By consideration of the information processing task faced by the central nervous system in estimating body spatial orientation and in controlling active body movement using an internal model referenced control strategy, a mathematical model for sensory conflict generation is developed. The model postulates a major dynamic functional role for sensory conflict signals in movement control, as well as in sensory-motor adaptation. It accounts for the role of active movement in creating motion sickness symptoms in some experimental circumstance, and in alleviating them in others. The relationship between motion sickness produced by sensory rearrangement and that resulting from external motion disturbances is explicitly defined. A nonlinear conflict averaging model is proposed which describes dynamic aspects of experimentally observed subjective discomfort sensation, and suggests resulting behaviours. The model admits several possibilities for adaptive mechanisms which do not involve internal model updating. Further systematic efforts to experimentally refine and validate the model are indicated.

/pdf/a-heuristic-mathematical-model-for-the-dynamics-of-sensory-4e5uay3fgz.pdf

A heuristic mathematical model for the dynamics of sensory conflict and motion sickness

The problem of solving tridmgonal linear systems on vector computers is considered. In particular, implementations of several direct and lterative methods are given for the Control Data Corporatlon STAR-100 computer. The direct methods considered are Gaussian elimination, a parallel method due to Stone, and cyclic reduction; the iteratlve methods considered are Jacobi's method, successive overrelaxaUon, and a parallel method due to Traub. In addition, timing formulas for the methods based on current information are included to provide a basis for comparison In general, the direct methods are found to be superior to the iterative methods. The choice of direct methods depends on the size of the system, but for more than 125 equations, cyclic reduction is the fastest algorithm.

The Solution of Tridiagonal Linear Systems on the CDC STAR 100 Computer

The concurrent solution of tridiagonal systems on linear and 2-dimensional arrays, complete binary trees, shuffle-exchange and perfect shuffle networks, and boolean cubes by elimination methods are devised and analyzed The methods can be obtained by symmetric permutations of some rows and columns, and amounts to cyclic reduction or a combination of Gaussian elimination and cyclic reduction (GECR) The ensembles have only local storage and no global control Synchronization is accomplished via message passing to neighboring processorsThe parallel arithmetic complexity of GECR for N equations on a K processor ensemble is $O({ N / K } + \log _2 K)$, and the communication complexity is $O(K)$ for the linear array, $O(\sqrt K )$ for the 2-dimensional mesh, and $O(\log _2 K)$ for the networks of diameter $O(\log _2 K)$ The maximum speed-up for the linear array is attained at $K \approx {({ N / \alpha })}^{1/2} $ and for the 2-d mesh at $K \approx ({N / 2\alpha })^{2/3} $, where $\alpha $ (the time to communicate one floating-point number)/(the time for a floating-point arithmetic operation) For the binary tree the maximum speed-up is attained at $K = N$, and for the perfect shuffle and boolean k-cube networks, $K = N/(1 + \alpha )$ yields the maximum speed-up The minimum time complexity is of order $O(N^{1/2} )$ for the linear array, of order $O(N^{1/3} )$ for the mesh, and of order $O(\log _2 N)$ for the binary tree, the shuffle-exchange, the perfect shuffle and the boolean k-cubeThe relative decrease in computational complexity due to a truncation of the reduction process in a highly concurrent system is much greater than on a uniprocessor The reduction in the arithmetic complexity is proportional to the number of steps avoided, if the number of processing elements equals the number of equations So also is the reduction in the communication complexity for ensembles configured as binary trees, shuffle-exchange and perfect shuffle networks, and boolean cubesPartitioning the ensemble into subsets of processors is shown to be more efficient for the solution of multiple independent problems than pipelining the solutions over the entire ensemble A balanced cyclic reduction algorithm is presented for the case where each system is spread uniformly over the processing elements, and its complexity is compared with Gaussian elimination

Solving tridiagonal systems on ensemble architectures

Large sparse linear systems of equations require hours to solve on conventional mainframe computers: however, with the advent of parallel architectures such as vector computers or arrays of microprocessors, these problems may be solved in less time. In addition, with hardware becoming cheaper, parallel algorithms for solving problems on these architectures may prove to be cost effective.
In this thesis, we develop algorithms for assembling in parallel the sparse system of linear equations that result from finite difference or finite element discretizations of elliptic partial differential equations, such as those that arise in structural engineering. Parallel linear stationary iterative algorithms and parallel preconditioned conjugate gradient algorithms are developed for solving these systems. In addition, a model for comparing parallel algorithms on array architectures is developed and results of this model for our algorithms are given.

/pdf/iterative-algorithms-for-large-sparse-linear-systems-on-4t99umuxrm.pdf

Iterative algorithms for large sparse linear systems on parallel computers

. A parallel algorithm for the solution of the general tridiagonal system is presented. The method is based on an efficient implementation of Cramer's rule, in which the only divisions are by the determinant of the matrix. Therefore, the algorithm is defined without pivoting for any nonsingular system. 0(n) storage is required for n equations and 0(log n) operations are required on a parallel computer with n processors. 0(n) operations are required on a sequential computer. Experimental results are presented from both the CDC 7600 and CRAY-1 computers.

/pdf/a-parallel-algorithm-for-solving-general-tridiagonal-k6sbfj9nx0.pdf

A parallel algorithm for solving general tridiagonal equations

Iterative methods for the solution of tridiagonal systems are considered, and a new iteration is presented, whose rate of convergence is comparable to that of the optimal two-cyclic Chebyshev iteration but which does not require the calculation of optimal parameters. The convergence rate depends only on the magnitude of the elements of the tridiagonal matrix and not on its dimension or spectrum. The theory also has a natural extension to block tridiagonal systems. Numerical experiments suggest that on a parallel computer this new algorithm is the best of the iterative algorithms considered.

Accelerated Iterative Methods for the Solution of Tridiagonal Systems on Parallel Computers

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Accelerated Iterative Methods for the Solution of Tridiagonal Systems on Parallel Computers