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#############################################################################
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## solveeq.gd CRISP Burkhard Höfling
##
## Copyright © 2000,2001 Burkhard Höfling
##
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##
#F LinearSystem(nrvars, nrsolutions, field, conv, convsol)
##
## LinearSystem returns a linear system of equations over the field <field>
## with <nrvars> variables. Each equation has <nrsolutions> right hand
## sides, which are treated in parallel.
##
## Initially, the linear system is empty. Equations can be added using
## AddEquation.
##
## conv and convsol are booleans. They determine whether the coefficients
## of the left hand side and the solutions on the right hand side will
## be stored as compressed vectors if possible.
##
DeclareGlobalFunction("LinearSystem");
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##
#F AddEquation(sys, row, sol)
##
## This function adds a new row to a system of linear equations <sys>
## obtained from LinearSystem. <row> is a vector containing the coefficients
## of the variables, <sol> is a vector containing the solutions.
##
## If <sys> has no solution, AddEquation simply returns fail.
##
## If <sys> has a solution, the new row is added to the system, and the
## system is triangulised. If the resulting system has a solution,
## AddEquation returns true, otherwise false.
##
## AddEquation may change sys, but does not change row or sol.
##
DeclareGlobalFunction("AddEquation");
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#F HasSolution(sys, n)
##
## This function returns true or false, depending whether sys has a solution
## or not, using the linear system whose right hand side consists of the
## <n>-th entries of the solutions added via AddEquation.
##
DeclareGlobalFunction("HasSolution");
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#F DimensionOfNullspace(sys)
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## This function returns the dimension of the nullspace of sys.
##
DeclareGlobalFunction("DimensionOfNullspace");
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#F OneSolution(sys, n)
##
## This function returns fail if sys has no solutions. Otherwise it returns
## a vector which is a solution for the linear system whose right hand side
## consists of the <n>-th entries of the solutions added via AddEquation.
##
DeclareGlobalFunction("OneSolution");
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#F BasisNullspaceSolution(sys)
##
## This function returns list of row vectors representing a basis of the
## vector space of all solutions of the homogeneous system corresponding
## to sys. Note that this also works if sys does not have a solution.
##
DeclareGlobalFunction("BasisNullspaceSolution");
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#E
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