-- -*- coding: utf-8 -*-
----------------------------------------------------
----------------------------------------------------
-- previous version: 0.2 30Jun08, submitted by Josephine Yu.
-- author: Mike Stillman --
-- core
-- author: Josephine Yu --
-- all remaining functions; documentation
-- Sonja Petrovic --
-- interface for windows; edited documentation; tests
--
-- latest major update: 6Jul08;
-- small revision in Documentation: 6oct08; SP.
----------------------------------------------------
----------------------------------------------------
newPackage(
"FourTiTwo",
Version => "1.0",
Date => "February 8, 2009",
Authors => {
{Name => "Mike Stillman", Email => "mike@math.cornell.edu"},
{Name => "Josephine Yu", Email => "jyu@math.mit.edu"},
{Name => "Sonja Petrovic", Email => "petrovic@psu.edu"}
},
Headline => "Interface to 4ti2",
Keywords => {"Interfaces"},
Configuration => { "path" => "",
"keep files" => true
},
DebuggingMode => false
)
export {
"toBinomial",
"getMatrix",
"putMatrix",
"toricMarkov",
"toricGroebner",
"toricCircuits",
"toricGraver",
"hilbertBasis",
"rays",
"InputType",
"toricGraverDegrees"
}
-- for backward compatibility
if not programPaths#?"4ti2" and FourTiTwo#Options#Configuration#"path" != ""
then programPaths#"4ti2" = FourTiTwo#Options#Configuration#"path"
fourTiTwo = null
run4ti2 = (exe, args) -> (
if fourTiTwo === null then
fourTiTwo = findProgram("4ti2", "markov -h",
Prefix => {(".*", "4ti2-"), -- debian
(".*", "4ti2_")}, -- suse
AdditionalPaths =>
{"/usr/lib/4ti2/bin", "/usr/lib64/4ti2/bin"}); -- fedora
runProgram(fourTiTwo, exe, args)
)
getFilename = () -> (
filename := temporaryFileName();
while fileExists(filename) or fileExists(filename|".mat") or fileExists(filename|".lat") do filename = temporaryFileName();
filename)
putMatrix = method()
putMatrix(File,Matrix) := (F,B) -> (
B = entries B;
numrows := #B;
numcols := #B#0;
F << numrows << " " << numcols << endl;
for i from 0 to numrows-1 do (
for j from 0 to numcols-1 do (
F << B#i#j << " ";
);
F << endl;
);
)
getMatrix = method()
getMatrix String := (filename) -> (
L := lines get filename;
l := toString first L;
v := value("{" | replace(" +",",",l)|"}");
dimensions := select(v, vi -> vi =!= null);
if dimensions#0 == 0 then ( -- matrix has no rows
matrix{{}}
) else(
L = drop(L,1);
--L = select(L, l-> regex("^[:space:]*$",l) === null);--remove blank lines
matrix select(
apply(L, v -> (w := value("{" | replace(" +",",",v)|"}"); w = select(w, wi -> wi =!= null))),
row -> row =!= {}
)
))
toBinomial = method()
toBinomial(Matrix,Ring) := (M,S) -> (
toBinom := (b) -> (
pos := 1_S;
neg := 1_S;
scan(#b, i -> if b_i > 0 then pos = pos*S_i^(b_i)
else if b_i < 0 then neg = neg*S_i^(-b_i));
pos - neg);
ideal apply(entries M, toBinom)
)
toricMarkov = method(Options=> {InputType => null})
toricMarkov Matrix := Matrix => o -> (A) -> (
filename := getFilename();
if debugLevel >= 1 then << "using temporary file name " << filename << endl;
if o.InputType === "lattice" then
F := openOut(filename|".lat")
else
F = openOut(filename|".mat");
putMatrix(F,A);
close F;
run4ti2("markov", rootPath | filename);
getMatrix(filename|".mar")
)
toricMarkov(Matrix,Ring) := o -> (A,S) -> toBinomial(toricMarkov(A,o), S)
toricGroebner = method(Options=>{Weights=>null})
toricGroebner Matrix := o -> (A) -> (
filename := getFilename();
if debugLevel >= 1 then << "using temporary file name " << filename << endl;
F := openOut(filename|".mat");
putMatrix(F,A);
close F;
if o.Weights =!= null then (
cost := concatenate apply(o.Weights, x -> (x|" "));
(filename|".cost") << "1 " << #o.Weights << endl << cost << endl << close;
);
run4ti2("groebner", rootPath | filename);
getMatrix(filename|".gro")
)
toricGroebner(Matrix,Ring) := o -> (A,S) -> toBinomial(toricGroebner(A,o), S)
toricCircuits = method()
toricCircuits Matrix := Matrix => (A ->(
filename := getFilename();
if debugLevel >= 1 then << "using temporary file name " << filename << endl;
F := openOut(filename|".mat");
putMatrix(F,A);
close F;
run4ti2("circuits", rootPath | filename);
getMatrix(filename|".cir")
))
toricGraver = method()
toricGraver Matrix := Matrix => (A ->(
filename := getFilename();
if debugLevel >= 1 then << "using temporary file name " << filename << endl;
F := openOut(filename|".mat");
putMatrix(F,A);
close F;
run4ti2("graver -q ", rootPath | filename);
getMatrix(filename|".gra")
))
toricGraver (Matrix,Ring) := Ideal => ((A,S)->toBinomial(toricGraver(A),S))
hilbertBasis = method(Options=> {InputType => null})
hilbertBasis Matrix := Matrix => o -> (A ->(
filename := getFilename();
if debugLevel >= 1 then << "using temporary file name " << filename << endl;
if o.InputType === "lattice" then
F := openOut(filename|".lat")
else
F = openOut(filename|".mat");
putMatrix(F,A);
close F;
run4ti2("hilbert", rootPath | filename);
getMatrix(filename|".hil")
))
rays = method()
rays Matrix := Matrix => (A ->(
filename := getFilename();
if debugLevel >= 1 then << "using temporary file name " << filename << endl;
F := openOut(filename|".mat");
putMatrix(F,A);
close F;
run4ti2("rays", rootPath | filename);
getMatrix(filename|".ray")
))
-- SP: the output command interface
-- I would like to have a command that gives the list of degrees of Graver/Groebner/Circuit/Markov file;
-- the way 4ti2 does this is you tell it the whatever.mar or whatever.cir file and it writes the degrees
-- to the screen.
-- On the other hand, it doesn't matter because you can ask M2 for those degrees directly!
toricGraverDegrees = method()
toricGraverDegrees Matrix := Matrix => (A ->(
filename := getFilename();
if debugLevel >= 1 then << "using temporary file name " << filename << endl;
F := openOut(filename|".mat");
putMatrix(F,A);
close F;
run4ti2("graver", rootPath | filename);
ret := run4ti2("output", "--degrees " | rootPath | filename|".gra");
print ret#"output"
))
beginDocumentation()
doc ///
Key
FourTiTwo
Headline
Interface for 4ti2
Description
Text
Interfaces most of the functionality of the software {\tt 4ti2} available at
@HREF"http://www.4ti2.de/"@.
(The user needs to have {\tt 4ti2} installed on his/her machine.)
A $d\times n$ integral matrix $A$ (with nonnegative entries) specifies a map from a polynomial
ring in d variables to a polynomial ring with n variables by specifying exponents of the variables indexing
its columns. For example, if $A$ is a matrix
$$\begin{pmatrix}
3&2&1&0\\
0&1&2&3
\end{pmatrix}$$
the map from $k[s,t]$ to $k[a,b,c,d]$ is given by
$(s,t) \mapsto \ (s^3,s^2t,st^2,t^3)$.
The toric ideal $I_A$ is the kernel of this map.
It is minimally generated by the 2-minors of the matrix
$$\begin{pmatrix}
x&y&z\\
y&z&w
\end{pmatrix}$$
Given the matrix $A$, one can compute its lattice basis ideal specified by the integral basis
of the lattice $A$, the toric ideal $I_A$, its Groebner bases, etc. In practice, however,
these are nontrivial computational tasks.
The software {\tt 4ti2} is very efficient in computing these objects.
For more theoretical details (and more generality), see the standard reference:
B. Sturmfels, {\bf Gr\"obner bases and convex polytopes.}
American Mathematical Society, University Lectures Series, No 8, Providence, Rhode Island, 1996.
{\bf Note for cygwin users:}
If a problem occurs during package installation and/or loading, it should be fixed
by setting the path inside the file {\tt .Macaulay2/init-FourTiTwo.m2} to whatever folder {\tt 4ti2} is installed.
For example, if {\tt 4ti2} has been installed in {\tt C:/cygwin/4ti2/win32}, then the line
inside the {\tt init-FourTiTwo.m2} file will look like this: {\tt "path" => "C:/cygwin/4ti2/win32/"} .
Alternately, the path for {\tt 4ti2} may be set when loading the package using the following command:
loadPackage("FourTiTwo", Configuration=>{"path"=>"C:/cygwin/4ti2/win32/"})
assuming that 4ti2 has been installed in C:/cygwin/4ti2/win32.
Caveat
If the package SimpleDoc is not found when installing {\tt FourTiTwo.m2},
see questions and answers 6, 7, and 8 on the Macaulay 2 web site.
///;
doc ///
Key
getMatrix
(getMatrix, String)
Headline
reads a matrix from a 4ti2-formatted input file
Usage
getMatrix s
Inputs
s:String
file name
Outputs
A:Matrix
Description
Text
The file should contain a matrix in the format such as
2 4\break
1 1 1 1\break
1 2 3 4\break
The first two numbers are the numbers of rows and columns.
SeeAlso
putMatrix
///;
doc ///
Key
putMatrix
(putMatrix,File,Matrix)
Headline
writes a matrix into a file formatted for 4ti2
Usage
putMatrix(F,A)
Inputs
F:File
A:Matrix
Description
Text
Write the matrix {\tt A} in file {\tt F} in {\tt 4ti2} format.
Example
A = matrix "1,1,1,1; 1,2,3,4"
s = temporaryFileName()
F = openOut(s)
putMatrix(F,A)
close(F)
getMatrix(s)
SeeAlso
getMatrix
///;
doc ///
Key
toBinomial
(toBinomial, Matrix, Ring)
Headline
creates a toric ideal from a given set of exponents of its generators
Usage
toBinomial(M,R)
Inputs
M: Matrix
R: Ring
ring with as least as many generators as the columns of {\tt M}
Outputs
I: Ideal
Description
Text
Equivalent to "output --binomials" in 4ti2.
Returns the ideal in the ring {\tt R} generated by the binomials corresponding to rows of {\tt M}.
Example
A = matrix "1,1,1,1; 1,2,3,4"
B = syz A
R = QQ[a..d]
toBinomial(transpose B,R)
///;
doc ///
Key
toricGroebner
(toricGroebner, Matrix)
(toricGroebner, Matrix, Ring)
[toricGroebner, Weights]
Headline
calculates a Groebner basis of the toric ideal I_A, given A; invokes "groebner" from 4ti2
Usage
toricGroebner(A) or toricGroebner(A,R)
Inputs
A:Matrix
whose columns parametrize the toric variety. The toric ideal $I_A$ is the kernel of the map defined by {\tt A}.
R:Ring
ring with as least as many generators as the columns of {\tt A}
Outputs
B:Matrix
whose rows give binomials that form a Groebner basis of the toric ideal of {\tt A}
I:Ideal
whose generators form a Groebner basis for the toric ideal
Description
Example
A = matrix "1,1,1,1; 1,2,3,4"
toricGroebner(A)
Text
Note that the output of the command is a matrix whose rows are the exponents of the binomials that for a Groebner basis of the
toric ideal $I_A$.
As a shortcut, one can ask for the output to be an ideal instead:
Example
R = QQ[a..d]
toricGroebner(A,R)
Text
{\tt 4ti2} offers the use of weight vectors representing term orders, as follows:
Example
toricGroebner(A,Weights=>{1,2,3,4})
Caveat
It seems that some versions of 4ti2 do not pick up on the weight vector. It may be better to run gb computation in M2 directly with specified weights.
///;
doc ///
Key
toricMarkov
(toricMarkov, Matrix)
(toricMarkov, Matrix, Ring)
[toricMarkov, InputType]
Headline
calculates a generating set of the toric ideal I_A, given A; invokes "markov" from 4ti2
Usage
toricMarkov(A) or toricMarkov(A, InputType => "lattice") or toricMarkov(A,R)
Inputs
A:Matrix
whose columns parametrize the toric variety; the toric ideal is the kernel of the map defined by {\tt A}.
Otherwise, if InputType is set to "lattice", the rows of {\tt A} are a lattice basis and the toric ideal is the
saturation of the lattice basis ideal.
InputType=>String
which is the string "lattice" if rows of {\tt A} specify a lattice basis
R:Ring
polynomial ring in which the toric ideal $I_A$ should live
Outputs
B:Matrix
whose rows form a Markov Basis of the lattice $\{z {\rm integral} : A z = 0\}$
or the lattice spanned by the rows of {\tt A} if the option {\tt InputType => "lattice"} is used
Description
Text
Suppose we would like to comput the toric ideal defining the variety parametrized by the following matrix:
Example
A = matrix"1,1,1,1;0,1,2,3"
Text
Since there are 4 columns, the ideal will live in the polynomial ring with 4 variables.
Example
R = QQ[a..d]
M = toricMarkov(A)
Text
Note that rows of M are the exponents of minimal generators of $I_A$. To get the ideal, we can do the following:
Example
I = toBinomial(M,R)
Text
Alternately, we might wish to give a lattice basis ideal instead of the matrix A. The lattice basis will be specified
by a matrix, as follows:
Example
B = syz A
N = toricMarkov(transpose B, InputType => "lattice")
J = toBinomial(N,R) -- toricMarkov(transpose B, R, InputType => "lattice")
Text
We can see that the two ideals are equal:
Example
I == J
Text
Also, notice that instead of the sequence of commands above, we could have used the following:
Example
toricMarkov(A,R)
///;
doc ///
Key
toricGraver
(toricGraver, Matrix)
(toricGraver, Matrix, Ring)
Headline
calculates the Graver basis of the toric ideal; invokes "graver" from 4ti2
Usage
toricGraver(A) or toricGraver(A,R)
Inputs
A:Matrix
whose columns parametrize the toric variety. The toric ideal $I_A$ is the kernel of the map defined by {\tt A}
R:Ring
polynomial ring in which the toric ideal $I_A$ should live
Outputs
B:Matrix
whose rows give binomials that form the Graver basis of the toric ideal of {\tt A}, or
I:Ideal
whose generators form the Graver basis for the toric ideal
Description
Text
The Graver basis for any toric ideal $I_A$ contains (properly) the union of all reduced Groebner basis of $I_A$.
Any element in the Graver basis of the ideal is called a primitive binomial.
Example
A = matrix "1,1,1,1; 1,2,3,4"
toricGraver(A)
Text
If we prefer to store the ideal instead, we may use:
Example
R = QQ[a..d]
toricGraver(A,R)
Text
Note that this last ideal equals the toric ideal $I_A$ since every Graver basis element is actually in $I_A$.
///;
doc ///
Key
toricGraverDegrees
(toricGraverDegrees, Matrix)
Headline
displays the degrees of all Graver basis elements for the toric ideal I_A
Usage
toricGraverDegrees(A)
Inputs
A:Matrix
whose columns parametrize the toric variety. The toric ideal $I_A$ is the kernel of the map defined by {\tt A}
Description
Text
Equivalent to "output --degrees foo.gra" in 4ti2.
Very often the Graver basis consists of too many binomials, and one is only interested in their degrees. In this case,
instead of looking at the Graver basis of $I_A$, we may just want to look for the degrees of binomials which show up:
Example
A = matrix "1,1,1,1; 1,2,3,4"
toricGraver(A) -- the Graver basis
toricGraverDegrees(A) -- just the degrees
Text
Note that these are all 1-norms of the vectors. Since $I_A$ is homogeneous, there are 3 binomials of degree 2 (norm 4)
and 2 binomials of degree 3 (norm 6).
Here is a more complicated example, where one may not want to see the Graver basis elements explicitly.
The toric ideal I_M is the ideal of the rational normal scroll S(3,2,3):
Example
M = matrix "1,1,1,1,1,1,1,1,1,1,1; 1,1,1,1,0,0,0,0,0,0,0; 0,0,0,0,1,1,1,0,0,0,0; 0,0,0,0,0,0,0,1,1,1,1; 1,2,3,4,1,2,3,1,2,3,4"
toricGraverDegrees(M)
Text
Here is another example where with many Graver basis elements. The following matrix is a design matrix for a particular statistical model for a 4-node p1 network; see Fienberg-Petrovic-Rinaldo.
Example
A = matrix "1,0,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0;0,1,1,0,0,0,0,0,0,1,0,1,1,0,1,0,0,0;0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,1,0,1;0,0,0,0,0,0,0,1,1,0,0,0,0,1,1,0,1,1;0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0;1,0,1,0,0,0,0,0,0,0,1,1,0,1,1,0,0,0;0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,0,1,1;0,0,0,0,0,0,1,0,1,0,0,0,1,0,1,1,0,1;0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1";
toricGraverDegrees(A)
///;
doc ///
Key
hilbertBasis
(hilbertBasis, Matrix)
[hilbertBasis, InputType]
Headline
calculates the Hilbert basis of the cone; invokes "hilbert" from 4ti2
Usage
hilbertBasis(A) or hilbertBasis(A, InputType => "lattice")
Inputs
A:Matrix
defining the cone $\{z : Az = 0, z \ge 0 \}$
Outputs
B:Matrix
whose rows form the Hilbert basis of the cone $\{z : Az = 0, z \ge 0 \}$
or the cone $\{z A : z {\rm {} is an integral vector and } z A \ge 0 \}$ if {\tt InputType => "lattice"} is used
Description
Example
A = matrix "1,1,1,1; 1,2,3,4"
B = syz A
hilbertBasis(transpose B)
hilbertBasis(A, InputType => "lattice")
///;
doc ///
Key
rays
(rays, Matrix)
Headline
calculates the extreme rays of the cone; invokes "rays" from 4ti2
Usage
rays(A)
Inputs
A:Matrix
defining the cone $\{z : Az = 0, z \ge 0 \}$
Outputs
B:Matrix
whose rows are the extreme rays of the cone $\{z : Az = 0, z \ge 0 \}$
Description
Example
A = matrix "1,1,1,1; 1,2,3,4"
B = syz A
rays(transpose B)
///;
doc ///
Key
toricCircuits
(toricCircuits, Matrix)
Headline
calculates the circuits of the toric ideal; invokes "circuits" from 4ti2
Usage
toricCircuits(A)
Inputs
A:Matrix
whose columns parametrize the toric variety. The toric ideal $I_A$ is the kernel of the map defined by {\tt A}
Outputs
B:Matrix
whose rows form the circuits of A
Description
Text
The circuits are contained in the Graver basis of $I_A$. In fact, they are precisely the primitive binomials in the ideal
with minimal support.
Example
A = matrix "1,1,1,1; 1,2,3,4"
C = toricCircuits A
Text
The ideal generated by the circuits of A in general differs from the toric ideal of A. For example:
Example
R = QQ[a..d]
Icircuit = toBinomial(toricCircuits(A), R) -- this is the circuit ideal of A
I = toBinomial(toricMarkov(A), R)
I==Icircuit
Text
The two ideals are not the same. There is a minimal generator of I which is not a circuit:
Example
a*d-b*c % I -- this binomial is in I:
a*d-b*c % Icircuit -- but not in Icircuit:
///;
doc ///
Key
InputType
Description
Text
Put {\tt InputType => "lattice"} as an argument in the functions toricMarkov and hilbertBasis
SeeAlso
toricMarkov
hilbertBasis
///;
TEST///
A = matrix "1,1,1,1; 1,2,3,4"
M = toricMarkov(A)
R = QQ[x_0,x_1,x_2,x_3]
I = toBinomial(M,R)
Irnc3 = ideal(x_0*x_2-x_1^2,x_1*x_3-x_2^2,x_0*x_3-x_1*x_2)
assert(I==Irnc3)
///
TEST ///
B = matrix "1,-2,1,0; 0,1,-2,1"
M = toricMarkov(B, InputType => "lattice")
R = QQ[x_0,x_1,x_2,x_3]
I = toBinomial(M,R)
Irnc3 = ideal(x_0*x_2-x_1^2,x_1*x_3-x_2^2,x_0*x_3-x_1*x_2)
assert(I== Irnc3)
///
TEST ///
R=CC[x_0,x_1,x_2,x_3]
A = matrix "1,1,1,1; 1,2,3,4"
C = toricCircuits(A) --circuits returned by 4ti2
Icir = toBinomial(C,R) -- circuit ideal returned by 4ti2
Ctrue = matrix{{0,1,-2,1},{1,-2,1,0},{1,0,-3,2},{2,-3,0,1}} --known: all circuits
IcirTrue = toBinomial(Ctrue,R) --known: circuit ideal
Irnc3 = ideal(x_0*x_2-x_1^2,x_1*x_3-x_2^2,x_0*x_3-x_1*x_2)
assert(Icir==IcirTrue)
shouldBeFalse = (Icir==Irnc3)
assert(shouldBeFalse==false)
assert(target C == target Ctrue)
assert(source C == source Ctrue)
///
TEST ///
B = matrix "1,-2,1,0; 0,1,-2,1"
R = QQ[a..d]
I = toBinomial(B,R)
assert(a*c-b^2 % I == 0)
assert(a*c-d^2 %I =!= 0)
assert(degree I == 4)
M = hilbertBasis B
assert(numrows M == 3)
assert(numcols M == 4)
M1 = rays B
assert(numrows M1 == 2)
assert(numcols M1 == 4)
///
TEST///
A = matrix "1,0,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0;0,1,1,0,0,0,0,0,0,1,0,1,1,0,1,0,0,0;0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,1,0,1;0,0,0,0,0,0,0,1,1,0,0,0,0,1,1,0,1,1;0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0;1,0,1,0,0,0,0,0,0,0,1,1,0,1,1,0,0,0;0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,0,1,1;0,0,0,0,0,0,1,0,1,0,0,0,1,0,1,1,0,1;0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1"
M = toricGroebner(A); --note this matrix is the design matrix for the p1 statistical model on 4 nodes using a constant rho. (see fienberg/rinaldo/petrovic; in prep-missing reference).
assert(numrows M == 137)
assert(numcols M == 18)
R = QQ[x_1..x_18]
I = toBinomial(M,R);
assert(degree I == 192)
A1 = matrix "3,2,1,0;0,1,2,3" --one more example
R=QQ[x_0..x_3]
G4ti2=gens toricGroebner(A1,R)
GM2 =gens gb toricMarkov(A1,R)
Gtrue=toList flatten entries GM2
G = toList flatten entries G4ti2
apply(0..#Gtrue-1, j-> (isSubset({Gtrue_j},G) )) --checking 4ti2's gb against M2's gb
assert(numrows GM2 == numrows G4ti2)
assert(numcols GM2 == numcols G4ti2)
Rwt=QQ[x_0..x_3,Weights=>{3,2,4,1}] --with wt vector
G4ti2=gens toBinomial(toricGroebner(A1,Weights=>{3,2,4,1}),Rwt)
GM2=gens gb toricMarkov(A1,Rwt)
Gtrue=toList flatten entries GM2
G = toList flatten entries G4ti2
assert( numrows GM2 == numrows G4ti2 )
assert( numcols GM2 == numcols G4ti2 )
apply(0..#Gtrue-1, j-> assert(isSubset({Gtrue_j},G)) ) --checking 4ti2's gb against M2's gb
///
TEST///
needsPackage "FourTiTwo" --testing graver
A1 = matrix "3,2,1,0;0,1,2,3"
R=QQ[x_0..x_3]
G = toricGraver(A1)
assert( numrows G==5)
assert(numcols G==4)
Gtrue = toBinomial(matrix{{1,-2,1,0},{0,1,-2,1},{1,-1,-1,1},{2,-3,0,1},{1,0,-3,2}},R) --known: Graver basis
Gtrue=toList flatten entries gens Gtrue
G = toList flatten entries gens toBinomial(G,R)
apply(0..#Gtrue-1, j-> assert(isSubset({Gtrue_j},G)) ) --testing 4ti2 output against by-hand calculation!
A = matrix "1,0,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0;0,1,1,0,0,0,0,0,0,1,0,1,1,0,1,0,0,0;0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,1,0,1;0,0,0,0,0,0,0,1,1,0,0,0,0,1,1,0,1,1;0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0,0,0;1,0,1,0,0,0,0,0,0,0,1,1,0,1,1,0,0,0;0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,0,1,1;0,0,0,0,0,0,1,0,1,0,0,0,1,0,1,1,0,1;0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1"
M = toricGraver(A); --note this matrix is the design matrix for the p1 statistical model on 4 nodes using a constant rho. (see fienberg/rinaldo/petrovic; in prep-missing reference).
assert(numrows M == 7462)
assert(numcols M == 18)
AS=matrix"1,1,1,1,1,1,1,1;1,1,1,0,0,0,0,0;1,2,3,1,2,3,4,5"--scroll S(2,4)
R=QQ[x_1..x_8]
G4ti2 = toList flatten entries gens toBinomial(toricGraver(AS),R)
assert(#G4ti2 == 82)
Gtrue=toList flatten entries gens toBinomial(matrix" 1,-2,1,0,0,0,0,0;1,-1,0,-1,1,0,0,0;2,-2,0,-1,0,1,0,0;3,-3,0,-1,0,0,1,0;4,-4,0,-1,0,0,0,1;3,-3,0,0,-1,0,0,1;2,-2,0,0,0,-1,0,1;1,-1,0,0,0,0,-1,1;2,-2,0,0,-1,0,1,0;1,-1,0,0,0,-1,1,0;1,-1,0,0,-1,1,0,0;0,1,-1,-1,1,0,0,0;0,1,-1,0,0,-1,1,0;0,1,-1,0,0,0,-1,1;0,1,-1,0,-1,1,0,0;1,0,-1,-1,0,1,0,0;1,0,-1,0,-1,0,1,0;1,0,-1,0,0,-1,0,1;2,-1,-1,-1,0,0,1,0;2,-1,-1,0,-1,0,0,1;3,-2,-1,-1,0,0,0,1;0,2,-2,0,-1,0,1,0;1,1,-2,-1,0,0,1,0;0,2,-2,-1,0,1,0,0;2,0,-2,-1,0,0,0,1;0,2,-2,0,0,-1,0,1;1,1,-2,0,-1,0,0,1;0,0,0,1,-1,-1,1,0;0,0,0,1,-1,0,-1,1;0,0,0,1,-2,1,0,0;1,-1,0,1,-2,0,1,0;1,-1,0,1,-1,-1,0,1;2,-2,0,1,-2,0,0,1;0,0,0,0,0,1,-2,1;0,0,0,0,1,-2,1,0;1,-1,0,-1,0,2,-1,0;0,0,0,0,1,-1,-1,1;1,-1,0,-1,0,1,1,-1;1,-1,0,0,-1,0,2,-1;2,-2,0,-1,0,0,2,-1;1,-1,0,0,1,-2,0,1;0,0,0,1,0,-2,0,1;0,1,-1,0,1,-2,0,1;0,1,-1,-1,0,1,1,-1;0,1,-1,-1,0,2,-1,0;1,0,-1,-1,0,0,2,-1;0,1,-1,0,-1,0,2,-1;0,1,-1,1,-1,-1,0,1;0,1,-1,1,-2,0,1,0;1,0,-1,1,-2,0,0,1;0,3,-3,-1,0,0,1,0;1,2,-3,-1,0,0,0,1;0,3,-3,0,-1,0,0,1;0,2,-2,1,-2,0,0,1;0,2,-2,-1,0,0,2,-1;0,4,-4,-1,0,0,0,1;0,0,0,1,-2,0,2,-1;0,0,0,2,-2,-1,0,1;0,0,0,2,-3,0,1,0;1,-1,0,2,-3,0,0,1;0,0,0,1,0,-3,2,0;0,0,0,1,0,-1,-2,2;1,-1,0,-1,0,0,3,-2;0,0,0,0,1,0,-3,2;0,0,0,0,2,-3,0,1;0,1,-1,-1,0,0,3,-2;0,1,-1,2,-3,0,0,1;0,0,0,3,-4,0,0,1;0,0,0,1,0,0,-4,3;1,0,-1,0,-1,1,-1,1;1,0,-1,0,-2,2,0,0;1,0,-1,-1,1,-1,1,0;1,0,-1,-1,1,0,-1,1;1,0,-1,-2,2,0,0,0;1,0,-1,0,0,-2,2,0;1,0,-1,0,0,0,-2,2;2,0,-2,0,-2,1,0,1;2,0,-2,-2,1,0,1,0;2,0,-2,-1,0,-1,2,0;2,0,-2,0,-1,0,-1,2;3,0,-3,-2,0,0,2,0;3,0,-3,0,-2,0,0,2" ,R)
apply(0..#Gtrue-1, j-> assert(isSubset({Gtrue_j},G4ti2)) ) -- checking 4ti2 output against by hand input!!
///
end
restart
--load "4ti2.m2"
installPackage ("FourTiTwo", RemakeAllDocumentation => true, UserMode=>true)
installPackage("FourTiTwo",UserMode=>true,DebuggingMode => true)
viewHelp FourTiTwo
check FourTiTwo
debug FourTiTwo
A = matrix{{1,1,1,1},{0,1,2,3}}
A = matrix{{1,1,1,1},{0,1,3,4}}
B = syz A
time toricMarkov A
A
toricMarkov(A, InputType => "lattice")
R = QQ[a..d]
time toricGroebner(A)
toBinomial(transpose B, R)
toricCircuits(A)
H = hilbertBasis(A)
hilbertBasis(transpose B)
toBinomial(H,QQ[x,y])
toricGraver(A)
A
toricMarkov(A)
7 9
A = matrix"
1,1,1,-1,-1,-1, 0, 0, 0;
1,1,1, 0, 0, 0,-1,-1,-1;
0,1,1,-1, 0, 0,-1, 0, 0;
1,0,1, 0,-1, 0, 0,-1, 0;
1,1,0, 0, 0,-1, 0, 0,-1;
0,1,1, 0,-1, 0, 0, 0,-1;
1,1,0, 0,-1, 0,-1, 0, 0"
transpose A
toricMarkov transpose A
hilbertBasis transpose A
toricGraver transpose A
toricCircuits transpose A
27 27
A = matrix"
1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0;
0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0;
0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0;
0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0;
0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0;
0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0;
0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0;
0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0;
0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1;
1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0;
0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0;
0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0;
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,0;
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0;
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1;
1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0;
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0;
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0,0,0,0;
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,0,0;
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1"
toricMarkov A
R = QQ[x_1..x_27]
toricMarkov(A,R)
toricGroebner(A,R)
gens gb oo
I = toBinomial(matrix{{}}, QQ[x])
gens I
gens gb I
-- Notes from Mike talking with Peter Malkin, 4/21/09
in 1.3.2 (and in 1.3.1):
These routines use the structure below:
groebner, markov,
hilbert, graver, zsolve
rays, circuits, qsolve
also: minimise, walk, normalform
a.mat: m by n
a.rel: 1 by m: symbols: >, =, < (means: >= 0, == 0, <= 0)
a.sign: 1 by n matrix: 0,1,-1,2
a.sign: 0: x_i unrestricted in sign
1: x_i >= 0
-1: x_i <= 0.
2: x_i is a Graver component
a.mat, a.rel, a.sign.
groebner, markov, zsolve: also can give a.rhs (1 x n matrix).
groebner does this:
Ax >= 0, x>=0.
Ax-Iy = 0, x >= 0, y >= 0.
doc on main page of 4ti2 web site: manual, and the slides.
.rhs doesn't work for qsolve though, possibly.
for hilbert, graver, zsolve, have the following filter:
.ub, .lb can be used to provide upper and lower bounds (lower only for Graver components, or for <= vars).
install glpk first, and make sure gmp is visible.
./configure --with-gmp=.... --with-glpk=....
call the different routines as: -p32, -p64, -parb
tests are in
4ti2-1.3.2/test
email Peter if I have more questions
4ti2 google group: joined.
glpk:open source linear programming