Parallel solution of triangular systems on distributed-memory multiprocessors
SIAM Journal on Scientific and Statistical Computing
Modified cyclic algorithms for solving triangular systems on distributed-memory multiprocessors
SIAM Journal on Scientific and Statistical Computing
Granularity issues for solving polynomial systems via globally convergent algorithms on a hypercube
C3P Proceedings of the third conference on Hypercube concurrent computers and applications - Volume 2
Distributed orthogonal factorization
C3P Proceedings of the third conference on Hypercube concurrent computers and applications - Volume 2
Algorithm 652: HOMPACK: a suite of codes for globally convergent homotopy algorithms
ACM Transactions on Mathematical Software (TOMS)
Algorithm 596: a program for a locally parameterized
ACM Transactions on Mathematical Software (TOMS)
A New Method for Solving Triangular Systems on Distributed Memory Message-Passing Multiprocessors
A New Method for Solving Triangular Systems on Distributed Memory Message-Passing Multiprocessors
Message Length Effects for Solving Polynomial Systems on a Hypercube
Message Length Effects for Solving Polynomial Systems on a Hypercube
Globally Convergent Parallel Algorithm for Zeros of Polynomial Systems
Globally Convergent Parallel Algorithm for Zeros of Polynomial Systems
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Probability-one homotopy methods are a class of methods for solving nonlinear systems of equations that are globally convergent from an arbitrary starting point. The essence of all such algorithms is the construction of an appropriate homotopy map and subsequent tracking of some smooth curve in the zero set of the homotopy map. Tracking a homotopy curve involves finding the unit tangent vectors at different points along the zero curve. Because of the way a homotopy map is constructed, the unit tangent vector at each point in the zero curve of a homotopy map &rgr;a(&lgr;,&khgr;) is in the kernel of the Jacobian matrix D&rgr;a(&lgr;,&khgr;). Hence, tracking the zero curve of a homotopy map involves finding the kernel of the Jacobian matrix D&rgr;a(&lgr;,&khgr;). The Jacobian matrix D&rgr;a is a n × (n+1) matrix with full rank. Since the accuracy of the unit tangent vector is very important, an orthogonal factorization instead of an LU factorization of the Jacobian matrix is computed. Two related orthogonal factorizations, namely QR and LQ factorization, are considered here. This paper presents computational results showing the performance of several different parallel orthogonal factorization/triangular system solving algorithms on a hypercube. Since the purpose of this study is to find ways to parallelize homotopy algorithms, it is assumed that the matrices have a special structure such as that of the Jacobian matrix of a homotopy map. In particular, we are interested in relatively small and dense Jacobians.