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spad
)abbrev package LINPROG LinearProgramming
++ Author: Kurt Pagani
++ Date Created: Sun Jun 03 02:05:55 CEST 2018
++ License: BSD
++ References:
++ Description:
++ TODO: description, more in test_linprog, eps, tol    
++ IDEA: linSolve standard method, try LU from jet
++ LINPROG  
++
LinearProgramming() : Exports == Implementation where
R ==> Float PI ==> PositiveInteger NNI ==> NonNegativeInteger MATR ==> Matrix R VECR ==> Vector R LI ==> List Integer RESR ==> Record(x:VECR,f:R,itn:NNI) RESRS ==> Record(X:VECR,objVal:R,Basis:LI) RLUP ==> Record(L:MATR,U:MATR,P:MATR) RXP ==> Record(X:VECR,P:MATR)
Exports == with
revisedSimplex : (MATR,VECR,VECR,LI) -> RESRS revisedSimplexLU : (MATR,VECR,VECR,LI) -> RESRS luDecompose : MATR -> RLUP luSolve : (MATR,VECR) -> RXP
Implementation == MATR add
eps:R:=0.00001 zeroTol:R:= 0.00001
revisedSimplex(A:MATR,b:VECR,c:VECR,B:LI):RESRS == numRows:PI := #b ::PI numVars:PI := #c ::PI --zeroTol:R:= 0.00001 N:LI := [i for i in 1..numVars | not member?(i,B)] --N:LI:=setDifference([i for i in 1..numVars],B) RA:LI:=[i for i in 1..nrows(A)] AB:MATR:=A(RA,B) AN:MATR:=A(RA,N) numNonbasicVars:=#N cN := vector [c.i for i in N] cB := vector [c.i for i in B] IAB:Union(MATR,"failed"):=inverse(AB) if IAB case MATR then bBAR:VECR := IAB*b else error "AB not invertible" objVal := dot(cB,bBAR) negRedCost := -1.0 while negRedCost < 0.0 repeat IAB:Union(MATR,"failed"):=inverse(AB) if IAB case MATR then wN:VECR := cN - (cB*IAB)*AN else error "AB not invertible" -- negRedCostIdx:NNI := 0 negRedCost := 0.0 -- for k in 1..numNonbasicVars repeat if wN.k < -zeroTol then negRedCost := wN.k --record the index negRedCostIdx := k break -- if negRedCost < 0.0 then IAB:Union(MATR,"failed"):=inverse(AB) if IAB case MATR then aBAR:VECR := IAB*column(AN,negRedCostIdx) else error "AB not invertible" minRatioVal := 1.0/zeroTol --inf minRatioIdx := 0 -- for k in 1..numRows repeat if aBAR.k > zeroTol then if bBAR.k / aBAR.k < minRatioVal then minRatioVal := bBAR.k / aBAR.k --record the index minRatioIdx:NNI := k -- bBAR := bBAR - minRatioVal * aBAR bBAR.minRatioIdx := minRatioVal -- tmpIdx := B.minRatioIdx B.minRatioIdx := N.negRedCostIdx N.negRedCostIdx := tmpIdx -- AB:MATR:=A(RA,B) AN:MATR:=A(RA,N) cN := vector [c.i for i in N] cB := vector [c.i for i in B] X:VECR:=zero(numVars) for i in 1..numRows repeat X.(B.i) := bBAR.i objVal := dot(cB,bBAR) return [X,objVal,B]
luDecompose(A:MATR):RLUP == not square?(A)$MATR => error "Input matrix is not square." (numRows, numVars) := (nrows A, ncols A) U:MATR:=copy A LK:MATR:=zero(numRows,numRows)$MATR I:MATR:=diagonalMatrix(vector [1 for i in 1..numRows]) P:MATR:=copy I LINV:=copy I RA:LI:=[i for i in 1..numRows] for i in 1..numVars repeat perm:LI:=[m for m in 1..numRows] -- don't use RA or i as index!! pivotEl := 0$R for j in i..numRows repeat if abs(U(j,i)) > eps then tmp := perm.i perm.i := perm.j perm.j := tmp U:=I(perm,RA)*U P:=I(perm,RA)*P pivotEl := U(i, i) break abs(pivotEl) <= eps => error("Matrix is singular") LK := I(perm,RA) * LK LK(i,i) := 1$R LINVK := copy I -- for k in i+1..numRows repeat if U(k, i) ~= 0$R then multiplier := U(k, i)/pivotEl U(k,RA) := U(k,RA) - multiplier * U(i,RA) LK(k,i) := multiplier LINVK(k, i) := - multiplier -- LINV := LINVK*LINV return [LK,U,P]
luSolve(A:MATR,b:VECR):RXP == not square?(A)$MATR => error "Input matrix is not square." (numRows, numVars) := (nrows A, ncols A) r:RLUP:=luDecompose(A) U:=copy r.U L:=copy r.L P:=copy r.P Y:=vector [0$R for m in 1..numRows] X:=vector [0$R for m in 1..numRows] rhs:VECR := P*b for i in 1..numVars repeat Y.i := rhs.i rhs := rhs - Y.i * column(L,i) for i in numVars..1 by -1 repeat X.i := Y.i / U(i,i) Y := Y - X.i * column(U,i) return [X,P]
revisedSimplexLU(A:MATR,b:VECR,c:VECR,B:LI):RESRS == numRows:PI := #b ::PI numVars:PI := #c ::PI --zeroTol:R:= 0.00001 N:LI := [i for i in 1..numVars | not member?(i,B)] --N:LI:=setDifference([i for i in 1..numVars],B) RA:LI:=[i for i in 1..nrows(A)] AB:MATR:=A(RA,B) AN:MATR:=A(RA,N) numNonbasicVars:=#N cN := vector [c.i for i in N] cB := vector [c.i for i in B] -- bBAR:VECR := luSolve(AB,b).X ---- luSolve objVal := dot(cB,bBAR) negRedCost := -1.0 while negRedCost < 0.0 repeat u:VECR:= luSolve(transpose(AB),cB).X wN:VECR := cN - u*AN -- negRedCostIdx:NNI := 0 negRedCost := 0.0 -- for k in 1..numNonbasicVars repeat if wN.k < -zeroTol then negRedCost := wN.k --record the index negRedCostIdx := k break -- if negRedCost < 0.0 then
aBAR:VECR := luSolve(AB,column(AN,negRedCostIdx)).X minRatioVal := 1.0/zeroTol --inf minRatioIdx := 0 -- for k in 1..numRows repeat if aBAR.k > zeroTol then if bBAR.k / aBAR.k < minRatioVal then minRatioVal := bBAR.k / aBAR.k --record the index minRatioIdx:NNI := k -- bBAR := bBAR - minRatioVal * aBAR bBAR.minRatioIdx := minRatioVal -- tmpIdx := B.minRatioIdx B.minRatioIdx := N.negRedCostIdx N.negRedCostIdx := tmpIdx -- AB:MATR:=A(RA,B) AN:MATR:=A(RA,N) cN := vector [c.i for i in N] cB := vector [c.i for i in B] X:VECR:=zero(numVars) for i in 1..numRows repeat X.(B.i) := bBAR.i objVal := dot(cB,bBAR) return [X,objVal,B]
spad
   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/5717432969159631359-25px001.spad
      using old system compiler.
   LINPROG abbreviates package LinearProgramming 
------------------------------------------------------------------------
   initializing NRLIB LINPROG for LinearProgramming 
   compiling into NRLIB LINPROG 
   compiling exported revisedSimplex : (Matrix Float,Vector Float,Vector Float,List Integer) -> Record(X: Vector Float,objVal: Float,Basis: List Integer)
Local variable minRatioIdx type redefined: (NonNegativeInteger) to (Float)
Time: 0.11 SEC.
compiling exported luDecompose : Matrix Float -> Record(L: Matrix Float,U: Matrix Float,P: Matrix Float) Time: 0.04 SEC.
compiling exported luSolve : (Matrix Float,Vector Float) -> Record(X: Vector Float,P: Matrix Float) Time: 0.02 SEC.
compiling exported revisedSimplexLU : (Matrix Float,Vector Float,Vector Float,List Integer) -> Record(X: Vector Float,objVal: Float,Basis: List Integer) Local variable minRatioIdx type redefined: (NonNegativeInteger) to (Float) Time: 0.03 SEC.
(time taken in buildFunctor: 0)
;;; *** |LinearProgramming| REDEFINED
;;; *** |LinearProgramming| REDEFINED Time: 0.01 SEC.
Warnings: [1] revisedSimplex: negRedCostIdx has no value [2] revisedSimplex: minRatioIdx has no value [3] revisedSimplex: bBAR has no value [4] revisedSimplex: X has no value [5] luDecompose: U has no value [6] luDecompose: LK has no value [7] luDecompose: P has no value [8] luSolve: U has no value [9] luSolve: L has no value [10] luSolve: P has no value [11] revisedSimplexLU: negRedCostIdx has no value [12] revisedSimplexLU: minRatioIdx has no value [13] revisedSimplexLU: bBAR has no value [14] revisedSimplexLU: X has no value
Cumulative Statistics for Constructor LinearProgramming Time: 0.21 seconds
--------------non extending category---------------------- .. LinearProgramming of cat (CATEGORY |package| (SIGNATURE |revisedSimplex| ((|Record| (|:| X (|Vector| (|Float|))) (|:| |objVal| (|Float|)) (|:| |Basis| (|List| (|Integer|)))) (|Matrix| (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|)) (|List| (|Integer|)))) (SIGNATURE |revisedSimplexLU| ((|Record| (|:| X (|Vector| (|Float|))) (|:| |objVal| (|Float|)) (|:| |Basis| (|List| (|Integer|)))) (|Matrix| (|Float|)) (|Vector| (|Float|)) (|Vector| (|Float|)) (|List| (|Integer|)))) (SIGNATURE |luDecompose| ((|Record| (|:| L (|Matrix| (|Float|))) (|:| U (|Matrix| (|Float|))) (|:| P (|Matrix| (|Float|)))) (|Matrix| (|Float|)))) (SIGNATURE |luSolve| ((|Record| (|:| X (|Vector| (|Float|))) (|:| P (|Matrix| (|Float|)))) (|Matrix| (|Float|)) (|Vector| (|Float|))))) has no (|MatrixCategory| (|Float|) (|Vector| (|Float|)) (|Vector| (|Float|))) finalizing NRLIB LINPROG Processing LinearProgramming for Browser database: --------constructor--------- --->-->LinearProgramming((revisedSimplex ((Record (: X (Vector (Float))) (: objVal (Float)) (: Basis (List (Integer)))) (Matrix (Float)) (Vector (Float)) (Vector (Float)) (List (Integer))))): Not documented!!!! --->-->LinearProgramming((revisedSimplexLU ((Record (: X (Vector (Float))) (: objVal (Float)) (: Basis (List (Integer)))) (Matrix (Float)) (Vector (Float)) (Vector (Float)) (List (Integer))))): Not documented!!!! --->-->LinearProgramming((luDecompose ((Record (: L (Matrix (Float))) (: U (Matrix (Float))) (: P (Matrix (Float)))) (Matrix (Float))))): Not documented!!!! --->-->LinearProgramming((luSolve ((Record (: X (Vector (Float))) (: P (Matrix (Float)))) (Matrix (Float)) (Vector (Float))))): Not documented!!!! ; compiling file "/var/aw/var/LatexWiki/LINPROG.NRLIB/LINPROG.lsp" (written 16 JUL 2018 07:35:26 PM):
; /var/aw/var/LatexWiki/LINPROG.NRLIB/LINPROG.fasl written ; compilation finished in 0:00:00.285 ------------------------------------------------------------------------ LinearProgramming is now explicitly exposed in frame initial LinearProgramming will be automatically loaded when needed from /var/aw/var/LatexWiki/LINPROG.NRLIB/LINPROG

Test flavours

fricas
-- Unittest .....: Package LinearProgramming
Author .......: Kurt Pagani -- Date .........: Sun Jun 17 02:14:10 CEST 2018
fricas
)set break resume
 
fricas
)expose UnittestCount UnittestAux Unittest
UnittestCount is now explicitly exposed in frame initial UnittestAux is now explicitly exposed in frame initial Unittest is now explicitly exposed in frame initial
--)co linprog
fricas
)expose LINPROG
LinearProgramming is already explicitly exposed in frame initial
-- Test Suite testsuite "Package: LinearProgramming"
All user variables and function definitions have been cleared. WARNING: string for testsuite should have less than 15 characters!
Type: Void
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-- Cases
testcase "revisedSimplex"
All user variables and function definitions have been cleared.
Type: Void
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-- testEquals("", "")
A:=matrix [[0.7,1.0,1.0,0.0,0.0,0.0],[0.5,(5/6)::Float,0.0,1.0,0.0,0.0], [1.0,(2/3)::Float,0.0,0.0,1.0,0.0],[0.1,0.25,0.0,0.0,0.0,1.0]]

\label{eq1}\left[ 
\begin{array}{cccccc}
{0.7}&{1.0}&{1.0}&{0.0}&{0.0}&{0.0}
\
{0.5}&{0.8333333333 \<u> 3333333333}&{0.0}&{1.0}&{0.0}&{0.0}
\
{1.0}&{0.6666666666 \</u> 6666666667}&{0.0}&{0.0}&{1.0}&{0.0}
\
{0.1}&{0.25}&{0.0}&{0.0}&{0.0}&{1.0}
(1)
Type: Matrix(Float)
fricas
c:=vector [-10.0,-9.0,0.0,0.0,0.0,0.0]

\label{eq2}\left[ -{10.0}, \: -{9.0}, \:{0.0}, \:{0.0}, \:{0.0}, \:{0.0}\right](2)
Type: Vector(Float)
fricas
b:=vector [630.0,600.0,708.0,135.0]

\label{eq3}\left[{630.0}, \:{600.0}, \:{708.0}, \:{135.0}\right](3)
Type: Vector(Float)
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B:=[3,4,5,6]

\label{eq4}\left[ 3, \: 4, \: 5, \: 6 \right](4)
Type: List(PositiveInteger?)
fricas
X1:Vector Float:= vector [540.0,252.0,0.0,120.0,0.0,18.0]

\label{eq5}\left[{540.0}, \:{252.0}, \:{0.0}, \:{120.0}, \:{0.0}, \:{18.0}\right](5)
Type: Vector(Float)
fricas
objVal1:Float:= - 7668.0

\label{eq6}-{7668.0}(6)
Type: Float
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Basis1:List Integer:= [2,4,1,6]

\label{eq7}\left[ 2, \: 4, \: 1, \: 6 \right](7)
Type: List(Integer)
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eps:=0.0000000001

\label{eq8}0.1 E - 9(8)
Type: Float
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floatNull(v:Vector Float):Boolean == sqrt dot(v,v) <= eps 
Function declaration floatNull : Vector(Float) -> Boolean has been added to workspace.
Type: Void
fricas
r1:=revisedSimplex( A, b, c, B)

\label{eq9}\begin{array}{@{}l}
\displaystyle
\left[{X ={\left[{540.0}, \:{252.0}, \:{0.0}, \:{120.0}, \:{0.0}, \:{18.0}\right]}}, \:{objVal = -{7668.0}}, \: \right.
\
\
\displaystyle
\left.{\hbox{\axiomType{Basis}\ } ={\left[ 2, \: 4, \: 1, \: 6 \right]}}\right] 
(9)
Type: Record(X: Vector(Float),objVal: Float,Basis: List(Integer))
fricas
testEquals("floatNull(r1.X-X1)", "true")
fricas
Compiling function floatNull with type Vector(Float) -> Boolean
Type: Void
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testEquals("abs(r1.objVal-objVal1)<eps", "true")
Type: Void
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testEquals("r1.Basis", "Basis1")
Type: Void
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r1lu:=revisedSimplexLU( A, b, c, B)

\label{eq10}\begin{array}{@{}l}
\displaystyle
\left[{X ={\left[{540.0}, \:{252.0}, \:{0.0}, \:{120.0}, \:{0.0}, \:{18.0}\right]}}, \:{objVal = -{7668.0}}, \: \right.
\
\
\displaystyle
\left.{\hbox{\axiomType{Basis}\ } ={\left[ 2, \: 4, \: 1, \: 6 \right]}}\right] 
(10)
Type: Record(X: Vector(Float),objVal: Float,Basis: List(Integer))
fricas
testEquals("floatNull(r1lu.X-X1)", "true")
Type: Void
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testEquals("abs(r1lu.objVal-objVal1)<eps", "true")
Type: Void
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testEquals("r1lu.Basis", "Basis1")
Type: Void
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--
--- A2 := matrix [[0,1,-1,0,0,0], [1,1,0,-1,0,0], [-0.5,1,0,0,1,0],[-1,1,0,0,0,1]]

\label{eq11}\left[ 
\begin{array}{cccccc}
{0.0}&{1.0}& -{1.0}&{0.0}&{0.0}&{0.0}
\
{1.0}&{1.0}&{0.0}& -{1.0}&{0.0}&{0.0}
\
-{0.5}&{1.0}&{0.0}&{0.0}&{1.0}&{0.0}
\
-{1.0}&{1.0}&{0.0}&{0.0}&{0.0}&{1.0}
(11)
Type: Matrix(Float)
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c2 := vector [2,-3,0,0,0,0]

\label{eq12}\left[ 2, \: - 3, \: 0, \: 0, \: 0, \: 0 \right](12)
Type: Vector(Integer)
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b2 := vector [1,2,8,6]

\label{eq13}\left[ 1, \: 2, \: 8, \: 6 \right](13)
Type: Vector(PositiveInteger?)
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B2 := [1,2,5,6]

\label{eq14}\left[ 1, \: 2, \: 5, \: 6 \right](14)
Type: List(PositiveInteger?)
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X2:= [4.0,10.0,9.0,12.0,0.0,0.0]

\label{eq15}\left[{4.0}, \:{10.0}, \:{9.0}, \:{12.0}, \:{0.0}, \:{0.0}\right](15)
Type: List(Float)
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objVal2:= - 22.0

\label{eq16}-{22.0}(16)
Type: Float
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Basis2:= [3,2,1,4]

\label{eq17}\left[ 3, \: 2, \: 1, \: 4 \right](17)
Type: List(PositiveInteger?)
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r2:=revisedSimplex( A2, b2, c2, B2)

\label{eq18}\begin{array}{@{}l}
\displaystyle
\left[{X ={\left[{4.0}, \:{10.0}, \:{9.0}, \:{12.0}, \:{0.0}, \:{0.0}\right]}}, \:{objVal = -{22.0}}, \: \right.
\
\
\displaystyle
\left.{\hbox{\axiomType{Basis}\ } ={\left[ 3, \: 2, \: 1, \: 4 \right]}}\right] 
(18)
Type: Record(X: Vector(Float),objVal: Float,Basis: List(Integer))
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testEquals("floatNull(r2.X-X2)", "true")
Type: Void
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testEquals("abs(r2.objVal-objVal2)<eps", "true")
Type: Void
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testEquals("r2.Basis", "Basis2")
Type: Void
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r2lu:=revisedSimplexLU( A2, b2, c2, B2)

\label{eq19}\begin{array}{@{}l}
\displaystyle
\left[{X ={\left[{4.0}, \:{10.0}, \:{9.0}, \:{12.0}, \:{0.0}, \:{0.0}\right]}}, \:{objVal = -{22.0}}, \: \right.
\
\
\displaystyle
\left.{\hbox{\axiomType{Basis}\ } ={\left[ 3, \: 2, \: 1, \: 4 \right]}}\right] 
(19)
Type: Record(X: Vector(Float),objVal: Float,Basis: List(Integer))
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testEquals("floatNull(r2lu.X-X2)", "true")
Type: Void
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testEquals("abs(r2lu.objVal-objVal2)<eps", "true")
Type: Void
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testEquals("r2lu.Basis", "Basis2")
Type: Void
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--
A3 := matrix [[0.4,0.5,1,0,0],[0,0.2,0,1,0],[0.6,0.3,0,0,1]]

\label{eq20}\left[ 
\begin{array}{ccccc}
{0.4}&{0.5}&{1.0}&{0.0}&{0.0}
\
{0.0}&{0.2}&{0.0}&{1.0}&{0.0}
\
{0.6}&{0.3}&{0.0}&{0.0}&{1.0}
(20)
Type: Matrix(Float)
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c3 := vector [-40,-30,0,0,0]

\label{eq21}\left[ -{40}, \: -{30}, \: 0, \: 0, \: 0 \right](21)
Type: Vector(Integer)
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b3 := vector [20,5,21]

\label{eq22}\left[{20}, \: 5, \:{21}\right](22)
Type: Vector(PositiveInteger?)
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B3 := [3,4,5]

\label{eq23}\left[ 3, \: 4, \: 5 \right](23)
Type: List(PositiveInteger?)
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X3:= [25.0,20.0,0.0,1.0,0.0]

\label{eq24}\left[{25.0}, \:{20.0}, \:{0.0}, \:{1.0}, \:{0.0}\right](24)
Type: List(Float)
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objVal3:= - 1600.0

\label{eq25}-{1600.0}(25)
Type: Float
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Basis3:= [2,4,1]

\label{eq26}\left[ 2, \: 4, \: 1 \right](26)
Type: List(PositiveInteger?)
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r3:=revisedSimplex(A3, b3, c3, B3)

\label{eq27}\begin{array}{@{}l}
\displaystyle
\left[{X ={\left[{25.0}, \:{20.0}, \:{0.0}, \:{1.0}, \:{0.0}\right]}}, \:{objVal = -{1600.0}}, \: \right.
\
\
\displaystyle
\left.{\hbox{\axiomType{Basis}\ } ={\left[ 2, \: 4, \: 1 \right]}}\right] (27)
Type: Record(X: Vector(Float),objVal: Float,Basis: List(Integer))
fricas
testEquals("floatNull(r3.X-X3)", "true")
Type: Void
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testEquals("abs(r3.objVal-objVal3)<eps", "true")
Type: Void
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testEquals("r3.Basis", "Basis3")
Type: Void
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r3lu:=revisedSimplexLU(A3, b3, c3, B3)

\label{eq28}\begin{array}{@{}l}
\displaystyle
\left[{X ={\left[{25.0}, \:{20.0}, \:{0.0}, \:{0.9999999999 \_ 9999999997}, \:{0.0}\right]}}, \: \right.
\
\
\displaystyle
\left.{objVal = -{1600.0}}, \:{\hbox{\axiomType{Basis}\ } ={\left[ 2, \: 4, \: 1 \right]}}\right] 
(28)
Type: Record(X: Vector(Float),objVal: Float,Basis: List(Integer))
fricas
testEquals("floatNull(r3lu.X-X3)", "true")
Type: Void
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testEquals("abs(r3lu.objVal-objVal3)<eps", "true")
Type: Void
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testEquals("r3lu.Basis", "Basis3")
Type: Void
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--
A4 := matrix [[1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0],_
              [0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],_
              [0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],_
              [0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0],_
              [3, 0, 6, 0, 5, 0, 7, 0, 1, 0, 0, 0, 0, 0, 1, 0],_
              [0, 2, 0, 4, 0, 10, 0, 4, 0, 1, 0, 0, 0, 0, 0, 1]]

\label{eq29}\left[ 
\begin{array}{cccccccccccccccc}
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \
3 & 0 & 6 & 0 & 5 & 0 & 7 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \
0 & 2 & 0 & 4 & 0 &{10}& 0 & 4 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 
(29)
Type: Matrix(NonNegativeInteger?)
fricas
b4 := vector [1, 1, 1, 1, 13, 10]

\label{eq30}\left[ 1, \: 1, \: 1, \: 1, \:{13}, \:{10}\right](30)
Type: Vector(PositiveInteger?)
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c4 := vector [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1]

\label{eq31}\left[ 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 1, \: 1, \: 1, \: 1, \: 1, \: 1 \right](31)
Type: Vector(NonNegativeInteger?)
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B4 := [11, 12, 13, 14, 15, 16]

\label{eq32}\left[{11}, \:{12}, \:{13}, \:{14}, \:{15}, \:{16}\right](32)
Type: List(PositiveInteger?)
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X4:= [1.0, 0.0, 0.25, 0.7500000000_0000000001, 0.3, 0.7, 1.0, 0.0, 0.0, 0.0,
      0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

\label{eq33}\begin{array}{@{}l}
\displaystyle
\left[{1.0}, \:{0.0}, \:{0.25}, \:{0.7500000000 \_ 0000000001}, \:{0.3}, \:{0.7}, \:{1.0}, \:{0.0}, \:{0.0}, \right.
\
\
\displaystyle
\left.\:{0.0}, \:{0.0}, \:{0.0}, \:{0.0}, \:{0.0}, \:{0.0}, \:{0.0}\right] (33)
Type: List(Float)
fricas
objVal4:= 0.0

\label{eq34}0.0(34)
Type: Float
fricas
Basis4:= [1,3,6,5,7,4]

\label{eq35}\left[ 1, \: 3, \: 6, \: 5, \: 7, \: 4 \right](35)
Type: List(PositiveInteger?)
fricas
r4 := revisedSimplex( A4, b4, c4, B4)

\label{eq36}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
X ={
\begin{array}{@{}l}
\displaystyle
\left[{1.0}, \:{0.0}, \:{0.25}, \:{0.7500000000 \_ 0000000001}, \:{0.3}, \:{0.7}, \:{1.0}, \: \right.
\
\
\displaystyle
\left.{0.0}, \:{0.0}, \:{0.0}, \:{0.0}, \:{0.0}, \:{0.0}, \:{0.0}, \:{0.0}, \:{0.0}\right] 
(36)
Type: Record(X: Vector(Float),objVal: Float,Basis: List(Integer))
fricas
testEquals("floatNull(r4.X-X4)", "true")
Type: Void
fricas
testEquals("abs(r4.objVal-objVal4)<eps", "true")
Type: Void
fricas
testEquals("r4.Basis", "Basis4")
Type: Void
fricas
r4lu := revisedSimplexLU( A4, b4, c4, B4)

\label{eq37}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
X ={
\begin{array}{@{}l}
\displaystyle
\left[{1.0}, \:{0.0}, \:{0.25}, \:{0.75}, \:{0.3}, \:{0.7}, \:{1.0}, \:{0.0}, \:{0.0}, \:{0.0}, \:{0.0}, \:{0.0}, \:{0.0}, \right.
\
\
\displaystyle
\left.\:{0.0}, \:{0.0}, \:{0.0}\right] 
(37)
Type: Record(X: Vector(Float),objVal: Float,Basis: List(Integer))
fricas
testEquals("floatNull(r4lu.X-X4)", "true")
Type: Void
fricas
testEquals("abs(r4lu.objVal-objVal4)<eps", "true")
Type: Void
fricas
testEquals("r4lu.Basis", "Basis4")
Type: Void
fricas
--
A5 := matrix [[1, -1, 1, 0, 0], [4, 9, 0, 1, 0], [-2, 4, 0, 0, 1]]

\label{eq38}\left[ 
\begin{array}{ccccc}
1 & - 1 & 1 & 0 & 0 
\
4 & 9 & 0 & 1 & 0 
\
- 2 & 4 & 0 & 0 & 1 
(38)
Type: Matrix(Integer)
fricas
c5 := vector [-1, -1, 0, 0, 0]

\label{eq39}\left[ - 1, \: - 1, \: 0, \: 0, \: 0 \right](39)
Type: Vector(Integer)
fricas
b5 := vector [2, 18, 4]

\label{eq40}\left[ 2, \:{18}, \: 4 \right](40)
Type: Vector(PositiveInteger?)
fricas
B5 := vector [3, 4, 5]

\label{eq41}\left[ 3, \: 4, \: 5 \right](41)
Type: Vector(PositiveInteger?)
fricas
X5 := [2.7692307692_307692308, 0.7692307692_3076923077, 0.0, 0.0,
       6.4615384615_384615385]

\label{eq42}\begin{array}{@{}l}
\displaystyle
\left[{2.7692307692 \<u> 307692308}, \:{0.7692307692 \</u> 3076923
077}, \:{0.0}, \: \right.
\
\
\displaystyle
\left.{0.0}, \:{6.4615384615 \_ 384615385}\right] 
(42)
Type: List(Float)
fricas
objVal5:= - 3.5384615384_615384615

\label{eq43}-{3.5384615384 \<u> 615384615}(43)
Type: Float
fricas
Basis5:= [1,2,5]

\label{eq44}\left[ 1, \: 2, \: 5 \right](44)
Type: List(PositiveInteger?)
fricas
r5 := revisedSimplex( A5, b5, c5, B5)

\label{eq45}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
X ={
\begin{array}{@{}l}
\displaystyle
\left[{2.7692307692 \<u> 307692308}, \:{0.7692307692 \</u> 3076923
077}, \: \right.
\
\
\displaystyle
\left.{0.0}, \:{0.0}, \:{6.4615384615 \_ 384615385}\right] 
(45)
Type: Record(X: Vector(Float),objVal: Float,Basis: List(Integer))
fricas
testEquals("floatNull(r5.X-X5)", "true")
Type: Void
fricas
testEquals("abs(r5.objVal-objVal5)<eps", "true")
Type: Void
fricas
testEquals("r5.Basis", "Basis5")
Type: Void
fricas
r5lu := revisedSimplexLU( A5, b5, c5, B5)

\label{eq46}\begin{array}{@{}l}
\displaystyle
\left[{
\begin{array}{@{}l}
\displaystyle
X ={
\begin{array}{@{}l}
\displaystyle
\left[{2.7692307692 \<u> 307692308}, \:{0.7692307692 \</u> 3076923
077}, \: \right.
\
\
\displaystyle
\left.{0.0}, \:{0.0}, \:{6.4615384615 \_ 384615385}\right] 
(46)
Type: Record(X: Vector(Float),objVal: Float,Basis: List(Integer))
fricas
testEquals("floatNull(r5lu.X-X5)", "true")
Type: Void
fricas
testEquals("abs(r5lu.objVal-objVal5)<eps", "true")
Type: Void
fricas
testEquals("r5lu.Basis", "Basis5")
Type: Void
fricas
-- Results/Statistics
fricas
)set output algebra on
 
fricas
)version
Value = "FriCAS 1.3.4 compiled at Wed Jun 27 14:53:01 UTC 2018"
fricas
)lisp (lisp-implementation-type)
Value = "SBCL"
fricas
)lisp (lisp-implementation-version)
Value = "1.1.1"
statistics()
============================================================================= General WARNINGS: * do not use ')clear completely' before having used 'statistics()' It clears the statistics without warning! * do not forget to pass the arguments of the testXxxx functions as Strings! Otherwise, the test will fail and statistics() will not notice! * testLibraryError does not prevent FriCAS from aborting the current block. Thus, if a block contains other test functions, they will not be executed and statistics() will not notice!
============================================================================= Testsuite: Package: LinearProgramming failed (total): 0 (1)
============================================================================= testsuite | testcases: failed (total) | tests: failed (total) Package: LinearProgramming 0 (1) 0 (30) ============================================================================= File summary. unexpected failures: 0 expected failures: 0 unexpected passes: 0 total tests: 30
Type: Void




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