PowerPoint-presentatie

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The CORS method Selecting the roots of a system
of polynomial equations
with combinatorial optimization H. Bekker E.P.
Braad B. Goldengorin University of Groningen, The
Netherlands
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Two implicit curves f1(x,y) and f2(x,y)
Problem calculate the intersections of f1(x,y)
and f2(x,y)
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Problem calculate the intersections of f1(x,y)
and f2(x,y)
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• Presentation outline
• Conventional methods
• What can go wrong
• The CORS method
• Results
• More dimensions?

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• Conventional method
• Eliminate x from f1(x,y) and f2(x,y) giving P(y)
• Solve P(y) numerically, giving the roots y1..yn
• Sustitute yi in f1(x,y) and f2(x,y) and solve for
  x
• Select the common root xj from these two
  equations
• The pair (xj,yi) is a solution of f1(x,y) and
  f2(x,y)

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• What can go wrong
• f1(x,yi) or f2(x,yi) may be zero for every x

• Numerical errors there are no identical
  solutions of f1(x,yi)0 and f2(x,yi)0

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• Remarks
• The conventional approach is not symmetric first
  x is eliminated, then y (or the other way
  around)
• Numerical errors f1(xi,yj) 0, f2(xi,yj)
  0
• We are dealing with a matching problem match
  every yi with an optimal xj

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The CORS method
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• The CORS method
• Eliminate x from f1(x,y) and f2(x,y) giving P(y)
• Solve P(y) numerically, giving the roots y1..yn
• Eliminate y from f1(x,y) and f2(x,y) giving Q(x)
• Solve Q(x) numerically, giving the roots x1..xn
• Determine the error of every pair (xi,yj)
• Construct a weighted bipartite graph G with n
  x-nodes
• and n y-nodes. The weight of edge (xi,yj) is
• Calculate the minimum weight matching of G. This
  gives the
• desired n solutions.

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The CORS method
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The essence of CORS The CORS method selects n
pairs (x,y) from n2 pairs, so that every
x-value and every y-value is used only once and
the total error is minimized.
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• Alternative CORS
• Instead of minimizing the sum of the errors it is
  possible to minimize the maximum error.
• Sort all solutions, based on nn errors, in
  increasing order
• Replace in G the errors with 1, 2, 4, 8,
  16, 32, ..
• Calculate the minimum weight matching of G.
  This gives the solutions
  with min-max error.

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CORS was implemented in C CORS was tested on a
computational geometry problem CORS results were
compared with SYNAPS and MAPLE Results CORS
allways found all solutions, SYNAPS missed
2 CORS was 30 times faster than SYNAPS
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Can CORS be used for more than two equations and
variables? Yes, but then a multidimensional
matching algorithm is needed. A conventional
branch and bound algorithm may be used to solve
the multidimensional matching problem. Currently
we are experimenting with a tolerance bases 3D
weighted matching algorithm.
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• Outline for solving the 2D LAP problem using
  tolerances
• Solve the relaxed LAP
• Use tolerances to remove multiple selections in
  columns until all
• multiple selection have been removed

EXAMPLE RLAP
solution Modified RLAP solution
3 3 2 7 3 3 2
7 2 8 4 7 2
8 4 7 4 6 2 1
4 6 2 1 8 5 3 9
8 5 3 9 This method may be
generalized to more than two dimensions
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Conclusion
Using combinatorial optimization to select the
roots of a system of polynomial equations (CORS)
is an efficient and robust method
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Questions?
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