J. M. Bull and T. L. Freeman, (1993) Parallel Globally Adaptive Quadrature on the KSR-1 , N. A. Report No. 228, Department of Mathematics, University of Manchester, submitted to Advances in Computational Mathematics.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....; QLAST n Gamma1 , E i ; ELAST 1 ; ELAST n Gamma1 RESULT : RESULT Q i P n Gamma1 j=1 QLAST j ; R : R E i P n Gamma1 j=1 ELAST j LAST : LAST n Gamma 1 Endwhile End Algorithm (Global Adaptive Quadrature) 1.2 Parallel global adaptive quadrature. Bull and Freeman [3] derived two algorithms for a KSR1 virtual shared memory parallel machine. In their second (by their comparisons, superior) algorithm, at the first stage the original interval is p sected and the subintervals are treated in parallel. At every subsequent stage the largest absolute error estimate, ....

Bull, J.M. and Freeman, T.L. (1994), "Parallel Globally Adaptive Quadrature on the KSR1", Advances in Computational Mathematics, to appear.

....3, an MPI version of the NAG PVM library [11] parallel implementation of lattice rank 1 rules is presented. The performance of these parallel algorithms is tested and analyzed in Section 4. Close relatives of the simplest of the global adaptive algorithms developed here have been discussed in [6, 1, 2, 3], including tests on a KSR1 virtual shared memory system and a Sequent Symmetry shared memory system. The concept of the level algorithm used here is similar to the sword algorithm of Rice [13] 2 Multidimensional Adaptive Algorithms We develop two versions of a multidimensional adaptive ....

Bull, J.M. and Freeman, T.L. (1994) "Parallel Globally Adaptive Quadrature on the KSR1", Advances in Computational Mathematics 2 pp.357-373.

....of the definite multi dimensional integral I = Z b 1 a 1 Z b 2 a 2 : Z bn an f(x 1 ; x 2 ; x n ) dx 1 dx 2 : dx n to some absolute accuracy ffl. We are particularly interested in implementing globally adaptive algorithms on parallel computers. In previous papers ([2] and [3] we have investigated parallel globally adaptive quadrature algorithms for one dimensional problems and here we seek to extend and adapt our ideas to the multi dimensional case. In recent years a number of authors have considered parallel algorithms for numerical integration. Some ....

....also discuss the implementation of our algorithms in the more traditional message passing and sharedmemory frameworks in Section 4.3. In [10] Genz describes a parallel adaptive quadrature algorithm (for multi dimensional problems) that does not depend on neighbour to neighbour communication. In [2] and [3] we develop similar parallel algorithms for one dimensional quadrature based on the routine D01AKF in the NAG library [19] and we find that one of these algorithms regularly outperforms Genz s algorithm. In this paper we extend the ideas of [2] and [3] to cubature over a hyper rectangle. ....

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J. M. Bull and T. L. Freeman, (1993) Parallel Globally Adaptive Quadrature on the KSR-1 , N. A. Report No. 228, Department of Mathematics, University of Manchester, submitted to Advances in Computational Mathematics.

....ALGORITHMS FOR SINGULAR INTEGRALS J. M. Bull and T. L. Freeman, Centre for Novel Computing and Department of Mathematics, University of Manchester, Manchester, M13 9PL, U.K. I. Gladwell, Department of Mathematics, Southern Methodist University, Dallas, Texas 75275, U.S.A. 1 Introduction In [1] we considered the problem of approximating, on a parallel computer, the definite integral I = Z b a f(x)dx; to some given absolute accuracy ffl and we described a number of parallel globally adaptive algorithms. Of these, the most successful, for the numerical experiments we performed on the ....

....be maintained in order. 2 Singular Integrands The DS algorithm can deal successfully with singular integrands, even though it takes no special action in the presence of singularities. However singularities affect the efficiency, and particularly the parallel efficiency, of the algorithm. In [1] we show that, in the presence of a strong singularity, the speed up of the DS algorithm on p processors is approximately bounded above by 2 log 2 p. Here, we describe enhancements to the algorithm to deal more effectively with singular problems. There are essentially two aspects to consider: ffl ....

Bull, J. M. and Freeman, T. L. (1994) Parallel Globally Adaptive Quadrature on the KSR-1 , to appear in Advances in Computational Mathematics.

....X i=1 e i ffl and r X i=1 e i ffl; 3. Select all the intervals with error estimates e r . The rationale behind this strategy is that it is certain that at least this many subintervals must be subdivided to satisfy the global error criterion. Strategy AL This strategy was suggested in [5] and explored further in [6] 1. Search the list of intervals for the largest error estimate, say e max , and 2. Select all the intervals with error estimates ffe max , for some ff satisfying 0 ff 1. The choice of a reasonable value for ff will be addressed in Section 5.3. Note that this ....

Bull, J.M. and T.L. Freeman, (1994) Parallel globally adaptive quadrature on the KSR-1, Advances in Computational Mathematics, vol. 2, pp. 357--373.

....ALGORITHMS FOR SINGULAR INTEGRALS J. M. Bull and T. L. Freeman, Centre for Novel Computing and Department of Mathematics, University of Manchester, Manchester, M13 9PL, U.K. I. Gladwell, Department of Mathematics, Southern Methodist University, Dallas, Texas 75275, U.S.A. 1 Introduction In [1] we considered the problem of approximating, on a parallel computer, the definite integral I = Z b a f(x) dx; to some given absolute accuracy ffl and we described a number of parallel globally adaptive algorithms. Of these, the most successful, for the numerical experiments we performed on the ....

....be maintained in order. 2 Singular Integrands The DS algorithm can deal successfully with singular integrands, even though it takes no special action in the presence of singularities. However singularities affect the efficiency, and particularly the parallel efficiency, of the algorithm. In [1] we show that, in the presence of a strong singularity, the speed up of the DS algorithm on p processors is approximately bounded above by 2 log 2 p. Here, we describe enhancements to the algorithm to deal more effectively with singular problems. There are essentially two aspects to consider: ffl ....

Bull, J. M. and Freeman, T. L. (1994) Parallel Globally Adaptive Quadrature on the KSR-1 , to appear in Advances in Computational Mathematics.

....X i=1 e i ffl and r X i=1 e i ffl; 3. Select all the intervals with error estimates e r . The rationale behind this strategy is that it is certain that at least this many subintervals must be subdivided to satisfy the global error criterion. Strategy AL This strategy was suggested in [5] and explored further in [6] 1. Search the list of intervals for the largest error estimate, say e max , and 2. Select all the intervals with error estimates ffe max , for some ff satisfying 0 ff 1. The choice of a reasonable value for ff will be addressed in Section 5.3. Note that this ....

Bull, J.M. and T.L. Freeman, (1994) Parallel globally adaptive quadrature on the KSR-1, Advances in Computational Mathematics, vol. 2, pp. 357--373.

....algorithms on both virtual shared memory (KSR 1) and distributed memory (iPSC 860) machines and investigate how the characteristics of the different architectures affect the choice of implementation and thereby the performances of the algorithms. 1 Introduction In a number of previous papers, [2], 3] 4] we have considered the implementation and numerical performance of parallel globally adaptive algorithms for numerical integration in one and several dimensions. Our target machine has been the KSR 1 at the University of Manchester (originally this machine had 32 processors, but it has ....

....we address in this paper are the approximation of either the onedimensional definite integral I = Z b a f(x)dx; or the multi dimensional definite integral I = Z b 1 a 1 Z b 2 a 2 : Z bn an f(x 1 ; x 2 ; x n ) dx 1 dx 2 : dx n to some specified absolute accuracy ffl. In [2], 3] 4] we describe a number of parallel algorithms for these numerical integration problems; the most successful is the DS (dynamic, synchronous) algorithm, which exploits parallelism at the relatively coarse granularity of concurrent applications of the quadrature cubature rules. For a ....

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Bull, J. M. and Freeman, T. L. (1994) Parallel Globally Adaptive Quadrature on the KSR-1 , to appear in Advances in Computational Mathematics.

....and numerical results are presented. The algorithm gives significant speedups on a range of hard problems, including ones with singular integrands. A number of alternatives for the interval selection strategy that is at the core of this algorithm are evaluated. 1 Introduction In an earlier paper [2] we suggest a number of parallel algorithms for approximating the definite integral I = Z b a f(x)dx; to some given absolute accuracy ffl. Here we examine the most successful of these algorithms (the DS algorithm) in more detail, focusing on the interval selection strategy that is at the heart ....

....subintervals to list update integral approximations and error estimates end do Fig. 1. Algorithm DS Different strategies for the interval selection calculation are considered in Section 4. The results for the DS algorithm presented in Section 3 are based on the following strategy, suggested in [2]: 1. Search the list of intervals for the largest error estimate, say Emax , and 2. identify all the intervals with error estimates ffE max , for some ff satisfying 0 ff 1. This strategy involves finding the interval with largest error estimate, and then selecting all intervals with error ....

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J. M. Bull and T. L. Freeman (1993) Parallel Globally Adaptive Quadrature on the KSR-1 , N. A. Report No. 228, Department of Mathematics, University of Manchester, to appear in Advances in Computational Mathematics.

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