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## Algorithm 8xx: a concise sparse Cholesky factorization package (2004)

Venue: | Univ. of Florida |

Citations: | 11 - 0 self |

### Citations

600 |
Computer Solution of Large Sparse Positive Definite Systems
- George, Liu
- 1981
(Show Context)
Citation Context ... does not achieve the same level of performance as methods based on dense matrix kernels (such as [12, 13]), its performance is competitive with column-by-column methods that do not use dense kernels =-=[3, 4, 5]-=-. Section 2 gives a brief description of the algorithm used in the symbolic and numeric factorization. A more detailed tutorial-level discussion may be found in [14]. Details of the ∗ Dept. of Compute... |

313 | An approximate minimum degree ordering algorithm
- Amestoy, Davis, et al.
- 1996
(Show Context)
Citation Context ...a few performance results on a Pentium 4 1 , to demonstrate that the performance of LDL is comparable to the column-by-column sparse Cholesky factorization algorithm. Each matrix is permuted with AMD =-=[1, 2]-=-. The chol function in MATLAB 7 computes L column-by-column, but as a result the columns are not sorted. It sorts the columns by transposing the matrix and returning R = L T instead. It thus uses twic... |

202 |
The role of elimination trees in sparse factorization
- Liu
- 1990
(Show Context)
Citation Context ...+1:n − Li+1:n,ixi end for The general result also governs the pattern of y in Algorithm 1. However, in this case L arises from a sparse Cholesky factorization, and is governed by the elimination tree =-=[10]-=-. A 2sgeneral graph traversal is not required. In the elimination tree, the parent of node i is the smallest j > i such that lji is nonzero. Node i has no parent if column i of L is completely zero be... |

163 | Sparse matrices in MATLAB: Design and implementation
- Gilbert, Moler, et al.
- 1992
(Show Context)
Citation Context ... does not achieve the same level of performance as methods based on dense matrix kernels (such as [12, 13]), its performance is competitive with column-by-column methods that do not use dense kernels =-=[3, 4, 5]-=-. Section 2 gives a brief description of the algorithm used in the symbolic and numeric factorization. A more detailed tutorial-level discussion may be found in [14]. Details of the ∗ Dept. of Compute... |

78 |
Sparse partial pivoting in time proportional to arithmetic operations
- Gilbert, Peierls
- 1988
(Show Context)
Citation Context ...indices of nonzero entries in x and b, respectively, in the lower triangular system Lx = b. To compute x efficiently the nonzero pattern X must be found first. In the general case when L is arbitrary =-=[7]-=-, the nonzero pattern X is the set of nodes reachable via paths in the graph GL from all nodes in the set B, and where the graph GL has n nodes and a directed edge (j, i) if and only if lij is nonzero... |

72 |
Algorithm 837: AMD, an approximate minimum degree ordering algorithm
- Amestoy, Davis, et al.
(Show Context)
Citation Context ...a few performance results on a Pentium 4 1 , to demonstrate that the performance of LDL is comparable to the column-by-column sparse Cholesky factorization algorithm. Each matrix is permuted with AMD =-=[1, 2]-=-. The chol function in MATLAB 7 computes L column-by-column, but as a result the columns are not sorted. It sorts the columns by transposing the matrix and returning R = L T instead. It thus uses twic... |

50 |
On the storage requirement in the out-of-core multifrontal method for sparse factorization
- Liu
- 1987
(Show Context)
Citation Context ...cted in topological order, so no stack is required. The run time of the symbolic analysis algorithm is thus proportional to the number of nonzeros in L. This is more costly than the optimal algorithm =-=[6, 9]-=-, which takes time essentially proportional to the number of nonzeros in A. The memory requirements are just the matrix A and a few size-n integer arrays. The result of the algorithm is the eliminatio... |

31 | An efficient algorithm to compute row and column counts for sparse cholesky factorization
- Gilbert, Ng, et al.
- 1994
(Show Context)
Citation Context ...cted in topological order, so no stack is required. The run time of the symbolic analysis algorithm is thus proportional to the number of nonzeros in L. This is more costly than the optimal algorithm =-=[6, 9]-=-, which takes time essentially proportional to the number of nonzeros in A. The memory requirements are just the matrix A and a few size-n integer arrays. The result of the algorithm is the eliminatio... |

24 |
A compact row storage scheme for Cholesky factors using elimination trees
- Liu
- 1986
(Show Context)
Citation Context .... Its primary purpose is to illustrate much of the basic theory of sparse matrix algorithms in as compact a code as possible, including an elegant method of sparse symmetric factorization (related to =-=[8, 11]-=-). The lower triangular factor L is computed rowby-row, in contrast to the conventional column-by-column method. Although it does not achieve the same level of performance as methods based on dense ma... |

24 | A supernodal Cholesky factorization algorithm for shared-memory multiprocessors
- Ng, Peyton
- 1993
(Show Context)
Citation Context ...r factor L is computed rowby-row, in contrast to the conventional column-by-column method. Although it does not achieve the same level of performance as methods based on dense matrix kernels (such as =-=[12, 13]-=-), its performance is competitive with column-by-column methods that do not use dense kernels [3, 4, 5]. Section 2 gives a brief description of the algorithm used in the symbolic and numeric factoriza... |

23 |
Efficient sparse matrix factorization on highperformance workstations: Exploiting the memory hierarchy
- Rothberg, Gupta
- 1991
(Show Context)
Citation Context ...r factor L is computed rowby-row, in contrast to the conventional column-by-column method. Although it does not achieve the same level of performance as methods based on dense matrix kernels (such as =-=[12, 13]-=-), its performance is competitive with column-by-column methods that do not use dense kernels [3, 4, 5]. Section 2 gives a brief description of the algorithm used in the symbolic and numeric factoriza... |

18 |
The design of a user interface for a sparse matrix package
- George, Liu
- 1979
(Show Context)
Citation Context ... does not achieve the same level of performance as methods based on dense matrix kernels (such as [12, 13]), its performance is competitive with column-by-column methods that do not use dense kernels =-=[3, 4, 5]-=-. Section 2 gives a brief description of the algorithm used in the symbolic and numeric factorization. A more detailed tutorial-level discussion may be found in [14]. Details of the ∗ Dept. of Compute... |

17 |
A generalized envelope method for sparse factorization by rows
- Liu
- 1988
(Show Context)
Citation Context .... Its primary purpose is to illustrate much of the basic theory of sparse matrix algorithms in as compact a code as possible, including an elegant method of sparse symmetric factorization (related to =-=[8, 11]-=-). The lower triangular factor L is computed rowby-row, in contrast to the conventional column-by-column method. Although it does not achieve the same level of performance as methods based on dense ma... |

4 | Algorithm 8xx: AMD, an approximate minimum degree ordering algorithm
- Amestoy, Davis, et al.
(Show Context)
Citation Context ...rsion 1.0 is available at http://www.cise.ufl.edu/research/sparse. The following table illustrates a few performance results on a Pentium 4 laptop with 1GB of memory. Each matrix is permuted with AMD =-=[1, 2]-=-. MATLAB’s chol computes L column-by-column using the conventional method, but as a result the columns are not sorted. It then sorts the columns by transposing the matrix and returning LT instead. It ... |

3 |
Building an old-fashioned sparse solver
- Stewart
- 2003
(Show Context)
Citation Context ...that do not use dense kernels [3, 4, 5]. Section 2 gives a brief description of the algorithm used in the symbolic and numeric factorization. A more detailed tutorial-level discussion may be found in =-=[14]-=-. Details of the ∗ Dept. of Computer and Information Science and Engineering, Univ. of Florida, Gainesville, FL, USA. email: davis@cise.ufl.edu. http://www.cise.ufl.edu/∼davis. This work was supported... |