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Solving Systems of Polynomial Equations
Results, Visualizations, and Conclusions
Eric Lee
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Existing Methods
• Groebner Bases (Analytic Solutions)
• Newton’s Method (Approximate Solutions)
• Homotopy Methods (Approximate Solutions)
• Optimization (Approximate Solutions)
• All of these are flawed in some way or another!
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Newton’s Method and Fractals
• Given a Polynomial System 𝑷(𝒙) and its Jacobian 𝑱(𝒙)
• 𝒛𝒏+𝟏 = 𝒛𝒏 − 𝑱−𝟏 𝒛𝒏 𝑷(𝒛𝒏)• 𝒛𝟎 is your initial guess
• Given an arbitrary initial guess, to which root does it converge?
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𝑧3 − 1
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Cyclic Newton’s Method
• Doesn’t require invertibility of the Jacobian
• Takes a Newton Step in each direction
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Heuristics
• Dynamical Systems• Iterations enter circular orbits around roots or explode to infinity
• To fix it, we take midpoint steps when the sign of the gradient changes drastically
• This leads to much more predictable convergence
• Initial Guesses• Take initial guesses on the surface of one or two equations
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A classes of cyclic iteration methods
• We can use higher order convergence methods, i.e. Halley’s method or Dr. Kalantari’s Basic Family (which have order of convergence 3 and 4…n respectively)
• We can also change up the order of alternation
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Linear Systems
• Linear systems are also systems of polynomial equations!• It turns out that this cyclic iteration works faster than Gaussian
Elimination in MATLAB (around 10 times faster)
• Whether not this holds in generality is not known.
• Also, Gaussian Elimination isn’t used that much anymore, and we haven’t tested it against modern methods i.e. Krylov Space methods
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Newton-Ellipsoid Methods
• Sort of like binary search/bisection method
• Given a search space, cut it by a factor until search space is the desired size
• Works for the monovariate polynomials (i.e. one single variable polynomial). We are working on generalizing it
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Conclusions
• Cyclic Iteration Methods work well, better and faster in many instances than existence methods
• The Newton-Ellipsoid Method is an excellent way of solving polynomial equations; while the behavior is much more erratic it converges to a root almost everywhere
• Solving nonlinear systems
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Thanks!