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fricas
)abbrev package DOPT DiscreteOptimalTransport
++ Author: Kurt Pagani
++ Date Created: Thu Jun 21 01:42:00 CEST 2018
++ License: BSD
++ References:
++ Description:
++
DiscreteOptimalTransport(R) : Exports == Implementation where
R:Join(Ring,IntegralDomain,OrderedSet)
PI ==> PositiveInteger NNI ==> NonNegativeInteger INT ==> Integer LLI ==> List List Integer MATR ==> Matrix R VECR ==> Vector R
null ==> empty?
Exports == with
constraintMatrix : (PI,PI) -> MATR northWest : (PI,PI,VECR,VECR) -> MATR ++ NW rule getBasis : MATR -> List List List Integer projRows : (LLI,INT) -> List Integer ++ Given a basis B and an integer c, return a list of integers ++ [r1,..,rn], such that each (ri,c) is in B, i=1..n. projCols : (LLI,INT) -> List Integer ++ Given a basis B and an integer r, return a list of integers ++ [c1,..,cn], such that each (r,ci) is in B, i=1..n. recSolve : (PI,PI,LLI,MATR) -> List VECR computeReducedCosts : (PI,PI,List VECR,LLI,MATR)-> MATR determineAlpha : (LLI,MATR,MATR) -> Record(alpha:R,ijmax:List INT) checkHoriz : (LLI,MATR,LLI,INT,INT,INT,INT) -> LLI checkVert : (LLI,MATR,LLI,INT,INT,INT,INT) -> LLI findPath : (MATR,LLI,INT,INT) -> LLI findDelta : (LLI,MATR) -> Record(delta0:R,ij0:List INT) computeSolutionMatrix : (PI,PI,VECR,VECR,MATR) -> MATR modiMethod :(VECR,VECR,MATR) -> Record(Sol:MATR,cost:R) --coerce : % -> OutputForm
Implementation == add
constraintMatrix(m:PI,n:PI):MATR == M:MATR:=new(m,n,0$R)$MATR e:VECR:=new(n,1$R)$VECR LM:List MATR:=[setRow!(copy M,i,e) for i in 1..m] U:MATR:=horizConcat(LM)$MATR E:MATR:=diagonalMatrix(e) LD:List MATR:=[E for i in 1..m] D:=horizConcat(LD)$MATR vertConcat(U,D)
northWest(m:PI,n:PI,a:VECR,b:VECR):MATR == X:MATR:=new(m,n,0$R)$MATR aa:VECR:=new(n,0$R)$VECR bb:VECR:=new(m,0$R)$VECR u:INT:=1 ; v:INT:=1 while u<=m and v<=n repeat if b.v - aa.v < a.u - bb.u then z:=b.v-aa.v X(u,v):=z aa.v := aa.v + z bb.u := bb.u + z v := v+ 1 else z:=a.u-bb.u X(u,v):=z aa.v := aa.v + z bb.u := bb.u + z u := u+ 1 return X
getBasis(X:MATR):List List List Integer == b:List List Integer:=[] c:List List Integer:=[] for i in 1..nrows(X) repeat for j in 1..ncols(X) repeat if X(i,j) = 0$R then c:=concat(c,[[i,j]]) else b:=concat(b,[[i,j]]) return [b,c]
projRows(b:LLI,nc:INT):List Integer == [first(t) for t in b | t.2=nc]
projCols(b:LLI,nr:INT):List Integer == [second(t) for t in b | t.1=nr]
recSolve(m:PI,n:PI,b:LLI,C:MATR):List VECR == u:VECR:=new(m,0$R)$VECR v:VECR:=new(n,0$R)$VECR ud:List INT:=[i for i in 1..m] vd:List INT:=[i for i in 1..n] bn:List Integer:=projRows(b,n) for i in bn repeat u.i:=C(i,n) ud:=remove!(i,ud) for j in projCols(b,i) repeat v.j:=C(i,j)-u.i vd:=remove!(j,vd) repeat for k in ud repeat for j in projCols(b,k) repeat if not member?(j,vd) then u.k:=C(k,j)-v.j ud:=remove!(k,ud) for l in vd repeat for j in projRows(b,l) repeat if not member?(j,ud) then v.l:=C(j,l)-u.j vd:=remove!(l,vd) if null(ud) and null(vd) then return [u,v]
computeReducedCosts(m:PI,n:PI,rs:List VECR,c:LLI,C:MATR):MATR == u:VECR:=rs.1 v:VECR:=rs.2 RC:MATR:=new(m,n,0$R)$MATR for t in c repeat RC(t.1,t.2):=u.(t.1)+v.(t.2) return RC
determineAlpha(c:LLI,RC:MATR,C:MATR):Record(alpha:R,ijmax:List INT) == mm:R:=RC(c.1.1,c.1.2)-C(c.1.1,c.1.2) tmax:List Integer:=c.1 for t in c repeat x:R:=RC(t.1,t.2)-C(t.1,t.2) if x>mm then mm:=x tmax:=t return [mm,tmax]
checkHoriz(path:LLI,X:MATR,bb:LLI,u:INT,v:INT,u1:INT,v1:INT):LLI == n:=ncols X for i in 1..n repeat if i ~= v and member?([u,i],bb) then if i=v1 then path:=concat(path,[u,i]) return path p := checkVert(path,X,bb,u,i,u1,v1) if not null p then path:=concat(p,[u,i]) return path return []::LLI
checkVert(path:LLI,X:MATR,bb:LLI,u:INT,v:INT,u1:INT,v1:INT):LLI == m:=nrows X for i in 1..m repeat if i ~= u and member?([i,v],bb) then p := checkHoriz(path,X,bb,i,v,u1,v1) if not null p then path:=concat(p,[i,v]) return path return []::LLI
findPath(X:MATR,bb:LLI,u:INT,v:INT):LLI == p0:LLI:=[[u,v]] path:LLI:=checkHoriz(p0,X,bb,u,v,u,v) return path
findDelta(cy:LLI,X:MATR):Record(delta0:R,ij0:List INT) == ncy:LLI:=[cy.j for j in 1..#cy | even? j] --nX:List R:=[X(t.1,t.2) for t in ncy] tmin:List Integer:=ncy.1 m:R:=X(tmin.1,tmin.2) for t in ncy repeat x:R:=X(t.1,t.2) if x<m then m:=x tmin:=t return [m,tmin]
maxIter:INT:=10
computeSolutionMatrix(m:PI,n:PI,a:VECR,b:VECR,C:MATR):MATR == X:MATR:=northWest(m,n,a,b) bl:List LLI:=getBasis(X) bb:LLI:=bl.1 cc:LLI:=bl.2 it:INT:=0 -- repeat it:=it+1 if it > maxIter then break rs:List VECR:=recSolve(m,n,bb,C) RC:MATR:=computeReducedCosts(m,n,rs,cc,C) -- da:=determineAlpha(cc,RC,C) -- da.alpha, da.ijmax if da.alpha > 0$R then path:=findPath(X,bb,da.ijmax.1,da.ijmax.2) d0:=findDelta(path,X) -- d0.delta0, d0.ij0 for j in 1..#path repeat p:=path.j if odd?(j) then X(p.1,p.2):=X(p.1,p.2)+d0.delta0 else X(p.1,p.2):=X(p.1,p.2)-d0.delta0 bb:=remove(d0.ij0,bb) bb:=concat(bb,da.ijmax) cc:=remove(da.ijmax,cc) cc:=concat(cc,d0.ij0) else return X X
modiMethod(a:VECR,b:VECR,C:MATR):Record(Sol:MATR,cost:R) == m0:NNI:=nrows(C) n0:NNI:=ncols(C) if m0>0 then m:PI:=m0::PI else error "1" if n0>0 then n:PI:=n0::PI else error "2" if #a~=m then error "3" if #b~=n then error "4" if reduce(_+,a) ~= reduce(_+,b) then error "5" S:MATR:=computeSolutionMatrix(m,n,a,b,C) CTS:MATR:=transpose(C)*S c:R:=reduce(_+,[CTS(i,i) for i in 1..n]) -- trace reqs too much return [S,c]</spad>
fricas
Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4555332396781246409-25px001.spad
      using old system compiler.
   DOPT abbreviates package DiscreteOptimalTransport 
------------------------------------------------------------------------
   initializing NRLIB DOPT for DiscreteOptimalTransport 
   compiling into NRLIB DOPT 
****** Domain: R already in scope
****** Domain: R already in scope
   compiling exported constraintMatrix : (PositiveInteger,PositiveInteger) -> Matrix R
Time: 0.02 SEC.
compiling exported northWest : (PositiveInteger,PositiveInteger,Vector R,Vector R) -> Matrix R Time: 0 SEC.
compiling exported getBasis : Matrix R -> List List List Integer Time: 0.01 SEC.
compiling exported projRows : (List List Integer,Integer) -> List Integer Time: 0 SEC.
compiling exported projCols : (List List Integer,Integer) -> List Integer Time: 0 SEC.
compiling exported recSolve : (PositiveInteger,PositiveInteger,List List Integer,Matrix R) -> List Vector R Time: 0.01 SEC.
compiling exported computeReducedCosts : (PositiveInteger,PositiveInteger,List Vector R,List List Integer,Matrix R) -> Matrix R Time: 0.02 SEC.
compiling exported determineAlpha : (List List Integer,Matrix R,Matrix R) -> Record(alpha: R,ijmax: List Integer) Time: 0 SEC.
compiling exported checkHoriz : (List List Integer,Matrix R,List List Integer,Integer,Integer,Integer,Integer) -> List List Integer Time: 0.01 SEC.
compiling exported checkVert : (List List Integer,Matrix R,List List Integer,Integer,Integer,Integer,Integer) -> List List Integer Time: 0 SEC.
compiling exported findPath : (Matrix R,List List Integer,Integer,Integer) -> List List Integer Time: 0 SEC.
compiling exported findDelta : (List List Integer,Matrix R) -> Record(delta0: R,ij0: List Integer) Time: 0 SEC.
compiling exported computeSolutionMatrix : (PositiveInteger,PositiveInteger,Vector R,Vector R,Matrix R) -> Matrix R Time: 0.05 SEC.
compiling exported modiMethod : (Vector R,Vector R,Matrix R) -> Record(Sol: Matrix R,cost: R) Time: 0 SEC.
(time taken in buildFunctor: 0)
;;; *** |DiscreteOptimalTransport| REDEFINED
;;; *** |DiscreteOptimalTransport| REDEFINED Time: 0 SEC.
Warnings: [1] getBasis: b has no value [2] getBasis: c has no value [3] recSolve: ud has no value [4] recSolve: vd has no value [5] determineAlpha: mm has no value [6] determineAlpha: tmax has no value [7] findDelta: m has no value [8] findDelta: tmin has no value [9] computeSolutionMatrix: alpha has no value [10] computeSolutionMatrix: ijmax has no value [11] computeSolutionMatrix: delta0 has no value [12] computeSolutionMatrix: ij0 has no value
Cumulative Statistics for Constructor DiscreteOptimalTransport Time: 0.16 seconds
finalizing NRLIB DOPT Processing DiscreteOptimalTransport for Browser database: --------constructor--------- --->-->DiscreteOptimalTransport((constraintMatrix ((Matrix R) (PositiveInteger) (PositiveInteger)))): Not documented!!!! --------(northWest ((Matrix R) (PositiveInteger) (PositiveInteger) (Vector R) (Vector R)))--------- --->-->DiscreteOptimalTransport((getBasis ((List (List (List (Integer)))) (Matrix R)))): Not documented!!!! --------(projRows ((List (Integer)) (List (List (Integer))) (Integer)))--------- --->-->DiscreteOptimalTransport((projRows ((List (Integer)) (List (List (Integer))) (Integer)))): Improper first word in comments: Given "Given a basis \\spad{B} and an integer \\spad{c},{} return a list of integers [\\spad{r1},{}..,{}\\spad{rn}],{} such that each (\\spad{ri},{}\\spad{c}) is in \\spad{B},{} \\spad{i=1}..\\spad{n}." --------(projCols ((List (Integer)) (List (List (Integer))) (Integer)))--------- --->-->DiscreteOptimalTransport((projCols ((List (Integer)) (List (List (Integer))) (Integer)))): Improper first word in comments: Given "Given a basis \\spad{B} and an integer \\spad{r},{} return a list of integers [\\spad{c1},{}..,{}\\spad{cn}],{} such that each (\\spad{r},{}\\spad{ci}) is in \\spad{B},{} \\spad{i=1}..\\spad{n}." --->-->DiscreteOptimalTransport((recSolve ((List (Vector R)) (PositiveInteger) (PositiveInteger) (List (List (Integer))) (Matrix R)))): Not documented!!!! --->-->DiscreteOptimalTransport((computeReducedCosts ((Matrix R) (PositiveInteger) (PositiveInteger) (List (Vector R)) (List (List (Integer))) (Matrix R)))): Not documented!!!! --->-->DiscreteOptimalTransport((determineAlpha ((Record (: alpha R) (: ijmax (List (Integer)))) (List (List (Integer))) (Matrix R) (Matrix R)))): Not documented!!!! --->-->DiscreteOptimalTransport((checkHoriz ((List (List (Integer))) (List (List (Integer))) (Matrix R) (List (List (Integer))) (Integer) (Integer) (Integer) (Integer)))): Not documented!!!! --->-->DiscreteOptimalTransport((checkVert ((List (List (Integer))) (List (List (Integer))) (Matrix R) (List (List (Integer))) (Integer) (Integer) (Integer) (Integer)))): Not documented!!!! --->-->DiscreteOptimalTransport((findPath ((List (List (Integer))) (Matrix R) (List (List (Integer))) (Integer) (Integer)))): Not documented!!!! --->-->DiscreteOptimalTransport((findDelta ((Record (: delta0 R) (: ij0 (List (Integer)))) (List (List (Integer))) (Matrix R)))): Not documented!!!! --->-->DiscreteOptimalTransport((computeSolutionMatrix ((Matrix R) (PositiveInteger) (PositiveInteger) (Vector R) (Vector R) (Matrix R)))): Not documented!!!! --->-->DiscreteOptimalTransport((modiMethod ((Record (: Sol (Matrix R)) (: cost R)) (Vector R) (Vector R) (Matrix R)))): Not documented!!!! ; compiling file "/var/aw/var/LatexWiki/DOPT.NRLIB/DOPT.lsp" (written 05 DEC 2024 07:58:51 PM):
; wrote /var/aw/var/LatexWiki/DOPT.NRLIB/DOPT.fasl ; compilation finished in 0:00:00.108 ------------------------------------------------------------------------ DiscreteOptimalTransport is now explicitly exposed in frame initial DiscreteOptimalTransport will be automatically loaded when needed from /var/aw/var/LatexWiki/DOPT.NRLIB/DOPT

Test flavours

fricas
--)co dopt
supply, demand a:=vector [4,7,4]$Vector INT

\label{eq1}\left[ 4, \: 7, \: 4 \right](1)
Type: Vector(Integer)
fricas
b:=vector [2,4,6,3]$Vector INT

\label{eq2}\left[ 2, \: 4, \: 6, \: 3 \right](2)
Type: Vector(Integer)
fricas
-- cost matrix
C:= matrix [[2,3,4,1],[5,4,2,3],[4,2,8,6]]

\label{eq3}\left[ 
\begin{array}{cccc}
2 & 3 & 4 & 1 
\
5 & 4 & 2 & 3 
\
4 & 2 & 8 & 6 
(3)
Type: Matrix(Integer)
fricas
X:=computeSolutionMatrix(3,4,a,b,C)

\label{eq4}\left[ 
\begin{array}{cccc}
2 & 0 & 0 & 2 
\
0 & 0 & 6 & 1 
\
0 & 4 & 0 & 0 
(4)
Type: Matrix(Integer)
fricas
cost:=trace(transpose(C)*X) -- 29

\label{eq5}29(5)
Type: PositiveInteger?
fricas
--2
C2 := matrix [[ 2, 1, 3, 3, 2, 5],[ 3, 2, 2, 4, 3, 4], _ [ 3, 5, 4, 2, 4, 1],[ 4, 2, 2, 1, 2, 2]]

\label{eq6}\left[ 
\begin{array}{cccccc}
2 & 1 & 3 & 3 & 2 & 5 
\
3 & 2 & 2 & 4 & 3 & 4 
\
3 & 5 & 4 & 2 & 4 & 1 
\
4 & 2 & 2 & 1 & 2 & 2 
(6)
Type: Matrix(Integer)
fricas
b2:= vector [ 30, 50, 20, 40, 30, 11]

\label{eq7}\left[{30}, \:{50}, \:{20}, \:{40}, \:{30}, \:{11}\right](7)
Type: Vector(PositiveInteger?)
fricas
a2:= vector [ 50, 40, 60, 31]

\label{eq8}\left[{50}, \:{40}, \:{60}, \:{31}\right](8)
Type: Vector(PositiveInteger?)
fricas
X2:=computeSolutionMatrix(4,6,a2,b2,C2)

\label{eq9}\left[ 
\begin{array}{cccccc}
{20}&{30}& 0 & 0 & 0 & 0 
\
0 &{20}&{20}& 0 & 0 & 0 
\
{10}& 0 & 0 &{39}& 0 &{11}
\
0 & 0 & 0 & 1 &{30}& 0 
(9)
Type: Matrix(Integer)
fricas
cost2:=trace(transpose(C2)*X2) -- 330

\label{eq10}330(10)
Type: PositiveInteger?
fricas
-- 3
C3:= matrix [[ 1, 2, 1, 4, 5, 2],[ 3, 3, 2, 1, 4, 3], _ [ 4, 2, 5, 9, 6, 2],[ 3, 1, 7, 3, 4, 6]]

\label{eq11}\left[ 
\begin{array}{cccccc}
1 & 2 & 1 & 4 & 5 & 2 
\
3 & 3 & 2 & 1 & 4 & 3 
\
4 & 2 & 5 & 9 & 6 & 2 
\
3 & 1 & 7 & 3 & 4 & 6 
(11)
Type: Matrix(Integer)
fricas
b3:= vector [ 20, 40, 30, 10, 50, 25]

\label{eq12}\left[{20}, \:{40}, \:{30}, \:{10}, \:{50}, \:{25}\right](12)
Type: Vector(PositiveInteger?)
fricas
a3:= vector [ 30, 50, 75, 20]

\label{eq13}\left[{30}, \:{50}, \:{75}, \:{20}\right](13)
Type: Vector(PositiveInteger?)
fricas
X3:=computeSolutionMatrix(4,6,a3,b3,C3)

\label{eq14}\left[ 
\begin{array}{cccccc}
{20}& 0 &{10}& 0 & 0 & 0 
\
0 & 0 &{20}&{10}&{20}& 0 
\
0 &{40}& 0 & 0 &{10}&{25}
\
0 & 0 & 0 & 0 &{20}& 0 
(14)
Type: Matrix(Integer)
fricas
cost3:=trace(transpose(C3)*X3) -- 430

\label{eq15}430(15)
Type: PositiveInteger?
fricas
-- 4 ??? sum(a)~sum(b)
C4:= matrix [[ 5, 3, 6, 2],[ 4, 7, 9, 1],[ 3, 4, 7, 5]]

\label{eq16}\left[ 
\begin{array}{cccc}
5 & 3 & 6 & 2 
\
4 & 7 & 9 & 1 
\
3 & 4 & 7 & 5 
(16)
Type: Matrix(Integer)
fricas
b4 := vector [ 16, 18, 30, 25]

\label{eq17}\left[{16}, \:{18}, \:{30}, \:{25}\right](17)
Type: Vector(PositiveInteger?)
fricas
a4 := vector [ 19, 37, 34]

\label{eq18}\left[{19}, \:{37}, \:{34}\right](18)
Type: Vector(PositiveInteger?)
fricas
X4:=computeSolutionMatrix(3,4,a4,b4,C4)

\label{eq19}\left[ 
\begin{array}{cccc}
0 &{18}& 1 & 0 
\
{12}& 0 & 0 &{25}
\
4 & 0 &{29}& 0 
(19)
Type: Matrix(Integer)
fricas
cost4:=trace(transpose(C4)*X4) -- 348

\label{eq20}348(20)
Type: PositiveInteger?
fricas
-- 5 WP TT
C5:= matrix [[6,4,3],[3,2,2]]

\label{eq21}\left[ 
\begin{array}{ccc}
6 & 4 & 3 
\
3 & 2 & 2 
(21)
Type: Matrix(Integer)
fricas
a5:= vector [16,14]

\label{eq22}\left[{16}, \:{14}\right](22)
Type: Vector(PositiveInteger?)
fricas
b5:= vector [15,5,10]

\label{eq23}\left[{15}, \: 5, \:{10}\right](23)
Type: Vector(PositiveInteger?)
fricas
X5:=computeSolutionMatrix(2,3,a5,b5,C5)

\label{eq24}\left[ 
\begin{array}{ccc}
1 & 5 &{10}
\
{14}& 0 & 0 
(24)
Type: Matrix(Integer)
fricas
cost5:=trace(transpose(C5)*X5) -- 98

\label{eq25}98(25)
Type: PositiveInteger?
fricas
-- 6 WP ZykM
C6:= matrix [[7,4,3],[5,5,6]]

\label{eq26}\left[ 
\begin{array}{ccc}
7 & 4 & 3 
\
5 & 5 & 6 
(26)
Type: Matrix(Integer)
fricas
a6:= vector [12,8]

\label{eq27}\left[{12}, \: 8 \right](27)
Type: Vector(PositiveInteger?)
fricas
b6:= vector [4,10,6]

\label{eq28}\left[ 4, \:{10}, \: 6 \right](28)
Type: Vector(PositiveInteger?)
fricas
X6:=computeSolutionMatrix(2,3,a6,b6,C6)

\label{eq29}\left[ 
\begin{array}{ccc}
0 & 6 & 6 
\
4 & 4 & 0 
(29)
Type: Matrix(Integer)
fricas
cost6:=trace(transpose(C6)*X6) -- 82

\label{eq30}82(30)
Type: PositiveInteger?
fricas
--- 7
C7:= matrix [[140,130,170,150,140],[150,160,140,180,130],[160,120,180,170,190]]

\label{eq31}\left[ 
\begin{array}{ccccc}
{140}&{130}&{170}&{150}&{140}
\
{150}&{160}&{140}&{180}&{130}
\
{160}&{120}&{180}&{170}&{190}
(31)
Type: Matrix(Integer)
fricas
a7:= vector [900,800,400]

\label{eq32}\left[{900}, \:{800}, \:{400}\right](32)
Type: Vector(PositiveInteger?)
fricas
b7:= vector [300,500,400,600,300]

\label{eq33}\left[{300}, \:{500}, \:{400}, \:{600}, \:{300}\right](33)
Type: Vector(PositiveInteger?)
fricas
X7:=computeSolutionMatrix(3,5,a7,b7,C7)

\label{eq34}\left[ 
\begin{array}{ccccc}
{200}&{100}& 0 &{600}& 0 
\
{100}& 0 &{400}& 0 &{300}
\
0 &{400}& 0 & 0 & 0 
(34)
Type: Matrix(Integer)
fricas
cost7:=trace(transpose(C7)*X7) -- 289000

\label{eq35}289000(35)
Type: PositiveInteger?
fricas
--- 8
C8:= matrix [[3,2],[1,5],[5,4]]

\label{eq36}\left[ 
\begin{array}{cc}
3 & 2 
\
1 & 5 
\
5 & 4 
(36)
Type: Matrix(Integer)
fricas
a8:= vector [45,60,5]

\label{eq37}\left[{45}, \:{60}, \: 5 \right](37)
Type: Vector(PositiveInteger?)
fricas
b8:= vector [50,60]

\label{eq38}\left[{50}, \:{60}\right](38)
Type: Vector(PositiveInteger?)
fricas
X8:=computeSolutionMatrix(3,2,a8,b8,C8)

\label{eq39}\left[ 
\begin{array}{cc}
0 &{45}
\
{50}&{10}
\
0 & 5 
(39)
Type: Matrix(Integer)
fricas
cost8:=trace(transpose(C8)*X8)  -- 210

\label{eq40}210(40)
Type: PositiveInteger?
fricas
--- 9
C9:= matrix [[5,5,2,7],[8,3,1,8],[2,4,4,5]]

\label{eq41}\left[ 
\begin{array}{cccc}
5 & 5 & 2 & 7 
\
8 & 3 & 1 & 8 
\
2 & 4 & 4 & 5 
(41)
Type: Matrix(Integer)
fricas
a9:= vector [9,15,14]

\label{eq42}\left[ 9, \:{15}, \:{14}\right](42)
Type: Vector(PositiveInteger?)
fricas
b9:= vector [10,12,9,7]

\label{eq43}\left[{10}, \:{12}, \: 9, \: 7 \right](43)
Type: Vector(PositiveInteger?)
fricas
X9:=computeSolutionMatrix(3,4,a9,b9,C9)

\label{eq44}\left[ 
\begin{array}{cccc}
0 & 0 & 6 & 3 
\
0 &{12}& 3 & 0 
\
{10}& 0 & 0 & 4 
(44)
Type: Matrix(Integer)
fricas
cost9:=trace(transpose(C9)*X9)  -- 112

\label{eq45}112(45)
Type: PositiveInteger?
fricas
modiMethod(a9,b9,C9) -- check sum a = sum b

\label{eq46}\begin{array}{@{}l}
\displaystyle
\left[{\hbox{\axiomType{Sol}\ } ={\left[ 
\begin{array}{cccc}
0 & 0 & 6 & 3 
\
0 &{12}& 3 & 0 
\
{10}& 0 & 0 & 4 
(46)
Type: Record(Sol: Matrix(Integer),cost: Integer)

Note: "null" must have been removed recently ?? WA: null ==> empty?




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