I. S. Du# and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM J. Matrix Anal. Appl., 22(4):973--996 (electronic), 2001.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....approximate minimumdeU05 (AMD) algorithm [1]applie tothe structure of A A # . UMFPACKuse a column AMD algorithm [10] topre ee ute onlythe columns of A andcompute a rowp ew utationbase on nume555w and sparsitycrite9w during factorization.The seet di#eoriz isthe use of a maximal matching algorithm [13] to peU ute the rows ofthe coewAA] t matrix tomaximize the product ofthe magnitude of its diagonal e trieo As shown in [6, 18] this cana#eL factorizationtime beUD9O it change the amount of structuralsymmetu andthe amount of numeL] w pivoting during factorization. WSMP use thisprewA cewA0 on ....

I. S. Duff and J. Koster, On algorithms for permuting large entries to the diagonal of a sparse matrix, SIAM J.Matrix Anal.Appl., 22 (2001), pp.973--996.

.... ll in is known to be an NP hard problem and several heuristics have been developed. Here we only consider symmetric reordering techniques. Those can also be applied to an unsymmetric matrix A by considering the structure of A A (after some column permutation for very unsymmetric matrices [7]) Two popular schemes for symmetric reordering are bottom up heuristics such as the minimum degree (AMD [1] MMD [12] or minimum ll (MMF [15, 19] and global or top down heuristics based on partitioning the graph of the matrix, such as nested dissection [10] The good properties of both ....

I. S. Du and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM Journal on Matrix Analysis and Applications, 22(4):973996, 2001.

....or equal to 1.0, must be placed in DPARM(11) The default value of pivoting threshold DPARM(11) is 0.01. Note 5.1 WGSMP uses several mechanisms, other than partial pivoting, to enhance the accuracy of the final solution. These include a static permutation of rows maximize the diagonal product [10, 13, 2, 12], scaling, and iterative refinement in double and quadruple precision. Therefore, it is recommended that the smallest pivoting threshold that yields a solution with acceptable accuracy should be used. Minimizing row interchanges associated with partial pivoting saves time and memory. Rook ....

....then the backward errors are computed with respect to the scaled system and not the original system. WGSMP can use a maximum weight matching on the bipartite graph induced by the sparse coefficient matrix to permute its row such that the product of the absolute values of the diagonal is maximized [13, 2, 12, 10]. By default, indicated by IPARM(8) 0, this option is turned on if WSMP detects that the structural symmetry of the coefficient matrix is less than 80 , otherwise this option is turned off. If IPARM(8) is 1, then this permutation is always performed and if IPARM(8) is 2, then this permutation is ....

Iain S. Duff and Jacko Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM Journal on Matrix Analysis and Applications, 22(4):973--996, 2001.

....order of magnitude, a good prescaling of the matrix can have a signi cant impact on the accuracy and performance of the sparse solver. In some cases it is also very bene cial to precede the ordering by performing an unsymmetric permutation to place large entries on the diagonal. Du and Koster [13] have designed algorithms to permute large entries onto the diagonal and have shown that it can very signi cantly improve the behaviour of multifrontal solvers. The multifrontal approach by Du and Reid [16] is used in the Harwell Subroutine Library code ma41 [2, 3] and in the distributed memory ....

....to precede the ordering by performing an unsymmetric permutation to place large entries on the diagonal and then scaling the matrix so that the diagonal entries are all of modulus one and the o diagonals have modulus less than or equal to one. We use the Harwell Subroutine Library code mc64 [13] to perform this preordering and scaling on all matrices of structural symmetry smaller than 55. When mc64 is not used, our matrices are always row and column scaled (each row column is divided by its maximum value) All results presented in this section have been obtained on one processor (R10000 ....

I. S. Du and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. Technical Report RAL-TR-1999-030, Rutherford Appleton Laboratory, 1999. To appear in SIAM Journal on Matrix Analysis and Applications.

....0.49 0.43 venkat01 SCHED 0.95 0.91 0.90 0.63 0.37 SLUD 0.93 0.75 0.74 0.56 0.47 wang4 SCHED 0.98 0.90 0.76 0.87 0.52 Table 3: Load balance results. For both algorithms, the preprocessing steps are the same. These include a step to permute large entries on the diagonal (using the routine MC64 [3]) followed by a symetric permutation to preserve the sparsity (using multiple minimum degree algorithm applied on A A [13] and the symbolic factorization to get the structures of L and U.Only the numerical factorization phase is di#erent in the two approaches. This includes the matrix ....

I. S. Du# and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM J. Matrix Analysis and Applications, 22(4):973--996, 2001.

....or equal to 1.0, must be placed in DPARM(11) The default value of pivoting threshold DPARM(11) is 0.01. Note 5.1 WGSMP uses several mechanisms, other than partial pivoting, to enhance the accuracy of the final solution. These include a static permutation of rows maximize the diagonal product [10, 13, 2, 12], scaling, and iterative refinement in double and quadruple precision. Therefore, it is recommended that the smallest pivoting threshold that yields a solution with acceptable accuracy should be used. Minimizing row interchanges associated with partial pivoting saves time and memory. ffl Rook ....

....are computed with respect to the scaled system and not the original system. ffl IPARM(8) type I: WGSMP can use a maximum weight matching on the bipartite graph induced by the sparse coefficient matrix to permute its row such that the product of the absolute values of the diagonal is maximized [13, 2, 12, 10]. By default, indicated by IPARM(8) 0, this option is turned on if WSMP detects that the structural symmetry of the coefficient matrix is less than 80 , otherwise this option is turned off. If IPARM(8) is 1, then this permutation is always performed and if IPARM(8) is 2, then this permutation is ....

Iain S. Duff and Jacko Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. Technical Report RAL-TR-

....applied to the structure of A A 0 . UMFPACK uses a column approximate minimum degree algorithm [9] to prepermute only the columns of A and computes a row permutation based on numerical and sparsity criteria during factorization. The second difference is the use of a maximal matching algorithm [12] to permute the rows of the coefficient matrix to maximize the product of the magnitudes of its diagonal entries. As shown in [4, 16] this can affect factorization times because it changes the amount of structural symmetry and the amount of numerical pivoting during factorization. WSMP uses this ....

Iain S. Duff and Jacko Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. Technical Report RAL-TR-

....order of magnitude, a good prescaling of the matrix can have a signi cant impact on the accuracy and performance of the sparse solver. In some cases it is also very bene cial to precede the ordering by performing an unsymmetric permutation to place large entries on the diagonal. Du and Koster [10]have designed algorithms to permute large entries onto the diagonal and have shown that it can very signi cantly improve the behaviour of multifrontal solvers. The multifrontal approachby Du and Reid [13] is used in the Harwell Subroutine Library code #### [2, 3] and in the distributed memory ....

....to precede the ordering by performing an unsymmetric permutation to place large entries on the diagonal and then scaling the matrix so that the diagonal entries are all of modulus one and the o diagonals have modulus less than or equal to one. We use the Harwell Subroutine Library code #### [10] to perform this preordering and scaling on all matrices of structural symmetry smaller than 55. When #### is not used, our matrices are always row and column scaled (eachrow column is divided by its maximum value) All results presented in this section, have been obtained on one processor ....

I. S. Du and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. Technical Report RAL-TR-1999-030, Rutherford Appleton Laboratory, 1999.

....order of magnitude, a good prescaling of the matrix can have a signi cant impact on the accuracy and performance of the sparse solver. In some cases it is also very bene cial to precede the ordering by performing an unsymmetric permutation to place large entries on the diagonal. Du and Koster [10] have designed algorithms to permute large entries onto the diagonal and have shown that it can very signi cantly improve the behaviour of multifrontal solvers. The multifrontal approach by Du and Reid [13] is used in the Harwell Subroutine Library code ma41 [2, 3] and in the distributed memory ....

....to precede the ordering by performing an unsymmetric permutation to place large entries on the diagonal and then scaling the matrix so that the diagonal entries are all of modulus one and the o diagonals have modulus less than or equal to one. We use the Harwell Subroutine Library code mc64 [10] to perform this preordering and scaling on all matrices of structural symmetry smaller than 55. When mc64 is not used, our matrices are always row and column scaled (each row column is divided by its maximum value) All results presented in this section, have been obtained on one processor ....

I. S. Du and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. Technical Report RAL-TR-1999-030, Rutherford Appleton Laboratory, 1999.

....SPAI preconditioning and with reduction to block triangular form. The one sided permutation and scaling routines are those implemented by Duff and Koster in Harwell Subroutine MC64. The permutations are based on algorithms for finding maximum transversals in bipartite graphs; see [43] 26] and [27] for a detailed description of these algorithms. Our experiments will indicate which, among the different algorithms implemented in MC64, are the most effective from the point of view of preconditioning. The use of one sided permutations to obtain a zero free diagonal for the purpose of ....

.... quote an early paper [54] on maximum product transversals (see section 3) A limited number of preliminary experiments suggesting the beneficial impact of one sided permutations of the kind considered here on ILU preconditioning were performed by the first author and reported in [26] see also [27]) Indeed, generating stable ILU type preconditioners was one of the applications envisaged by the authors of MC64, and was already mentioned in [43] Here we report on a large number of experiments targeting matrices from several different applications: chemical engineering, economics linear ....

[Article contains additional citation context not shown here]

I. S. Duff and J. Koster, On Algorithms for Permuting Large Entries to the Diagonal of a Sparse Matrix, Rutherford Appleton Laboratory Technical Report RAL-TR-1999-030, April 1999.

....the ll in is known to be an NP hard problem and several heuristics have been developed. Here we only consider symmetric reordering techniques. Those can also be applied to an unsymmetric matrix A by considering the structure of A A (after some column permutation for very unsymmetric matrices [7]) Two popular schemes for symmetric reordering are bottom up heuristics such as the minimum degree (AMD [1] MMD [12] or minimum ll (MMF [15, 19] and global or top down heuristics based on partitioning the graph of the matrix, such as nested dissection [10] The good properties of both ....

I. S. Du and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM Journal on Matrix Analysis and Applications, 22(4):973996, 2001.

....rows (with some respect to numerical stability) aims at reducing the amount of ll in. Here we only consider symmetric reordering techniques which can also be applied to an unsymmetric matrix A by considering the structure of A A (after some column permutation for very unsymmetric matrices [8]) Two popular schemes for symmetric reordering are bottom up heuristics such as the minimum degree (AMD [1] MMD [15] or minimum ll (MMF [17, 21] and global or top down heuristics based on partitioning the graph of the matrix, such as nested dissection [12] A new class of algorithms has been ....

I. S. Du and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM Journal on Matrix Analysis and Applications, 22(4):973996, 2001.

....= 0, see Section 4) matrix A, the system Ax = b or x = b can be solved during the solve stage. This is controlled by ICNTL(9) see Section 4.5) 2.4 Pre processing and post processing facilities MUMPS offers pre processing and post processing facilities. Permutations for a zero free diagonal [14, 15, 16] can be applied to very unsymmetric matrices and can help reduce fill in and arithmetic (controlled by ICNTL(6) see Section 4.5) Prescaling of the input matrix can help reduce fill in during factorization and can improve the numerical accuracy. A range of classical scalings are provided and can ....

....components N, NELT, ELTPTR, ELTVAR, and A ELT. ICNTL(6) has default value 7 for unsymmetric matrices and 0 for symmetric matrices. It is only accessed by the host and only during the analysis phase. If ICNTL(6) 1, 2, 3, 4, 5, 6, 7 a column permutation based on the public domain code MC64 (see [15, 16] for more details) is applied to the original matrix. Column permutations are then applied to the original matrix to get a zero free diagonal. Possible values of ICNTL(6) are: 0 : No column permutation is computed. 1 : The permuted matrix has as many entries on its diagonal possible. The ....

I. S. Duff and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM Journal on Matrix Analysis and Applications, 22(4):973--996, 2001.

....ordering by performing an unsymmetric permutation to place large entries on the diagonal and then scaling the matrix so that the diagonals are all of modulus one and the off diagonals have modulus less than or equal to one. We use the MC64 code of HSL [22] to perform this preordering and scaling [14] and indicate clearly when this is done in the forthcoming results. The effect of using this preordering of the matrix is discussed in detail in Section 4.1. Finally, when MC64 is not used, our matrices are always row and column scaled (each row column is divided by its maximum value) In both ....

....the relative influence of these orderings on the performance of the solvers in Section 4.2. We will also comment on the relative cost of the analysis phase of the two solvers. 4. 1 Use of a preordering to place large entries onto the diagonal and comments on the cost of the analysis phase In [14], Duff and Koster developed an algorithm for permuting a sparse matrix so that the diagonal is large relative to the off diagonals. More precisely, when the matrix is reordered and scaled, the resulting matrix has diagonal entries all equal to one in modulus with off diagonal entries all of ....

[Article contains additional citation context not shown here]

I. S. Duff and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. Technical Report RAL-TR-

....the ordering by performing an unsymmetric permutation to place large entries on the diagonal and then scaling the matrix so that the diagonals are all of modulus one and the o diagonals have modulus less than or equal to one. We use the MC64 code of HSL to perform this preordering and scaling [13] and indicate clearly when this is done. The e ect of using this preordering of the matrix is discussed in detail in Section 4.1. Finally, when MC64 is not used, our matrices are always scaled. In both approaches, a pivot order is de ned by the analysis and symbolic factorization stages. In ....

....cost of the analysis phase of the two solvers. 4. 1 Use of a preordering to place large entries onto the diagonal and the cost of the analysis phase Du and Koster developed an algorithm for permuting a sparse matrix so that the diagonal entries are large relative to the o diagonal entries [13]. They have also written a computer code, MC64 (available from HSL [21] to implement this algorithm. Here, we use option 5 of MC64 which maximizes the product of the modulus of the diagonal entries and then scales the permuted matrix so that it has diagonal entries of modulus one and all ....

[Article contains additional citation context not shown here]

I. S. Du and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. Technical Report RAL-TR-1999-030, Rutherford Appleton Laboratory, 1999. To appear in SIAM Journal on Matrix Analysis and Applications.

....can be very beneficial (in different ways) to precede the ordering by performing an unsymmetric permutation to place large entries on the diagonal and then scaling the matrix so that the diagonals are all of modulus one and the off diagonals have modulus less than or equal to one. We use the MC64 (Duff and Koster 1999) code of HSL (HSL 2000) to perform this preordering and scaling (Amestoy, Duff, L Excellent and Li 2000b) Both approaches use Level 3 BLAS to perform the elimination operations. However, in MUMPS the frontal matrices are always square. It is shown in Amestoy and Puglisi (2000) how one can detect ....

....Roman and Amestoy 1999) depending on which performs better on each particular problem. For the matrix twotone it is very beneficial to precede the ordering by an unsymmetric permutation to place large entries on the diagonal. We use the MC64 code of HSL to perform this preordering and scaling (Duff and Koster 1999). When the best ordering for MUMPS is different from that for SuperLU, results with both orderings are provided. We see that MUMPS is usually faster than SuperLU and is significantly so on a small number of processors. We believe there are two reasons. First, MUMPS handles symmetric and more ....

Duff, I. S. and Koster, J. (1999), On algorithms for permuting large entries to the diagonal of a sparse matrix, Technical Report RAL-TR-1999-030, Rutherford Appleton Laboratory. Also appeared as Report TR/PA/99/13, CERFACS, Toulouse, France. To appear in SIAM Journal on Matrix Analysis and Applications.

....the ordering by performing an unsymmetric permutation to place large entries on the diagonal and then scaling the matrix so that the diagonals are all of modulus one and the off diagonals have modulus less than or equal to one. We use the MC64 code of HSL to perform this preordering and scaling (Duff and Koster 1999) and indicate clearly when this is done. The effect of using this preordering of the matrix is discussed in detail in Section 4.1. Finally, when MC64 is not used, our matrices are always scaled. In both approaches, a pivot order is defined by the analysis and symbolic factorization stages. In ....

....is used and we study the relative influence of these orderings on the performance of the solvers in Section 4.2. We also comment on the relative cost of the analysis phase of the two solvers. 4. 1 Use of a preordering to place large entries onto the diagonal and the cost of the analysis phase Duff and Koster (1999) developed an algorithm for permuting a sparse matrix so that the diagonal entries are large relative to the off diagonal entries. They have also written a computer code, MC64 (available from HSL (2000) to implement this algorithm. Here, we use option 5 of MC64 which maximizes the product of the ....

[Article contains additional citation context not shown here]

Duff, I. S. and Koster, J. (1999), On algorithms for permuting large entries to the diagonal of a sparse matrix, Technical Report RAL-TR-1999-030, Rutherford Appleton Laboratory. Also appeared as Report TR/PA/99/13, CERFACS, Toulouse, France. To appear in SIAM Journal on Matrix Analysis and Applications.

....ordering by performing an unsymmetric permutation to place large entries on the diagonal and then scaling the matrix so that the diagonals are all of modulus one and the o diagonals have modulus less than or equal to one. We use the MC64 code of HSL [22] to perform this preordering and scaling [14] and indicate clearly when this is done in the forthcoming results. The e ect of using this preordering of the matrix is discussed in detail in Section 4.1. Finally, when MC64 is not used, our matrices are always row and column scaled (each row column is divided by its maximum value) In both ....

....study the relative in uence of these orderings on the performance of the solvers in Section 4.2. We will also comment on the relative cost of the analysis phase of the two solvers. 4. 1 Use of a preordering to place large entries onto the diagonal and comments on the cost of the analysis phase In [14], Du and Koster developed an algorithm for permuting a sparse matrix so that the diagonal is large relative to the o diagonals. More precisely, when the matrix is reordered and scaled, the resulting matrix has diagonal entries all equal to one in modulus with o diagonal entries all of modulus ....

[Article contains additional citation context not shown here]

I. S. Du and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. Technical Report RAL-TR-1999-030, Rutherford Appleton Laboratory, 1999.

....of the difficulty of handling dynamic data structures efficiently, most distributed memory approaches do not perform numerical pivoting during the factorization phase. Numerical pivoting can clearly be avoided for symmetric positive definite matrices. On unsymmetric matrices, Duff and Koster [17, 18] have designed algorithms to permute large entries onto the diagonal and have shown that this can significantly reduce numerical pivoting. Demmel and Li [11] have shown that, if one preprocesses the matrix using the code of Duff and Koster, then static pivoting (with possibly modified diagonal ....

I. S. Duff and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. Technical Report RAL-TR-1999-030, Rutherford Appleton Laboratory, 1999.

....is important that sparse codes offer the possibility of iterative refinement both to obtain a more accurate answer and to provide a measure of the backward error. Demmel and Li [69] try to avoid the dynamic data structures required 9 by numerical pivoting by using the algorithm of Duff and Koster [41, 42] to permute large entries to the diagonal prior to starting the factorization. They also suggest computing in increased precision to avoid some of the problems from this compromise to stability pivoting. An important aspect of these approaches is that the parallelism is obtained directly because ....

I. S. Duff and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. Technical Report RAL-TR-1999-030, Rutherford Appleton 17 Laboratory, 1999. Also appeared as Report TR/PA/99/13, CERFACS, Toulouse, France.

....pivoting during the factorization phase. Instead, they are based on a static mapping of the tasks and data and do not allow task migration during numerical factorization. Numerical pivoting can clearly be avoided for symmetric positive definite matrices. For unsymmetric matrices, Duff and Koster [18, 19] have designed algorithms to permute large entries onto the diagonal and have shown that this can significantly reduce numerical pivoting. Demmel and Li [12] have shown that, if one preprocesses the matrix using the code of Duff and Koster, static pivoting (with possibly modified diagonal values) ....

I. S. Duff and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. Technical Report RAL-TR-1999-030, Rutherford Appleton Laboratory, 1999.

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I. S. Du# and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM J. Matrix Anal. Appl., 22(4):973--996 (electronic), 2001.

No context found.

I. S. Du and J. Koster. On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM Journal on Matrix Analysis and Applications, 22(4):973996, 2001.

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