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Homotopy Continuation Methods For Solving Polynomial Systems
, 1996
"... Homotopy continuation methods have been proven to be reliable for computing numerically approximations to all isolated solutions of polynomial systems. The performance of... ..."
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Cited by 19 (1 self)
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Homotopy continuation methods have been proven to be reliable for computing numerically approximations to all isolated solutions of polynomial systems. The performance of...
HOMOTOPY CONTINUATION METHODS FOR NONLINEAR COMPLEMENTARITY PROBLEMS
, 1991
"... A complementarity problem with a continuous mapping f from Rn into itself can be written as the system of equations F(x, y) = 0 and (x, y)> 0. Here F is the mapping from R ~ " into itself defined by F(x, y) = ( xl y,, x2yZ,..., x, ~ ye, y ffx)). Under the assumption that the mapping f is ..."
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Cited by 44 (3 self)
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is a P,,function, we study various aspects of homotopy continuation methods that trace a trajectory consisting of solutions of the family of systems of equations F(x, y) = t(a, b) and (x, y) 8 0 until the parameter t> 0 attains 0. Here (a, b) denotes a 2ndimensional constant positive vector. We
Numerical Polynomial Homotopy Continuation Method and String Vacua
 Adv.High Energy Phys
, 2011
"... iv ..."
PHoMpara  parallel implementation of the Polyhedral Homotopy continuation Method for polynomial systems
, 2006
"... The polyhedral homotopy continuation method is known to be a successful method for finding all isolated solutions of a system of polynomial equations. PHoM, an implementation of the method in C++, finds all isolated solutions of a polynomial system by constructing a family of modified polyhedral hom ..."
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Cited by 10 (1 self)
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The polyhedral homotopy continuation method is known to be a successful method for finding all isolated solutions of a system of polynomial equations. PHoM, an implementation of the method in C++, finds all isolated solutions of a polynomial system by constructing a family of modified polyhedral
PHoMpara  Parallel Implementaion of the Polyhedral Homotopy Continuation Method for Polynomial Systems
"... The polyhedral homotopy continuation method is known to be a successful method for finding all isolated solutions of a system of polynomial equations. PHoM, implementation of the method in C++, finds isolated solutions of a polynomial system by costructing a family of polyhedrallinear homotopy func ..."
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The polyhedral homotopy continuation method is known to be a successful method for finding all isolated solutions of a system of polynomial equations. PHoM, implementation of the method in C++, finds isolated solutions of a polynomial system by costructing a family of polyhedrallinear homotopy
A review on Homotopy Continuation Methods for Polynomial Systems
"... ❖ A polynomial p ∈ C[x] is a finite sum of terms cax a. Each term is the product of a coefficient ca ∈ C and a monomial xa = x a1 1 xa2 2 · · ·xan n. Exponent a = [a1, a2,...,an] are n nonnegative integers. a1 + a2 + · · · + an is the degree of term cax a. The degree of a polynomial is the bigges ..."
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❖ A polynomial p ∈ C[x] is a finite sum of terms cax a. Each term is the product of a coefficient ca ∈ C and a monomial xa = x a1 1 xa2 2 · · ·xan n. Exponent a = [a1, a2,...,an] are n nonnegative integers. a1 + a2 + · · · + an is the degree of term cax a. The degree of a polynomial is the biggest degree of its terms. A homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have the same degree. ❖ We consider a polynomial system f(x) = 0 of N equations f = (f1, f2,...,fN) in n variables x = (x1, x2,...,xn) with complex coefficients, fi(x) ∈ C[x], for i = 1, 2,..., N. ❖ The Jacobian matrix of the system f(x) = 0 is the matrix of all first partial, for i = 1, 2,...,N and j = 1, 2,...,n. derivatives, denoted by ∂f ∂x =
Numerical Stability of Path Tracing in Polyhedral Homotopy Continuation Methods
, 2003
"... The reliability of polyhedral homotopy continuation methods for solving a polynomial system becomes increasingly important as the dimension of the polynomial system increases. High powers of the homotopy continuation parameter t and illconditioned Jacobian matrices encountered in tracing of homotop ..."
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Cited by 6 (1 self)
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The reliability of polyhedral homotopy continuation methods for solving a polynomial system becomes increasingly important as the dimension of the polynomial system increases. High powers of the homotopy continuation parameter t and illconditioned Jacobian matrices encountered in tracing
PHoM  a Polyhedral Homotopy Continuation Method for Polynomial Systems
 Computing
, 2003
"... PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedrallinear homotopy functions, based on the polyhedral ..."
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Cited by 39 (10 self)
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PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedrallinear homotopy functions, based on the polyhedral
On Homotopy Continuation Method for Computing Multiple Solutions to the Henon Equation
, 2006
"... Motivated by numerical examples in solving semilinear elliptic PDEs for multiple solutions, some properties of Newton homotopy continuation method, such as its continuation on symmetries, the Morse index, and certain functional structures, are established. Those results provide useful information on ..."
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Cited by 1 (0 self)
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Motivated by numerical examples in solving semilinear elliptic PDEs for multiple solutions, some properties of Newton homotopy continuation method, such as its continuation on symmetries, the Morse index, and certain functional structures, are established. Those results provide useful information
Balancing the Lifting Values to Improve the Numerical Stability of Polyhedral Homotopy Continuation Methods
, 1997
"... Polyhedral homotopy continuation methods exploit the sparsity of polynomial systems so that the number of solution curves to reach all isolated solutions is optimal for generic systems. The numerical stability of tracing solution curves of polyhedral homotopies is mainly determined by the height of ..."
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Cited by 10 (3 self)
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Polyhedral homotopy continuation methods exploit the sparsity of polynomial systems so that the number of solution curves to reach all isolated solutions is optimal for generic systems. The numerical stability of tracing solution curves of polyhedral homotopies is mainly determined by the height
Results 1  10
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