R.Barret,M.Berry,T.Chan,J.Demmel,J.Donato,J. Dongarra, V. Eijkhout, R. Pozo, C.Romine, and H. van derr Vost, Templates for the solutions of linear systems: Building blocks for iterative methods,2 nd ed., Philadelphia (PA), SIAM, 1994  Home/Search   Document Not in Database   Summary   Related Articles   Check

....even more expensive to solve the linear system. 4 On the other hand, our iterative method is much more e#cient. It requires O(mN) operations for each iteration. We have applied Bi CGSTAB (biconjugate gradient stabilized) 8] to our method, although other modern Krylov subspace iterative methods [9] can also be used. To accelerate the convergence, Eq. 4) is written as B # G, for B = I A 1 11 A 12 A 1 22 A 21 I # # G = A 1 11 A 1 22 # F . 5) This is equivalent to using the diagonal blocks of A as a preconditioner. The matrix B has a smaller condition number leading to a ....

....errors of the reflected wave at z = 0 and the transmitted wave at z = z m are listed for various values of p and q. It is clear that the chosen rational approximants have produced very accurate solutions. We have implemented the Krylov subspace method Bi CGSTAB [8] following the template in [9] and set the error tolerance for convergence to be 10 8 . This implies that BU G G 10 8 , where U is the approximate solution and is the vector 2 norm. For all these calculations, only 3 iterations are needed. We also solved the problem for di#erent number of ....

R. Barret et al, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, PA, 1994. 9

.... 3 A Quantitative Model of Locality Let N be the number of rows or columns of a square sparse matrix A and N Z the number of non zero elements (entries) Figure 1 shows a code for the row oriented sparse matrix vector product when the matrix is stored in the Compressed Row Storage (CRS) format [2]. X and Y are arrays of N elements. DA, IND and PTR are the three arrays (data, column indexes and row pointers) used by the CRS storage format. The number of memory references required by the different data structures in order to perform the product is 3 Theta N Z 2 Theta N . A detailed ....

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM Press, 1994.

.... conventional matrix multiplication with split matrix matrix vector multiplication of a partially sparsified matrix as regards the number of floating point operations (FLOPs) required for each of the example matrices (the cost of the indirect addressing required by sparse matrix multiplication [1] is ignored) For the conventional approach, the number of FLOPs is 2n . The split matrix approach requires 2n nz FLOPs for the first of the steps defined in Section 2, the sparse matrix vector multiplication of A 1 ; approximately #n(1 2 d ) for the second, where d is the depth of the ....

R. Barret et al. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, 1994.

.... D) 1 and it converges whenever #(D) 1. Results in [5] indicate that this method converges for a wider range of configurations. However, the convergence may be slow, if n 0 and n 1 are not very close. Our approach is to replace the classical Neumann series by a modern Krylov subspace method [12]. To be precise, we use the bi conjugate gradient stabilized method (BiCGSTAB) 11] as the iterative scheme. The operators L 1 0 or L 1 1 or L 1 1 2 are still multiplied to the two sides of (1) so that the condition number of the coe#cient matrix is reduced. Our implementation follows the ....

....we use the bi conjugate gradient stabilized method (BiCGSTAB) 11] as the iterative scheme. The operators L 1 0 or L 1 1 or L 1 1 2 are still multiplied to the two sides of (1) so that the condition number of the coe#cient matrix is reduced. Our implementation follows the templates given in [12]. 3 Numerical results For simplicity, we use the same reference refractive index n and the same rational approximant for L 0 and L 1 . When the square root operator 1 X j is approximated by S j , Eq. 1) for the reflected wave is replaced by (S 0 S 1 )u = S 0 S 1 )u Six ....

[Article contains additional citation context not shown here]

R. Barret et al, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, PA, 1994.

....k 8 Comparison with other methods Our algorithms, in the case without breakdown, compare well with other known algorithms with respect to the computational kernel and to the storage requirements. In Table 1, we will show a summary of operations 16 for iteration k for selected algorithms (see [4]) and our algorithms and also the storage requirements for each algorithm (without preconditioning) For the case of breakdown, the exact summary of operations and storage requirements needed in some look ahead versions given in the literature on the subject is not known. Thus we will show, in ....

R. Barret, M. Berry et al, Templates for the Solution of Linear Systems: Building blocks for Iterative Methods, SIAM, Philadelphia, PA, 1994.

....to the solution x of the linear system of equations Ax = b, such that x spanfb#Ab#A b# : # A bg. The subspace K t (A#b) spanfb#Ab#A bg is called a Krylov subspace. Krylov subspace methods exist for eigenproblems as well. For recent surveys of Krylov subspace methods, see [11, 48]. Most Krylov subspace methods involve several types of operations in every iteration, namely multiplication of a vector by the matrix A, vector operations (additions and multiplication by scalars) and inner products. The iteration may also include preconditioning, which consists of solving a ....

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadeplphia, PA, 1993.

....not reported here. Thanks to Haim Kaplan for referring us to Tarjan s paper on augmented union find data structures. 6 CHAPTER 1 Background This chapter provides background material for the next two chapters, which describe original research results. This chapter is based on material from [4], 5] 10] 12] and from Sivan Toledo s lecture notes on high performance computing. 1.1. Iterative Solvers The term iterative methods refers in this thesis to a wide range of techniques that use successive approximations to obtain more accurate solutions to a linear system Ax = b at each ....

....has a physical justification [4, Section 3.4. 2] More importantly, it has been shown that when A represents a regular 5 point discretization of Laplace s equation with Dirichlet boundary conditions, a no fill modified incomplete factorization reduced the condition number from O(n) to O( # n) see [4, 6, 16] and their reference) An unmodified factorization reduces the condition number by a constant factor, but not asymptotically [4, 16] 1.5. REORDERING MATRICES FOR SPARSITY 9 1.4. Matrices And Graphs We normally think of matrices as two dimensional arrays of numbers or as representations of linear ....

[Article contains additional citation context not shown here]

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia, PA, 1993.

....periodic boundary conditions. In Section 4 we describe the Lifting preconditioner and some algorithmical details of the involved basis transforms. We nish this paper with Section 5, where numerical experiments are presented. Here, the preconditioners are used within a BiCGstab iterative solver [34]. They also seem to be very e ective for situations which are not covered by our analysis, e.g. for adaptive grids. 2 Biorthogonal Petrov Galerkin and nitedi erence discretizations on sparse grids We de ne some generic notation for general biorthogonal Wavelets rst and then describe the ....

Barret R., Berry M., Chan T. et al., Templates for the solution of linear systems: Building Blocks for Iterative Methods; SIAM, Philadelphia,

....significant improvement in computational performance. Iterative methods for linear systems can be good alternatives, and extensive research in the area has resulted in a large number of efficient algorithms, such as the conjugate gradient method [14, 18] and its wide ranging variants (see e.g. [1, 3, 24, 32]) Another notable method is GMRES [31] which in principle is intended for non symmetric matrices but is widely used to solve indefinite or nearly semi definite symmetric problems. The key to the success of these iterative methods is often the use of efficient preconditioners 3 aimed at ....

R. Barret, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst. Templates for the solutions of linear systems: building blocks for iterative methods. SIAM, 1994. 19

.... 700 800 900 0 50 100 150 200 250 300 number of iterations in GMRES(m) m=10 l=1 50 nonzero per row 7 nonzero per row Figure 1: Cost evaluation for GMRES(m) and DEFLGMRES(m,l) 4 Numerical Results We have tested the algorithm using Matlab and the template of GMRES provided in netlib [18]. The matrices are taken from [4] and have the form A = SDS Gamma1 with A; S; D 2 IR 100 Theta100 and with S = 1; fi) a bidiagonal matrix with 1 on the diagonal and fi on the upper subdiagonal. The system Ax = b is solved for right hand sides b = 1; 1) T and GMRES starts with x 0 ....

R. Barret M. Berry T. Chan J. Demmel J. Donato J. Dongarra V. Eijkhout R. Pozo C. Romine H. van der Vorst. Templates for the solution of linear systems: building blocks for iterative methods. SIAM / netlib, 1993.

....e#cient parallel code for distributed memory systems. They are intended to be used by experts on the one hand, and by non experts who are either unfamiliar with parallel systems or unwilling to cope with new machines, environments and languages, on the other. Several other projects (eg templates [7], skeletons [8, 9, 10, 11, 12] clips [13, 14] or the BACS approach [15, 16] have been initiated to develop new and more sophisticated ways for supporting the programming of distributed memory systems via libraries of basic algorithms, data structures and programming templates. Like LEDA [17] ....

....in a PASCAL like language. It is supposed to be the optimal tool for experts of parallel machines for testing new algorithms. We couldn t use this approach, since we don t want the user to be an expert, nor do we want him to learn a new complex language (see above) The group of K. M. Chandy [7] develops a multi media system that provides users of parallel systems with a library of programming templates. The focus of this project is on software engineering technology to (i) get a parallel program up and running with minimal programming e#ort (ii) reasoning about correctness and ....

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, TEMPLATES for the Solution of Linear Systems: Building Blocks for Iterative Methods, Tech. Rep., CS-Dept., Univ. of Tennessee, 1993. http://www.netlib.org/templates/templates.ps

....of ecient parallel code for distributed memory systems. They are intended to be used by experts on the one hand, and by non experts who are either unfamiliar with parallel systems or unwilling to cope with new machines, environments and languages, on the other. Several other projects (eg templates [7], skeletons [8, clips [13, 14] or the BACS approach [15, 16] have been initiated to develop new and more sophisticated ways for supporting the programming of distributed memory systems via libraries of basic algorithms, data structures and programming templates. Like LEDA [17] for the ....

....in a PASCAL like language. It is supposed to be the optimal tool for experts of parallel machines for testing new algorithms. We couldn t use this approach, since we don t want the user to be an expert, nor do we want him to learn a new complex language (see above) The group of K. M. Chandy [7] develops a multi media system that provides users of parallel systems with a library of programming templates. The focus of this project is on software engineering technology to (i) get a parallel program up and running with minimal programming e ort (ii) reasoning about correctness and ....

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst, TEMPLATES for the Solution of Linear Systems: Building Blocks for Iterative Methods, Tech. Rep., CS-Dept., Univ. of Tennessee, 1993. http://www.netlib.org/templates/templates.ps

....re dundant work is performed. Also, note that the speed up achieved is lower than that which could be expected due to the reduction in the number of floating point operations. This is because GMRES uses dense storage, while the new proposals use a sparse storage, compress column storage (CCS) [8]. CCS storage formats are the most general: they make absolutely no assumptions about the sparsity structure of the matrix, and they do not store any unnecessary elements. However, they are not so efficient, requiring an indirect addressing step for every single scalar operation in the ....

R. Barret et al. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, 1994.

....matrix can be overcome in most cases. For example, when the same matrix has to be used to carry out a LU factorization and a series of SpM ThetaV , it is useful to store a reordered copy of the matrix. This reordered matrix will only be used for the SpM ThetaV . Often, such as in iterative methods [1], several products are performed sequentially. The only term that changes in the sequential products is the vector. Thus the computational cost of reordering is globally overcome. 3 A Qualitative Study of Locality within SpM ThetaV Let N be the number of rows or columns of a sparse matrix and NZ ....

....PTR(I) PTR(I 1) Gamma 1 REG = REG DA(J) X(Index(J) ENDDO Y(I) REG ENDDO (a) Sequential algorithm A = X Y = b) Data accesses Fig. 1. Algorithm for the product of a sparse matrix by a vector matrix vector product when the matrix is stored in Compressed Row Storage (CRS) format [1]. DA, Index and PTR are the three vectors (data, columns and row pointer) corresponding to this storage format. It provides simplicity in the access to data and flexibility to distribute data stored in this format among different processors. The accesses required by the different data structures ....

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM Press, 1994.

No context found.

R. Barret, M.Berry, T.F.Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. Van der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia, PA, 1994.

....ordering in the (x; y) plane to number the unknowns. The resulting system matrix A is a nonsingular block tridiagonal, irreducibly diagonally dominant, Stieltjes matrix. The preconditioned conjugate gradient (PCG) method, which is given in Fig. 1, is a common method for such systems (see, e.g. [3,16]) As a preconditioner B, we use the standard incomplete factorization with fill level ( 27,28] The algorithm for the construction of B (LPL t version) is shown in Fig. 2. The symbol D refers to the set of discarded fill in entries: D = f (k; i) j lev(l k;i ) g ; where denotes a user ....

R.F. Barret, M. Berry, T.F. Chan, J. Demmel, J. Donato, J.J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst. Templates for the Solution of Linear Systems : Building Blocks for Iterative Methods. SIAM, Philadelphia, 1994.

No context found.

R.Barret,M.Berry,T.Chan,J.Demmel,J.Donato,J. Dongarra, V. Eijkhout, R. Pozo, C.Romine, and H. van derr Vost, Templates for the solutions of linear systems: Building blocks for iterative methods,2 nd ed., Philadelphia (PA), SIAM, 1994

No context found.

Barret, R., Berry, M., Chan, T., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C. & van der Vost, H. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, Siam Press, 1994.

No context found.

BARRET,R.,BERRY,M.,CHAN,T.,DEMMEL,J.,DONATO,J.,DONGARRA,J.,EIJKHOUT, V. POZO,R.,ROMINE,C.,AND VAN DER VORST,H. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, 1994.

No context found.

Richard Barret, Michael Berry, Tony F. Chan, James Demmel, June Donato, Jack Dongarra, Victor Eijkhout, Roldan Pozo, Charles Romine, and Henk van der Vonst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia, 1994. Available from netlib.

No context found.

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. van der Vorst. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. Siam Press, 1994. 231

No context found.

R. Barret, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1994. (Available from Netlib at URL http://www.netlib.org/templates/index.html).

No context found.

R. Barret, M. Berry, T. F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1994. (Available from Netlib at URL http://www.netlib.org/templates/index.html).

No context found.

Barret R., Berry M., Chan T., Demmel J., Donato J., Dongarraa J., Eijkhout V., Pozo R., Romine C., and van der Vorst H. (1994) Templates for the solution of linear systems : building blocks for iterative methods. SIAM.

No context found.

R. Barret, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Soc. Indus.-Appl. Math., Philadelphia 1993).

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC