color
-
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.
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HSV
A color described as a Hue-Saturation-Value triple. Is convertible to and from an RGB representation.
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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.