FUNCTIONS

USAGE : printMat(M); M matrix

ASSUME : M is a matrix.

THEORY : This is the print function used by Singular to print a matrix.

KEYWORDS: matrix

Example :

ring R=0,(x),lp;
matrix M[2][3]=1,243,3,4,522222,6;
printMat(M);

USAGE : printGraph(G); G graph

ASSUME : G is a graph.

THEORY : This is the print function used by Singular to print a graph.

KEYWORDS: graph

Example :

graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(2,4),list(3,4),list(1),list(2),list(3),list(4)));
G;

USAGE : printLabeledGraph(G); G labeledgraph

ASSUME : G is a labeled graph.

THEORY : This is the print function used by Singular to print a labeled graph.

KEYWORDS: Feynman graph

Example :

ring R=(0),q(1..6),dp;
labeledgraph G = makeLabeledGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(1,2),list(2,4),list(3,4),list(3,4)),R, list (q(1),q(2),q(3),q(4),q(5),q(6)),R);
G;

USAGE : printIBP(I); I oneIBP

ASSUME : I is an IBP identity computed using computeIBP.

THEORY : This is the print function used by Singular to print an IBP relation.

KEYWORDS: Feynman graph

Example :

USAGE : printIBP(I); I setIBP

ASSUME : I is the set of IBP identities computed using computeIBP.

THEORY : This is the print function used by Singular to print setIBP.

KEYWORDS: Feynman graph

Example :

USAGE : makeGraph(v,e); v list, e list

ASSUME : v is a list of integers, e is a list of two element lists of v.

RETURN : graph with vertices v and edges e

THEORY : Creates a graph from a list of vertices and edges. The vertices can be any type. The data structure respects the ordering of vertices of edges, so can be used for directed graphs, KEYWORDS: graph

Example :

graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(1,2),list(2,4),list(3,4),list(3,4)));
G;

USAGE : makeLabeledGraph(v,e,R,l,P); v list, e list, R ring, l list, P ring

ASSUME : v is a list of integers, e is a list of two element lists of pairwise different elements of v, R is a ring, l is a list of labels, P is a ring

RETURN : labeled graph with vertices v and edges e with labels of the edges in R with infinite edges being constants

THEORY :

KEYWORDS: Feynman graph

Example :

ring R=(0),q(1..6),dp;
labeledgraph G = makeLabeledGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(1,2),list(2,4),list(3,4),list(3,4)),R, list (q(1),q(2),q(3),q(4),q(5),q(6)),R);
G;

USAGE : labelGraph(G); G graph

ASSUME : G is a graph and ch is either zero or a prime.

RETURN : labeled graph with polynomial variables qi at the bounded edges and function field variables pi at the unbounded edges over a prime field of characteristic ch

THEORY :

KEYWORDS: Feynman graph

Example :

graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(2,4),list(3,4),list(1),list(2),list(3),list(4)));
labeledgraph lG = labelGraph(G,0);
lG;

USAGE : balancingIdeal(G); G labeledgraph

ASSUME : G is a labeled graph

RETURN : ideal of balancing condition of the graph, basering is assumed to be G.over

THEORY :

KEYWORDS: Feynman graph

Example :

graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(2,4),list(3,4),list(1),list(2),list(3),list(4)));
labeledgraph lG = labelGraph(G,0);
def R= lG.over;
setring R;
ideal I = balancingIdeal(lG);
I;

USAGE : eliminateVariables(G); G labeledgraph

ASSUME : G is a labeled graph

RETURN : labeled graph with variables of the bounded edges eliminated according to balancing condition and using an ordering $q[i]>p[j]$.

THEORY :

KEYWORDS: Feynman graph

Example :

graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(2,4),list(3,4),list(1),list(2),list(3),list(4)));
labeledgraph lG = labelGraph(G,0);
eliminateVariables(lG);

USAGE : removeVariable(R); R ring

ASSUME : R is a polynomial ring

RETURN : polynomial ring with j-th variable removed

THEORY :

KEYWORDS: ring

Example :

ring R=0,(x,y,z),(lp(2),dp(1));
def S= removeVariable(R,2);
S;

USAGE : removeParameter(R); R ring

ASSUME : R is a polynomial ring

RETURN : polynomial ring with j-th variable removed

THEORY :

KEYWORDS: ring

Example :

ring R=(0,p(1),p(2),p(3)),(x,y,z),(lp(2),dp(1));
def S= removeParameter(R,2);
S;

USAGE : substituteGraph(G); G labeledgraph

ASSUME : G is a labeled graph

RETURN : substitute the variable a in the labeling by b

THEORY :

KEYWORDS: Feynman graph

Example :

USAGE : feynmanDenominators(G); G labeledgraph

ASSUME : G is a labeled graph

RETURN : ideal containing the propagators in the Feynman integral

THEORY :

KEYWORDS: Feynman graph

Example :

graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(2,4),list(3,4),list(1),list(2),list(3),list(4)));
labeledgraph lG = labelGraph(G,0);
labeledgraph lGelim = eliminateVariables(lG);
def R = lGelim.over;
setring R;
ideal I = feynmanDenominators(lGelim);
I;

USAGE : propagators(G); G labeledgraph

ASSUME : G is a labeled graph

RETURN : ideal, containing the denominators in the Feynman integral

THEORY :

KEYWORDS: Feynman graph

Example :

graph G = makeGraph(list(1,2,3,4),list(list(1,3),list(1,2),list(2,4),list(3,4),list(1),list(2),list(3),list(4)));
labeledgraph lG = labelGraph(G,0);
labeledgraph lGelim = eliminateVariables(lG);
def R = lGelim.over;
setring R;
ideal I = propagators(lGelim);
I;

USAGE : ISP(G); G labeledgraph

ASSUME : G is a labeled graph

RETURN : ideal, containing the irreducible scalar products, that is, those scalar product which are not linearly dependent on the propagators.

THEORY :

KEYWORDS: Feynman graph

Example :

graph G = makeGraph(list(1,2,3,4,5,6),list(list(1,2),list(3,6),list(4,5),list(1,6),list(2,3),list(5,6),list(3,4),list(1),list(2),list(5),list(4)));
labeledgraph lG = labelGraph(G,0);
labeledgraph G1 = eliminateVariables(lG);
G1;
ring R= G1.over;
setring R;
R;
ISP(G1);

USAGE : removeElimVars(G); G labeledgraph

ASSUME : G is a labeled graph

RETURN : Removes the variables from G.elimvars. This key is generated by the procedure eliminateVariables.

THEORY :

KEYWORDS: Feynman graph

Example :

graph G = makeGraph(list(1,2,3,4,5,6),list(list(1,2),list(3,6),list(4,5),list(1,6),list(2,3),list(5,6),list(3,4),list(1),list(2),list(5),list(4)));
labeledgraph lG = labelGraph(G,0);
labeledgraph G1 = eliminateVariables(lG);
labeledgraph G2 = removeElimVars(G1);
G2;
ring R= G2.over;
setring R;
R;
G2;

USAGE : computeBaikovMatrix(G); G labeledgraph, or G graph

ASSUME : G is a Graph, or G is a labeled graph where redundant variables have been eliminated by the procedure eliminateVariables, and deleted from the ring by the procedure removeElimVars.

RETURN : a labeled graph G1, computes the Baikov matrix of G defined in G1.baikovover and stores it in G1.baikovmatrix

THEORY :

KEYWORDS: Feynman graph

Example :

graph G = makeGraph(list(1,2,3,4,5,6),list(list(6,1),list(4,6),list(1,2),list(3,5),list(4,3),list(2,5),list(5,6),list(1),list(2),list(3),list(4)));
labeledgraph G1=computeBaikovMatrix(G);
ring RB= G1.baikovover;
setring RB;
RB;
matrix B = G1.baikovmatrix;
printMat(B);

USAGE : computeM1(G0); G labeledgraph, or G graph

ASSUME : G is a Graph, or G is a labeled graph where redundant variables have been eliminated by the procedure eliminateVariables, and deleted from the ring by the procedure removeElimVars.

RETURN : The module M1 over G1.baikovover that requires to compute IBP identities

THEORY :

KEYWORDS: Feynman graph

Example :

graph G = makeGraph(list(1,2,3,4,5,6),list(list(6,1),list(4,6),list(1,2),list(3,5),list(4,3),list(2,5),list(5,6),list(1),list(2),list(3),list(4)));
labeledgraph G1=computeBaikovMatrix(G);
ring RB=G1.baikovover;
RB;
module ML=computeM1(G1);
ML;

USAGE : computeM2(G,Nu); G labeledgraph, or G graph

ASSUME : G is a Graph, or G is a labeled graph where redundant variables have been eliminated by the procedure eliminateVariables, and deleted from the ring by the procedure removeElimVars. Nu is the seed.

RETURN : The module M2 over G1.baikovover that requires to compute IBP identities

THEORY :

KEYWORDS: Feynman graph

Example :

graph G = makeGraph(list(1,2,3,4,5,6),list(list(6,1),list(4,6),list(1,2),list(3,5),list(4,3),list(2,5),list(5,6),list(1),list(2),list(3),list(4)));
labeledgraph G1=computeBaikovMatrix(G);
ring RB=G1.baikovover;
RB;
module M2=computeM2(G1,list(1,1,1,0,0,1,0,0,0));
M2;
module M2=computeM2(G1, list(1,1,1,1,1,1,1,-5,0));
M2;

USAGE : computeIBP(G0,Nu); G labeledgraph, or G graph

ASSUME : G is a Graph, or G is a labeled graph where redundant variables have been eliminated by the procedure eliminateVariables, and deleted from the ring by the procedure removeElimVars. Nu is the seed.

RETURN : The set of IBPS correspond to G0 and given Nu.

THEORY :

KEYWORDS: Feynman graph

Example :

graph G = makeGraph(list(1,2,3,4,5,6),list(list(6,1),list(4,6),list(1,2),list(3,5),list(4,3),list(2,5),list(5,6),list(1),list(2),list(3),list(4)));
labeledgraph G1=computeBaikovMatrix(G);
setIBP S=computeIBP(G1,list(1,1,0,1,0,1,0,1,0));
ring R=S.over;
setring R;
S;
oneIBP I=S.IBP[1];
I;

USAGE : getSector(l); l list

ASSUME : l is a list of integer indices of a Feynman integral

RETURN : list L, L[1]=s The sector (a list of 1s and 0s) that the corresponding integral belongs L[2]=n The sector in that the integral belongs

THEORY :

KEYWORDS: Feynman graph

Example :

list l=list(1,2,-3,-4,0,1);
list s=getSector(l);
s;

USAGE : listCombintions(L,r); L list, r int

ASSUME : r is a positive integer such that r < size(L)

RETURN : list of r-combinations of the elements in the list L

THEORY :

KEYWORDS: feynman graph

Example :

ring R=0,(x,y,z),dp;
list L=listCombinations(list(1,2,3,4),3);
L[1];

USAGE : generateWebSectors(seed);seed list

ASSUME : seed is a list of integer values.

RETURN : Web structure of the sectors L, where L is the list and L[1] is the sector that correspond to the given seed and L[i] contain the subsectors of the sectors in L[i-1]. Note that sector maps between the sectors have not been setted.

THEORY :

KEYWORDS: feynman graph

Example :

ring R=0,(x,y,z),dp;
list l=list(1,-1,0,1,2,-2);
list w=generateWebSectors(l);

USAGE : isSubList(l1,l2); l1 list, l2 list

ASSUME : l1 and l2 are list of positive integers

RETURN : 1 if elements in l1 contain in l2 0 if elements in l1 do not contain in l2

THEORY :

KEYWORDS: Feynman graph

Example :

ring R=0,(x,y,z),dp;
list l1=list(1,2,3,4,5,6,7);
list l2=list(1,4,6);
list l3=list(1,2,8);
list l4=list(1,4,6);
isSubList(l2,l1);
isSubList(l3,l1);
isSubList(l1,l2);
isSubList(l2,l4);

USAGE : getSectorMap(L1,L2); L1 list, L2 list, sector

ASSUME : L1 and L2 are lists of sectors where the lab field of each sector in both lists are filled(i.e. two layers of a sector web)

RETURN : L1 where sectorMap of each sector in the list L1 is filled.

THEORY :

KEYWORDS: sector,graph,feynman,setIBP

Example :

ring R=0,(x,y,z),dp;
list l=list(1,-1,0,1,2,-2);
list w=generateWebSectors(l);
list w1=getSectorMap(w[1],w[2]);
w1[1].sectorMap;
list w2=getSectorMap(w[2],w[3]);
w2[2].sectorMap;

USAGE : setSectorMap(sectorWeb); sectorWeb list, sector

ASSUME : sectorWeb is an output produced by the function @*generateWebSectors

RETURN : sectorWeb where the field sectorMap field of each sector in sectorWeb is filled.

THEORY :

KEYWORDS: sector, generateWebSectors, getSectorMap

Example :

ring R=0,(x,y,z),dp;
list l=list(1,-1,0,1,2,-2);
list w=generateWebSectors(l);
list w1=setSectorMap(w);

USAGE : findSector(sectorWeb,currentPosition,L); sectorWeb list,currentPosition list, L list,

ASSUME : sectorWeb is an output produced by the function generateWebSectors, L is an output produced by the function getSector@

RETURN : position of the sector in the sectorWeb, where the L belongs. -1, if the sector is not found

THEORY :

KEYWORDS: sector, generateWebSectors, getSectorMap

Example :

ring R=0,(x,y,z),dp;
list l=list(1,-1,0,1,2,-2);
list w=generateWebSectors(l);
list w1=setSectorMap(w);
list oneInt=list(4,-1,-1,0,-1,-2);
list L=getSector(oneInt);
def pos=findSector(w1,list(1,1),L[2]);
isSubList(w[pos[1]][pos[2]].lab,L[2])==1 && isSubList(L[2],w[pos[1]][pos[2]].lab); //this returns 1, since the given integral is in the sector at pos.

//example for a integral that is not in the set
list oneInt=list(4,1,0,-1,-2,3);
list L=getSector(oneInt);
def pos=findSector(w1,list(1,1),L[2]);
pos;

USAGE : updateOneSector(sectorWeb,currentPosition,oneInt); sectorWeb list, sector

ASSUME : sectorWeb is an output produced by the function generateWebSectors, oneInt is a list of indeces of the denominators associated to an integral correspond to a given feynman graph. Also assume the sectorweb isalso associated to the same feynman graph.

RETURN : updated sectorWeb, where the oneInt is assigned to the targetInts field of the seector correspond to provided oneInt

THEORY :

KEYWORDS: sector, generateWebSectors, getSectorMap,updateWeb,findSector

Example :

ring R=0,(x,y,z),dp;
list l=list(1,-1,0,1,2,-2);
list w=generateWebSectors(l);
list w1=setSectorMap(w);
list oneInt=list(4,-1,-1,0,-1,-2);
list w2=updateOneSector(w1,list(1,1),oneInt);
list L=getSector(oneInt);
L[2];
w2[3][2].lab;

USAGE : updateWeb(sectorWeb,currentPosition,setInt); sectorWeb list, sector

ASSUME : sectorWeb is an output produced by the function generateWebSectors, setInt is a list of indeces of the denominators associated to integrals correspond to a given feynman graph. Also assume the sectorweb is also associated to the same feynman graph.

RETURN : list (sectorWeb,MasterInt,notInWeb) where, sectorWeb is the updated web by assingning integrals to correspondng sectors, masterInt is the list integrals belong to the sector at currentPosition notInWeb is the list of integrals that are not belong the integral family associated the SectorWeb.

THEORY :

KEYWORDS: generateWebSectors, getSector,findSector

Example :

Example 1:

ring R=(0,(t,D)),(x,y,z),dp;
list l=list(1,2,1);
list w=generateWebSectors(l);
list w1=setSectorMap(w);
list setInt=list(list(1,2,3),list(-1,1,2),list(1,1,-1),list(-1,0,-2));
list setInt=list(list(1,2,3));
list setInt=list(l);
  
list L1=pickHighestSector(setInt);
list w2=updateWeb(w1,list(1,1),L1[1]);
w2[2]; //master integrals
w2[3];//integrals not in the web

Example 2:

graph G = makeGraph(list(1,2,3,4,5,6),list(list(6,1),list(4,6),list(1,2),list(3,5),list(4,3),list(2,5),list(5,6),list(1),list(2),list(3),list(4)));
labeledgraph G1=computeBaikovMatrix(G);
ring RZ= G1.baikovover;
printMat(G1.baikovmatrix);
  
list setInt=list(list(1,1,1,-1,-3,1,-1,-1,-1),list(1,-1,1,-1,-3,-1,-1,-4,-1));
list web=generateWebSectors(setInt[1]);
list w1=setSectorMap(web); 
web=w1;
list L1=pickHighestSector(setInt);  
  
list w2=updateWeb(web,list(1,1),L1[1]); //updateWeb returns a list w3 with w3[1]=sectorWeb,w3[2]=list of master Integrals, w3[3]=list of integrals that not belong to the current web
web=w2[1]; 
setIBP S=computeIBP(G1,L1[1][1]);
ring R=S.over;
setring R;
list L=getRedIBPs(S,101); //L[1]=list of independent IBPs,L[2]=list of master integrals
list independIBPs=L[1];
list masterAndTailIntgrals=L[2];
size(independIBPs) < size(S.IBP); //number of linearly independent set of IBPs are less than the number of orginal IBPs. So this returns true
  
oneIBP I1=independIBPs[18];     //Here is an example for one IBP i
I1;
list w3=updateWeb(web,list(1,1),masterAndTailIntgrals); //updateWeb returns a list w3 with w3[1]=sectorWeb,w3[2]=list of master Integrals, w3[3]=list of integrals that not belong to the current web
web=w3[1];   
size(web[1][1].targetInts);

USAGE : pickHighestSector(targetInt); G is a list of list of integers of same length

ASSUME : targetInt is the list of target integrals

RETURN : the intgral that belong to the heighest sector, if all integrals belong to the same sector web; otherwise, it returns a list of collection of integrals each need to be handled using different sector webs,

THEORY :

KEYWORDS: Feynman graph

Example :

USAGE : pickHighestSector(targetInt); G is a list of list of integers of same length

ASSUME : targetInt is the list of target integrals

RETURN : the intgral that belong to the heighest sector, if all integrals belong to the same sector web; otherwise, it returns a list of collection of integrals each need to be handled using different sector webs,

THEORY :

KEYWORDS: Feynman graph

Example :

setInt=list(list(-1,1,2),list(1,1,-1),list(-1,0,-2),list(1,2,3)); //here we can do the reduction using one web
list L=pickHighestSector(setInt);
size(L);

list setInt=list(list(-1,1,2),list(1,1,-1),list(-1,0,-2)); //here we need more than one web
list L=pickHighestSector(setInt);
size(L);

USAGE : getSortMeasures(l); l list,

ASSUME : l is a list of integers (i.e a seed).

RETURN : list of sort measures that are used in Laporta Algorithm

THEORY :

KEYWORDS: Feynman graph

Example :

setInt=list(list(-1,1,2),list(1,1,-1),list(-1,0,-2),list(1,2,3)); 
getSortMeasures(l);

USAGE : extractCoef(I,ind,l); I oneIBP,ind list,l list,

ASSUME : ind is the output of getSortedIntegrals, and l is the list of values over the base field I.baikovover and size(l)=npars(I.baikovover)

RETURN : list of values where, the i-th element is the evaluation of coefficient function at values in the list l of the IBP relation oneIBP, whose index is i=ind[i][1].

THEORY :

KEYWORDS: feynman graph,IBPs

Example :

graph G = makeGraph(list(1,2,3,4,5,6),list(list(6,1),list(4,6),list(1,2),list(3,5),list(4,3),list(2,5),list(5,6),list(1),list(2),list(3),list(4)));
labeledgraph G1=computeBaikovMatrix(G);
setIBP S=computeIBP(G1,list(1,1,0,1,0,1,0,1,0));
ring R=S.over;
setring R;
list ind = getSortedIntegrals(S);
oneIBP I=S.IBP[1];
I;
list rowCorrespondToI=extractCoef(I,ind,list(1,2,9)); 
  
// the nonzero coefficient of the IBP relation correspond to integral I(1,1,0,1,0,0,0,1,0).
// when we use lex ordering to order the used integrals in set of IBPs, this integral correspond to the 82th place.
// so we get only a nonzero value at position 82 and the below, the output will be -14.

rowCorrespondToI[82]; //output will be -14

USAGE : setMat(S,val); S setIBP,val list

ASSUME : size(val)=npars(I.baikovover) and val list of integers and ind is the output of getSortedIntegrals(S)

RETURN : atrix,where i-th row correspond to the evaluation of coefficient functions of i-th IBP in setIBP. Columns of the matrix correspond to the all used indices in the setIBP which are ordered with respect to the output ofgetSortMeasures.

THEORY :

KEYWORDS: feynman graph,IBPs

Example :

graph G = makeGraph(list(1,2,3,4,5,6),list(list(6,1),list(4,6),list(1,2),list(3,5),list(4,3),list(2,5),list(5,6),list(1),list(2),list(3),list(4)));
labeledgraph G1=computeBaikovMatrix(G);
setIBP S=computeIBP(G1,list(1,1,0,1,0,1,0,1,0));
ring R=S.over;
setring R;
list ind = getSortedIntegrals(S);
matrix N=setMat(S,list(1,2,3),ind);

USAGE : getRedIBPs(S,p);

ASSUME : S is setIBP, and p is a prime number.

RETURN : list L, L[1]=indIBP, L[2]=seed where, indIBP contain the linearly independent IBP relations of setIBP which are obtained by finite field row reduction over the field Fp. seed contain the indeces correspond to the non-free columns in rref.

THEORY :

KEYWORDS: feynman graph,IBPs

Example :

graph G = makeGraph(list(1,2,3,4,5,6),list(list(6,1),list(4,6),list(1,2),list(3,5),list(4,3),list(2,5),list(5,6),list(1),list(2),list(3),list(4)));
labeledgraph G1=computeBaikovMatrix(G);
setIBP S=computeIBP(G1,list(1,1,0,1,0,1,0,1,0));
ring R=S.over;
setring R;
list L=getRedIBPs(S,101);
size(L[1])<size(S.IBP);

USAGE : getSortedIntegrals(I); I setIBP,

ASSUME :

RETURN : list ind where each entry is a pair (indv,sortmeasures), indv is the list of indices(seed) appered in the setIBP and sortmeasures is the output of getSortMeasures(indv). The function getSortedIntegrals extract the seeds appeared in the IBP identities of the setIBP, sort them lexicographically based on the values got from getSortMeasures and return the output.

THEORY :

KEYWORDS:

Example :

graph G = makeGraph(list(1,2,3,4,5,6),list(list(6,1),list(4,6),list(1,2),list(3,5),list(4,3),list(2,5),list(5,6),list(1),list(2),list(3),list(4)));
labeledgraph G1=computeBaikovMatrix(G);
setIBP S=computeIBP(G1,list(1,1,0,1,0,1,0,1,0));
ring R=S.over;
setring R;
list L=getSortedIntegrals(S); //L list of pair of sorted integrals and the corresponding sorting measures
L[1];

USAGE : computeManyIBP(G0,setNu); G0 graph,

ASSUME : setNu is a list of seeds correspond to the graph G0 which are belong to the same sector

RETURN : setIBP S, where it contains all the IBP relations obtained by module intersection and seeding

THEORY :

KEYWORDS:

Example :

graph G = makeGraph(list(1,2,3,4,5,6),list(list(6,1),list(4,6),list(1,2),list(3,5),list(4,3),list(2,5),list(5,6),list(1),list(2),list(3),list(4)));
labeledgraph G1=computeBaikovMatrix(G);

//here we compute set of IBPs correspond to two seeds seperately
setIBP IBP1=computeIBP(G1,list(1,1,0,1,0,1,0,-1,0));
setIBP IBP2=computeIBP(G1,list(1,1,0,1,0,1,0,-3,0));
size(IBP1.IBP);
size(IBP2.IBP);
  
//here we compute set of IBPs correspond both seeds simultaneously 
//We can use this only when both integrals belongs to the same sector

setIBP S=computeManyIBP(G,list(list(1,1,0,1,0,1,0,-1,0),list(1,1,0,1,0,1,0,-3,0)));
size(S.IBP); 

USAGE : getReducedIBPSystem(G,targetInt); targetInt list,G graph,

ASSUME : G is a graph and targetInt is a list of seeds of target integrals.

RETURN : ist (reducedIBPs,MI) where reducedIBPs::setIBP, MI::list. reducedIBPs contain the reduced IBP system for the target integrals MI contain the master integrals

THEORY :

KEYWORDS: Feynman graph,IBPs

Example :

graph G = makeGraph(list(1,2,3,4,5,6),list(list(6,1),list(4,6),list(1,2),list(3,5),list(4,3),list(2,5),list(5,6),list(1),list(2),list(3),list(4)));
list targetInt=list(list(1,1,1,-1,-3,-1,-1,-1,-1),list(1,-1,1,-1,-3,-1,-1,-4,-1));
list finalset=getReducedIBPSystem(G,targetInt);
setIBP S=finalset[1];
ring R=S.over;
setring R;
oneIBP I=S.IBP[5];
I;
size(finalset[2]);