Browsing by Author "Irani, Kashmira M."

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    Experiments with Conjugate Gradient Algorithms for Homotopy Curve Tracking
    Irani, Kashmira M.; Kamat, Manohar P.; Ribbens, Calvin J.; Walker, Homer F.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1990)
    There are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points, i.e., with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map. HOMPACK is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve, and has separate routines for dense and sparse Jacobian matrices. The HOMPACK algorithms for sparse Jacobian matrices use a preconditioned conjugate gradient algorithm for the computation of the kernal of the homotopy Jacobian matrix, a required linear algebra step for homotopy curve tracking. Here variants of the conjugate gradient algorithm are implemented in the context of homotopy curve tracking and compared with Craig's preconditioned conjugate gradient method used in HOMPACK. The test problems used include actual large scale, sparse structural mechanics problems.
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    High Dimensional Homotopy Curve Tracking on a Shared Memory Multiprocessor
    Allison, Donald C. S.; Irani, Kashmira M.; Ribbens, Calvin J.; Watson, Layne T. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1991)
    Results are reported for a series of experiments involving numerical curve tracking on a shared memory parallel computer. Several algorithms exist for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points, i.e., with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then the tracking of some smooth curve in the zero set of this homotopy map. HOMPACK is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve, and has separate routines for dense and sparse Jacobian matrices. A parallel version of HOMPACK is implemented on a shared memory parallel computer with various levels and degrees of parallelism (e.g., linear algebra, function and Jacobian matrix evaluation), and a detailed study is presented for each of these levels with respect to the speedup in execution time obtained with the parallelism, the time spent implementing the parallel code and the extra memory allocated by the parallel algorithm.
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    Preconditioned Conjugate Gradient Algorithms for Homotopy CurveTracking
    Irani, Kashmira M.; Ribbens, Calvin J.; Walker, Homer F.; Watson, Layne T.; Kamat, Manohar P. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1989)
    These are alogorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points, i.e., with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map. HOMPACK is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve, and has separate routines for dense and sparse Jacobian Matrices. The HOMPACK alogorithms for sparse Jacobian matrices use a preconditioned conjugate gradient algorithm for the computation of the kernel of the homotopy Jacobian matrix, a required linear algebra step for homotopy curve tracking. Here variants of the conjugate gradient algorithms are implemented in the context of homotopy curve tracking and compared with Craig's preconditioned conjugate gradient method used in HOMPACK. The test problems used include actual large scale, sparse structural mechanics problems.
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    Preconditioned Iterative Methods for Homotopy Curve Tracking
    DeSa, Colin; Irani, Kashmira M.; Ribbens, Calvin J.; Watson, Layne T.; Walker, Homer F. (Department of Computer Science, Virginia Polytechnic Institute & State University, 1991-05-01)
    Homotopy algorithms are a class of methods for solving systems of nonlinear equations that are globally convergent with probability one. All homotopy algorithms are based on the construction of an appropriate homotopy map and then the tracking of a curve in the zero set of this homotopy map. The curve-tracking algorithms used here require the solution of a series of very special systems. In particular, each (n + 1) x (n + 1) system is in general nonsymmetric but has a leading symmetric indefinite n x n submatrix (typical of large structural mechanics problems, for example). Furthermore, the last row of each system may by chosen (almost) arbitrarily. The authors seek to take advantage of these special properties. The iterative methods studied here include Craig's variant of the conjugate gradient algorithm and the SYMMLQ algorithm for symmetric indefinite problems. The effectiveness of various preconditioning strategies in this context are also investigated, and several choices for the last row of the systems to be solved are explored.
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    Preconditioned sequential and parallel conjugate gradient algorithms for homotopy curve tracking
    Irani, Kashmira M. (Virginia Tech, 1990-10-14)
    There are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points, i.e., with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map. HOMPACK is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve, and has separate routines for dense and sparse Jacobian matrices. The HOMPACK algorithms for sparse Jacobian matrices use a preconditioned conjugate gradient algorithm for the computation of the kernel of the homotopy Jacobian matrix, a required linear algebra step for homotopy curve tracking. Variants of the conjugate gradient algorithm along with different preconditioners are implemented in the context of homotopy curve tracking and compared with Craig's preconditioned conjugate gradient method used in HOMPACK. In addition, a parallel version of Craig's method with incomplete LU factorization preconditioning is implemented on a shared memory parallel computer with various levels and degrees of parallelism (e.g., linear algebra, function and Jacobian matrix evaluation, etc.). An in-depth study is presented for each of these levels with respect to the speedup in execution time obtained with the parallelism, the time spent implementing the parallel code and the extra memory allocated by the parallel algorithm.