Z. Zlatev, Computational Methods for General Sparse Matrices. Mathematics and Its Applications. 65 Kluwer Academic Publishers, Dordrecht The Netherlands. (1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....more non zero bands in the L and U matrices (i.e. larger stencils) 63, 8, 76] Another variant is to replace the drop by position strategy in (2) by a drop by size one. That is, a fill is dropped if its absolute value is below a certain tolerance. This drop tolerance strategy was proposed by [84, 110, 11]. For applications to fluid flow problems, see [109, 40, 41] A slightly different approach is to allow more than one level of fills [63, 76] The level of a particular fill is defined recursively. Fills caused by original entries of A are defined to have level 1. The level of other fills are ....

Z. Zlatev. Computational methods for general sparse matrices. Kluwer Acad. Pub., Dordrecht, Boston, London, 1991. 23

....pivot node [4] Another one uses a cost function: at step k, r i (k) is the number of non zero elements within row i and c j (k) is the number of non zeroes within column j. Markowitz version chooses to eliminate with the row and columns that minimize the function (r i (k) 1) c j (k) 1) Zlatev [36], Du and Reid [8] or Amestoy and Davis [2] have conducted many experiments on this subject. There is a third class of heuristics, namely the nested dissection [1, 17, 22] This method reduces ll in from 20 to 30 in some cases when compared to minimum degree ordering. But it needs a costly ....

Z. Zlatev. Computational Methods for General Sparse Matrices, chapter Pivotal Strategies for Gaussian Elimination, pages 67-86. Kluwer Academic Publishers, Norwell, MA, 1992.

....a method for solving least squares problems by preconditioned CG. Their algorithm first computes the incomplete orthogonal factorization A = QDt q E using the Gentleman version [28] of Givens rotation with a numerical dropping technique which has been apply in the Y12M sparse linear system solver [65]. DR is used as a preconditioner for CG applied to the normal equations. In this incomplete orthogonalization, E is from the contribution of the dropped elements whose magnitude are smaller than a given positive number called the dropping tolerance. From the test results reported in the paper, if ....

....of selecting P is to use a dynamic strategy. By dynamic we mean that the set P is unknown until the factorization is complete. Certain criteria are imposed on the value of a matrix element and at some point during the factorization a decision is made to keep or drop it based on the criteria (see [65] for a detailed discussion of this topic) Such a strategy has the advantage that it can adapt to numerical values encountered during the factorization and not just precondition based on its sparsity pattern. In the case of IMGS and ICGS, after each step a test is made on the rij whose final ....

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Z. Zlatev. Computational Methods for General S'parse Matrices. Kluwer Academic Publishers, 1991.

....is that, for an arbitrary graph, the problem of minimizing the ll in is NP complete [Yan81] Therefore we use heuristics. In [LO91] a Structured Gaussian Elimination is proposed to solve large sparse linear systems arising in integer factorization. Some more general methods are discussed in [Zla92] and, in 3 particular, methods using the Markowitz cost function are proposed. Another commonly used heuristic is the so called minimum degree ordering which at a step k will eliminate a vertex of current minimum degree. We implement a simpler version of this (using a simpli ed Markowitz ....

Zahari Zlatev. { Computational Methods for General Sparse Matrices, chapitre Pivotal Strategies for Gaussian Elimination, pages 67-86. { Norwell, MA, Kluwer Academic Publishers, 1992. 17

....Therefore these requirements should not always be imposed. The main purpose of this paper is to show how to avoid them (when this is appropriate) A direct solver, where one attempts to exploit coarse grain parallelism without imposing the above two requirements, is described and tested in [30, 54]. This solver is based on partitioning the matrix into an upper block triangular form with rectangular diagonal blocks. A reordering algorithm, by which as many as possible zero elements are obtained in the lower left corner of the matrix, is to be applied before the partitioning. After that the ....

....approximately the same number of rows during the partitioning (because it is allowed to use rectangular diagonal blocks) This is why we concentrate our attention on the initial reordering. In the remaining of this chapter we discuss an improvement of the reordering algorithm proposed in [30] and [54], and its application in the solution of systems of linear algebraic equations by Gaussian elimination. Throughout, n and NZ denote respectively the order of the matrix and the number of its non zero elements. More details about the algorithm can be found in [24] 3.1 The Locally Optimized ....

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Z. Zlatev, "Computational methods for general sparse matrices". Kluwer Academic Publishers, Dordrecht-Toronto-London, 1991.

....J loop, and dense storage of the remaining diagonals becomes desirable. 2.3 Storage Access patterns that are dense are stored in fullsized arrays. Sparse access patterns are stored as sparse vectors in data structures (D1) or (L1) illustrated in figure 2 (inspired on existing data structures [13, 24, 26, 30]) In these implementations, the 4 numerical values of entries are stored in an array AVAL, while corresponding column or row indices are stored in a parallel integer array AIND. The implementations differ in the way each access pattern is stored: consecutively (D1) or as a linked list (L1) and ....

....The most general and flexible initialization method is from file construction, since it can be used for arbitrary sparse matrices. In order to keep the input storage scheme simple, coordinate scheme storage is used, where each file consists of an integer nz, followed by nz triples (i; j; a ij ) [13, 14, 15, 18, 30]. In [9] it is discussed how the compiler can generate appropriate initializing routines. 5 Examples Consider code generation for the first fragment of section 3.4, where the diagonal of a sparse matrix A is dense. For C i A1 , data structure ADNS(N) is used (note that I 2 GammaI 1 = 0 ....

Zahari Zlatev. Computational Methods for General Sparse Matrices. Kluwer Academic Publishers, 1991. 15

....ADIAG6(J) B(J) ENDDO DO J = 1, 900 C(J 100) C(J 100) ADIAG7(J) B(J) ENDDO Obviously, the same kind of code can be derived automatically for other matrices having only a few nonzero diagonals. In table 2, we present some timings on an HP UX 9000 720 for an element 15 of the E(n; c) class [28] and some matrices of the Harwell Boeing Sparse Matrix Collection with a limited number of nonzero diagonals (indicated by d in the table) We present the execution time of the dense code in which the I and J loop have been interchanged to enhance spatial locality, a general sparse row wise ....

Zahari Zlatev. Computational Methods for General Sparse Matrices. Kluwer Academic Publishers, 1991. 21

....for most Fortran77 implementations. Formats for special, more regular sparsity patterns, such as for band matrices, block sparse matrices, or skyline matrices, are not considered here. The abbreviations of format names are partially adapted from [13] More details can be found in [13] 35] and [36]. ffl DNS (dense storage format) uses a two dimensional array A(N,M) to store all elements. Due to the symmetric access structure of the two dimensional array, a leading dimension flag ld tells us whether the matrix is stored row major or column major. In the following, we summarize all data ....

Z. Zlatev. Computational Methods for General Sparse Matrices. Kluwer Academic Publisher, 1991.

....when using multiprocessors. To take full advantage of the features of the high performance supercomputers, dense LAPACK [1] kernels are used in most of the floating point computations. This results in a larger amount of arithmetics to be done, compared to classical sparse techniques ( 3] 5] [12]) but results also in more efficiency on multiprocessors due to higher data locality. The approach generally gives good results for matrices, in which most of the nonzeros are packed in a band around the main diagonal (or matrices that can be reordered in such form) The bandwidth imposes certain ....

....these bounds. This will guarantee preserving the nonzeros within the band. 3. The i th row is scanned again and all the nonzeros outside the interval [l i ; r i ] are dropped (i.e. considered to be zeros) This procedure is consistent with the classical numerical dropping strategy (see [5] [12]) as all the nonzeros larger (by absolute value) than i are kept. The elements in between the boundaries, however, are never dropped irrespective of their numerical value. As the matrix will be processed as dense blocks later, no saving will be realized even if the elements are dropped. Saving ....

Z. Zlatev, Computational Methods for General Sparse Matrices, Kluwer Academic Publishers, Dordrecht-Toronto-London, 1991.

.... COO (coordinate format) MSR (a CSR extension) CUR (unsorted CSR variant) XSR XUR (sorted unsorted CSR extension) CSC (column compressed format) JAD (jagged diagonal format) and LNK (linked list format) are explained in [19] The format names are partially adapted from [32] See also [2] and [38]. There are also many possibilities for slight modifications and extensions of these data structures. For instance, a flag may indicate symmetry of a matrix. Such changes are quite ad hoc, and it seems generally not sensible to define a new family of concepts for each such modification. For ....

Z. Zlatev. Computational Methods for General Sparse Matrices. Kluwer 1991.

....are smaller than a threshold d (with default value 0) Normally, we expect iterative refinement to be used when this option is active, but it is quite possible to use this option to obtain a preconditioner for a more powerful iterative method. In this case, a higher value for d may be possible. Zlatev (1991, chapter 11) points out that conjugate gradient type methods may be very effective in this context. In some applications, only a small number of entries differ between successive factorizations. If the changed entries are confined to columns late in the pivot sequence, the factorization ....

Zlatev, Z. (1991). Computational methods for general sparse matrices. Kluwer Academic Publishers, Dordrecht, Boston, and London.

....Scalar Pub NETLIB sympattern ) MUPS [1] MA42 [8] MF, threshold, BLAS 3 Frontal, BLAS 3 Com HSL Com HSL sym. MA27 [7] MA47 [5] MF, LDL T , BLAS 1 Com HSL s.p.d. Ng Peyton [16] LL, BLAS 3 Pub Author Shared Memory Algorithms nonsym. SuperLU LL, partial, BLAS 2.5 Pub UCB nonsym. PARASPAR [19, 20] RL, Markowitz, BLAS 1, SD Res Author sym MUPS [2] MF, threshold, BLAS 3 Res Author pattern nonsym. George Ng [9] RL, partial, BLAS 1 Res Author s.p.d. Gupta, Rothberg, LL, BLAS 3 Com SGI Ng Peyton [11] Pub Author s.p.d. SPLASH [13] RL, 2 D block, BLAS 3 Pub Stanford Distributed Memory ....

Zahari Zlatev. Computational methods for general sparse matrices. Kluwer Academic, Dordrecht; Boston, 1991.

....is that, for an arbitrary graph, the problem of minimizing the fill in is NP complete [Yan81] Therefore we use heuristics. In [LO91] a Structured Gaussian Elimination is proposed to solve large sparse linear systems arising in integer factorization. Some more general methods are discussed in [Zla92] and, in particular, methods using the Markowitz cost function are proposed. Another commonly used heuristic is the so called minimum degree ordering which at a step k will eliminate a vertex of current minimum degree. We implement a simpler version of this (using a simplified Markowitz ....

Zahari Zlatev. Computational Methods for General Sparse Matrices, chapter Pivotal Strategies for Gaussian Elimination, pages 67--86. Kluwer Academic Publishers, Norwell, MA, 1992.

....velocitypressure formulation fall in this category. 1.2. Related work. Several robust preconditioning techniques targeted at general sparse systems have been proposed in recent years. These include preconditioners based on incomplete orthogonalization, such as incomplete LQ [45] and incomplete QR [55], 8] row projection methods [1] 2] 14] and sparse approximate inverse techniques based on adaptive Frobenius norm minimization [19] 34] 17] 35] However, all these techniques are considerably more expensive than standard ones based on incomplete factorizations, and furthermore, as ....

Z. Zlatev, Computational Methods for General Sparse Matrices, Kluwer Academic Publishers, Dordrecht, 1991.

....were invariant in one execution of the concurrentized loop. This is true if no assignments to elements of the corresponding matrix occur (as in e.g. triangular solve) or if all assignments are performed within a static nonzero structure (as in e.g. LU decomposition after symbolic factorization [10, 28]) In general, however, the nonzero structure might change during execution of the loop due to insertions and deletions of entries (referred to as creation and cancellation respectively) Generation of an identical guard in two different iterations is only valid if changes in value between both ....

....are carried by the outermost loop, concurrentization of the J loop is a common optimization. However, if the code is converted into sparse code, the amount of concurrency can drastically decrease because many iterations of the J loop get disabled, since only so called target rows are considered [28]. Therefore, concurrentization of an outermost loop might be useful, which also introduce less startup overhead [17, 25, 27] Loop interchanging yields the JIK version, in which S 2 ffi S 1 and S 2 ffi =S 2 hold. The first dependence is caused by A(J,K) and A(I,I) Consequently, the value of ....

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Zahari Zlatev. Computational Methods for General Sparse Matrices. Kluwer Academic Publishers, 1991.

.... for e mail: tto cs.purdue.edu y e mail: sameh cs.purdue.edu z e mail: sarin cs.purdue.edu GRE216B in Table 1 in the last section of this paper) That is why the preconditioning technique is at least as important as the iterative method, a lot of research has been conducted in this area [3, 6, 8, 14, 15, 16, 17]. More details regarding construction of our preconditioners is given in Section 2. The Balance method (its original projection based version is described in [10] is used to perform the preconditioning. After block row partitioning and factorization of the blocks, it eventually leads to the ....

....when using multiprocessors. To take full advantage of the features of the high performance supercomputers, dense LAPACK [1] kernels are used in most of the floating point computations. This results in a larger amount of arithmetics to be done, compared to classical sparse techniques ( 4] 7] [17]) but results also in more efficiency on multiprocessors due to higher data locality. The approach generally gives good results for matrices, in which most of the nonzeros are packed in a band around the main diagonal (or matrices that can be reordered in such form) The bandwidth imposes certain ....

[Article contains additional citation context not shown here]

Z. Zlatev, Computational Methods for General Sparse Matrices, Kluwer Academic Publishers, Dordrecht-Toronto-London, 1991.

....sparse identically structured matrices is a common subproblem to solving systems of ODEs via an implicit method. Furthermore, these systems are frequently encountered in applications that include nuclear magnetic resonance spectroscopy, computational 28 chemistry, and computational biology [147]. More specifically, Miranker provides examples of systems of nonlinear ODEs with the circuit simulation of tunnel diodes commonly used in high speed circuits, thermal decomposition of ozone, and the behavior of a catalytic fluidized bed [107] Electronic circuit simulations are another common ....

Zahari Zlatev. Computational Methods for General Sparse Matrices. Kluwer Academic Publishers, Dordrecht, Boston, London, 1991.

.... matrices are: huib 209, a small circuit simulation problem; poisson, the Poisson equation on a unit square; qcgstab1 7, problem 1 and 7 from [1] saad3, problem 3 from [16] tfqmr1, problem 1 from [7] vdvorst2 3, problems 2 and 3 from [20] All matrices are stored in sparse column wise format [23], no symmetry is exploited. The matrices contain real entries only. Some of them are supplied with a right hand side. If there was no right hand side available, a random one was generated. The H0 reordering [8] was applied prior to factorization to make sure that the diagonal did not contain any ....

Z. Zlatev. Computational Methods for General Sparse Matrices. Mathematics And Its Applications. Kluwer Academic Publishers, P.O. Box 322, 3300AH Dordrecht, The Netherlands, 1991.

....a 55 a 22 A a 11 a 15 a 43 a 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 VAL A ROW A COL A NNZ A a 15 a 11 a 22 a 43 a 55 2 2 1 5 5 5 4 3 1 1 Figure 1: Dense vs. Sparse Storage operates on dense matrices by only storing nonzero elements and avoiding redundant operations on zero elements (see e.g. [28, 29, 30, 31, 32, 34]) EXAMPLE: Below, two FORTRAN fragments performing the operation b b A x are given. In the dense fragment, a two dimensional array A is used to store all elements of the matrix A, whereas a more complex sparse storage scheme (data structure) is used in the sparse fragment to avoid ....

....of the sparse fragment is only executed NNZ A times. Keeping the storage requirements as well as the amount of work truly proportional to the number of nonzero elements in a sparse matrix is one of the most important objectives in sparse matrix computations [25] 30, ch2] 31, ch2] 32, p1 3][34]. As already illustrated by this small example, however, achieving this objective may be a complex and cumbersome task for the programmer. The use of complicated sparse storage schemes usually obscures the actual functionality of the code, making both the development and maintenance of sparse ....

Zahari Zlatev. Computational Methods for General Sparse Matrices. Kluwer, Dordrecht, 1991.

....column number j is stored in the same position, position J , of a one dimensional integer array CNQR. There are two integer arrays (of length m) which contain pointers for the row starts and row ends in array AQR. This structure is the same as the structure used in the codes discussed in Zlatev, [12] and very similar to that used in MA28; see Duff et al. 4] and Duff and Reid, 6] Some additional ordering is performed in order to facilitate some modifications. Consider any two non zero elements a ij and a ik in row i, i = 1; 2; m. If j k, then a ij is located, in array AQR, ....

....but also their column indices are in an increasing order within each row. The information about the storage of the non zero elements, which has been sketched above, is quite sufficient for the purposes in this paper; more details can be found in Duff et al. 4] Duff and Reid, 6] and Zlatev, [12]. 4 SCATTERING THE NON ZERO ELEMENTS IN A TWO DIMENSIONAL ARRAY The first task that is to be solved after the call of LORA is to determine a large block (that will be treated by using dense matrix technique) An obvious choice is to take the block row containing the first dense block of the ....

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Z. Zlatev: "Computational methods for general sparse matrices". Kluwer Academic Publishers, Dordrecht-Toronto-London, 1991.

.... b 2 R m , find x 2 R n that minimizes the Euclidean norm of the residual vector r = b Gamma Ax : kb Gamma Axk 2 = min x2R n kb Gamma Axk 2 (1) or equivalently, solve the system of linear algebraic equations Ax = b Gamma r A T r = 0 : 2) Following Duff et al. 7] and Zlatev [30] it is assumed that the matrix A is large, sparse (most of its elements are equal to zero) and has full column rank (i.e. rank(A) n) Furthermore, it is also assumed that A has neither any special property (such as symmetry or positive definiteness) nor any special structure (banded, block ....

....on the Givens method, i.e. on a series of elementary plane rotations. A single element of the active submatrix is annihilated in each plane rotation. This method is as stable as the Householder method (cf. 18] but it preserves in general much better the sparsity of the original matrix A (cf. [30]) Only two rows take part in each rotation and a number of rotations can be performed concurrently (assuming that each row takes part in at most one rotation) which means that there is a lot of potential parallelism in this method, which may be an important factor in many situations, but it is ....

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Z. Zlatev, Computational Methods for General Sparse Matrices, Kluwer Academic Publishers, Dordrecht-Toronto-London, 1991. 29

.... of the least squares problem explain why several well developed methods for its solution have gradually been developed, for example: The method of normal equations The QR factorization method The Peters Wilkinson method The augmentation method Iterative methods (see [3, 4, 19]) One of them, the QR factorization method, is discussed in more detail below. 1.2 The Method of Orthogonal Factorization One of the most popular methods for solving numerically the sparse linear least squares problem is by orthogonal factorization, A = QR ; 3) where Q 2 R m Thetan has ....

....techniques for sparse matrix factorization, exploiting small dense submatrices [1, 15, 16] The Givens method uses elementary plane rotations. By each rotation a single element of the active submatrix is annihilated. It is as stable as Householder s method and best in preserving the sparsity [19]. Two rows only take part in each rotation, which means that there is a lot of potential parallelism in this method. These reasons were the most essential in the decision to select this method for our parallel algorithm. There are two versions Classical Givens [13] and GivensGentleman [9] The ....

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Z. Zlatev, Computational Methods for General Sparse Matrices, Kluwer Academic Publishers, Dordrecht-Toronto-London, 1991. This article was processed using the L a T E X macro package with LLNCS style

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Z. Zlatev, Computational Methods for General Sparse Matrices. Mathematics and Its Applications. 65 Kluwer Academic Publishers, Dordrecht The Netherlands. (1991.

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Z. Zlatev, 1991. Computational Methods for General Sparse Matrices, ch. 16. Kluwer Acad. Publ., 1991.

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Zahari Zlatev. Computational Methods for General Sparse Matrices. Kluwer, Dordrecht, 1991.

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