#### DMCA

## Dynamic supernodes in sparse Cholesky update/downdate and triangular solves (2006)

### Cached

### Download Links

- [www.cise.ufl.edu]
- [www.cise.ufl.edu]
- [www.math.ufl.edu]
- DBLP

### Other Repositories/Bibliography

Venue: | ACM Trans. Math. Software |

Citations: | 30 - 10 self |

### Citations

861 |
A set of level 3 basic linear algebra subprograms
- Dongarra, Croz, et al.
- 1990
(Show Context)
Citation Context ...n. Improved locality also enables its use on parallelcomputers, but only sequential algorithms are considered here. The use of dense matrix kernels (the BLAS [Lawson et al. 1979; Dongarra et al.1988; =-=Dongarra et al. 1990-=-]) is a common technique for improving the performance of sparse matrix factorization and the solution of the subsequent triangular systemsthat are required to solve Ax = b for a general matrix A. Sup... |

639 |
Basic linear algebra subprograms for Fortran usage
- Lawson, Hanson, et al.
- 1979
(Show Context)
Citation Context ...se submatrices in the sparse factorization. Improved locality also enables its use on parallelcomputers, but only sequential algorithms are considered here. The use of dense matrix kernels (the BLAS [=-=Lawson et al. 1979-=-; Dongarra et al.1988; Dongarra et al. 1990]) is a common technique for improving the performance of sparse matrix factorization and the solution of the subsequent triangular systemsthat are required ... |

600 |
Computer Solution of Large Sparse Positive Definite Systems
- George, Liu
- 1981
(Show Context)
Citation Context ...imination tree are then used to find the fundamental supernodes. Consider the jth column of L. Its nonzero pattern is related to the nonzero patterns of the children of node j in the elimination tree =-=[15]-=-, ⎛ Lj = Aj ∪ {j} ∪ ⎝ � ⎞ Lc \ {c} ⎠, (2) j=parent(c) 1 The number of nonzeros in matrix or vector x, or the size of a set x, is denoted as |x|. 3swhere Aj is the nonzero pattern of the jth column of ... |

517 | An extended set of Fortran basic linear algebra subroutines
- Dongarra, Croz, et al.
- 1988
(Show Context)
Citation Context ...se submatrices in the sparse factorization. Improved locality also enables its use on parallel computers, but only sequential algorithms are considered here. The use of dense matrix kernels (the BLAS =-=[22, 14, 13]-=-) is a common technique for improving the performance of sparse matrix factorization and the solution of the subsequent triangular systems that are required to solve Ax = b for a general matrix A. Sup... |

381 |
LINPACK User’s Guide
- Bunch, Dongarra, et al.
- 1979
(Show Context)
Citation Context ...ang's [Bischof et al. 1993] combination of Carlson's update [Carlson 1973] and Pan's downdate[Pan 1990]. All of these methods modify the LLT factorization instead, as doesStewart's method in LINPACK [=-=Dongarra et al. 1978-=-; Stewart 1979; 1998] (the method used by cholupdate in MATLAB). To update LLT factorization, the in-nermost loop requires 5 floating-point operations instead of 4 for the LDLT case.The memory traffic... |

202 |
The role of elimination trees in sparse factorization
- Liu
- 1990
(Show Context)
Citation Context ...cisely, a supernode is defined by a chain of nodes in the elimination tree, and the sparsity pattern of the corresponding columns of L. The elimination tree of an n-by-n matrix A is a tree of n nodes =-=[23, 24, 30]-=-. The parent of node j in the tree is given by the first off-diagonal nonzero entry lij in column j, parent(j) = min{i | i > j and lij �= 0}. (1) If this set is empty, then node j is a root of the eli... |

183 |
Direct Methods for Sparse Linear Systems
- Davis
(Show Context)
Citation Context ...hile the multiple rank case is in [8]. It is assumed that A has already been permuted by a fill-reducing ordering; this is a large and critical topic in itself which is beyond the scope of this paper =-=[6]-=-. The supernodal Cholesky factorization method [2, 11, 21, 26, 28, 29] exploits dense matrix kernels during the factorization and solution of the resulting triangular systems. It is based on supernode... |

158 |
Methods for modifying matrix factorizations
- Gill, Golub, et al.
- 1974
(Show Context)
Citation Context ...in L that change [7]. This includes the time required to modify the nonzero pattern 5sof L, if the pattern needs to change. For additional background on the update/downdate problem, see (for example) =-=[3, 4, 17, 31, 32, 27]-=-. A simple rank-1 update/downdate of a sparse LL T factorization that does not change the nonzero pattern of L is discussed in detail in [6]; that algorithm is a mere 35 lines of C. The rank-1 update/... |

111 | Updating the inverse of a matrix
- Hager
- 1989
(Show Context)
Citation Context ...amic supernodal solve in which the supernodes are detected as the solve progresses. Update/downdate problems such as these arise in optimization algorithms, sensitivity analysis, and many other areas =-=[20]-=-. The sparse rank-1 update when L is not a supernodal Cholesky factorization is discussed in [7], while the multiple rank case is in [8]. It is assumed that A has already been permuted by a fill-reduc... |

107 | RAJAMANICKAM S.: Algorithm 887: Cholmod, supernodal sparse cholesky factorization and update/downdate - CHEN, DAVIS, et al. |

101 |
Block Sparse Cholesky Algorithms on Advanced Uniprocessor Computers
- Ng, Peyton
- 1993
(Show Context)
Citation Context ...ed that A has already been permuted by a fill-reducing ordering; this is a large and critical topic in itself which is beyond the scope of this paper [6]. The supernodal Cholesky factorization method =-=[2, 11, 21, 26, 28, 29]-=- exploits dense matrix kernels during the factorization and solution of the resulting triangular systems. It is based on supernodes, which are adjacent columns of L with identical nonzero pattern stor... |

67 |
de Geijn, On Reducing TLB Misses in Matrix Multiplication
- Goto, van
- 2002
(Show Context)
Citation Context ...a conventional supernodal triangular solver. These results were obtained on a Intel Pentium 4 (3.2GHz clock frequency, 4 GB RAM (DDR 333 Mhz), 512 KB cache, an 800 MHz memory bus, the Goto BLAS v1.05 =-=[18]-=-, and running Linux). The theoretical peak performance of the computer is 6.4 GFlops. The gcc compiler was used (version 3.3.5, with -O3 optimization). 5.1 Dynamic supernodal update/downdate Three mat... |

62 |
A new implementation of sparse Gaussian elimination
- Schreiber
- 1982
(Show Context)
Citation Context ...cisely, a supernode is defined by a chain of nodes in the elimination tree, and the sparsity pattern of the corresponding columns of L. The elimination tree of an n-by-n matrix A is a tree of n nodes =-=[23, 24, 30]-=-. The parent of node j in the tree is given by the first off-diagonal nonzero entry lij in column j, parent(j) = min{i | i > j and lij �= 0}. (1) If this set is empty, then node j is a root of the eli... |

51 | SPOOLES: An object-oriented sparse matrix library
- Ashcraft, Grimes
(Show Context)
Citation Context ...A has alreadybeen permuted by a fill-reducing ordering; this is a large and critical topic in itself which is beyond the scope of this paper [Davis 2006].The supernodal Cholesky factorization method [=-=Ashcraft and Grimes 1999-=-; Dobrian et al. 2000; H'enon et al. 2002; Ng and Peyton 1993; Rothberg and Gupta1991; Rotkin and Toledo 2004] exploits dense matrix kernels during the factorization and solution of the resulting tria... |

49 | Modifying a sparse Cholesky factorization
- Davis, Hager
- 1996
(Show Context)
Citation Context ...te problems such as these arise in optimization algorithms, sensitivity analysis, and many other areas [20]. The sparse rank-1 update when L is not a supernodal Cholesky factorization is discussed in =-=[7]-=-, while the multiple rank case is in [8]. It is assumed that A has already been permuted by a fill-reducing ordering; this is a large and critical topic in itself which is beyond the scope of this pap... |

47 | A numerical evaluation of sparse direct solvers for the solution of large sparse, symmetric linear systems of equations
- Gould, Hu, et al.
- 2005
(Show Context)
Citation Context ...7.2, x=A\b and chol use the above supernodal Cholesky factorization method, as implemented in CHOLMOD. The performance of CHOLMOD and many other sparse Cholesky factorization packages is discussed in =-=[19]-=-. 2.3 Supernodal solve Consider the triangular system Lx = b, ⎡ ⎢ ⎣ L11 L21 L22 L31 L32 L33 ⎤ ⎡ ⎤ ⎡ x1 ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ x2 ⎦ = ⎣ x3 b1 b2 b3 ⎤ ⎥ ⎦ , (5) where L is partitioned the same as in (3), and x is ... |

43 |
PaStiX: A high-performance parallel direct solver for sparse symmetric definite systems
- Hénon, Ramet, et al.
(Show Context)
Citation Context ...ed that A has already been permuted by a fill-reducing ordering; this is a large and critical topic in itself which is beyond the scope of this paper [6]. The supernodal Cholesky factorization method =-=[2, 11, 21, 26, 28, 29]-=- exploits dense matrix kernels during the factorization and solution of the resulting triangular systems. It is based on supernodes, which are adjacent columns of L with identical nonzero pattern stor... |

41 |
On finding supernodes for sparse matrix computations
- Liu, Ng, et al.
- 1993
(Show Context)
Citation Context ...ntical, and j - 1 need not be the only child of j, for example.Supernodes can be found without constructing the nonzero pattern of L, in timethat is essentially linear in the number of nonzeros of A [=-=Liu et al. 1993-=-]. First, theelimination tree of A is computed in nearly O(|A|) time [Liu 1986; 1990].1 Moreprecisely the time is O(| A|ff(|A|, n)) where ff is the inverse Ackerman function, afunction that grows extr... |

34 | The design and implementation of a new out-of-core sparse Cholesky factorization method
- Rotkin, Toledo
(Show Context)
Citation Context ...ed that A has already been permuted by a fill-reducing ordering; this is a large and critical topic in itself which is beyond the scope of this paper [6]. The supernodal Cholesky factorization method =-=[2, 11, 21, 26, 28, 29]-=- exploits dense matrix kernels during the factorization and solution of the resulting triangular systems. It is based on supernodes, which are adjacent columns of L with identical nonzero pattern stor... |

31 | The effect of rounding error on an algorithm for downdating a Cholesky factorization
- Stewart
- 1979
(Show Context)
Citation Context ...ed to modify the nonzero pattern of L, if the patternneeds to change. For additional background on the update/downdate problem, see (for example) [Bischof et al. 1993; Carlson 1973; Gill et al. 1974; =-=Stewart 1979-=-;1998; Pan 1990]. A simple rank-1 update/downdate of a sparse LLT factorizationthat does not change the nonzero pattern of L is discussed in detail in [Davis 2006];that algorithm is a mere 35 lines of... |

31 | An efficient algorithm to compute row and column counts for sparse cholesky factorization
- Gilbert, Ng, et al.
- 1994
(Show Context)
Citation Context ...tion tree (parent) and its postordering (q) using CHOLMOD. Once the tree is found and postordered, the number of entries in each column of L is found, using an algorithm that takes nearly O(|A|) time =-=[16]-=-. In MATLAB, this is computed by the routine symbfact, also using CHOLMOD. If count=symbfact(A), then count(j) = |Lj|. The column counts and the elimination tree are then used to find the fundamental ... |

31 |
Matrix Algorithms Volume 1: Basic Decompositions
- Stewart
- 1998
(Show Context)
Citation Context ...in L that change [7]. This includes the time required to modify the nonzero pattern 5sof L, if the pattern needs to change. For additional background on the update/downdate problem, see (for example) =-=[3, 4, 17, 31, 32, 27]-=-. A simple rank-1 update/downdate of a sparse LL T factorization that does not change the nonzero pattern of L is discussed in detail in [6]; that algorithm is a mere 35 lines of C. The rank-1 update/... |

24 |
A compact row storage scheme for Cholesky factors using elimination trees
- Liu
- 1986
(Show Context)
Citation Context ...cisely, a supernode is defined by a chain of nodes in the elimination tree, and the sparsity pattern of the corresponding columns of L. The elimination tree of an n-by-n matrix A is a tree of n nodes =-=[23, 24, 30]-=-. The parent of node j in the tree is given by the first off-diagonal nonzero entry lij in column j, parent(j) = min{i | i > j and lij �= 0}. (1) If this set is empty, then node j is a root of the eli... |

23 |
Efficient sparse matrix factorization on highperformance workstations: Exploiting the memory hierarchy
- Rothberg, Gupta
- 1991
(Show Context)
Citation Context |

18 | The design of sparse direct solvers using objectoriented techniques
- DOBRIAN, KUMFERT, et al.
(Show Context)
Citation Context |

17 | Algorithm 8xx: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate - Chen, Davis, et al. - 2006 |

13 | Multiple-rank modifications of a sparse cholesky factorization. SIAM Journal on Matrix Analysis and Applications 22, 4, 997–1013. 16: A gallery of digital paper models. Models were computed with the algorithm described in Section 5 with scans of real pape
- DAVIS, HAGE
(Show Context)
Citation Context ... and W. W. Hager many other areas [Hager 1989]. The sparse rank-1 update when L is not a su-pernodal Cholesky factorization is discussed in [Davis and Hager 1999], while the multiple rank case is in [=-=Davis and Hager 2001-=-]. It is assumed that A has alreadybeen permuted by a fill-reducing ordering; this is a large and critical topic in itself which is beyond the scope of this paper [Davis 2006].The supernodal Cholesky ... |

12 | On reducing tlb misses in matrix multiplication - Goto, Geijn - 2002 |

11 |
A modification to the LINPACK downdating algorithm
- Pan
- 1990
(Show Context)
Citation Context ...in L that change [7]. This includes the time required to modify the nonzero pattern 5sof L, if the pattern needs to change. For additional background on the update/downdate problem, see (for example) =-=[3, 4, 17, 31, 32, 27]-=-. A simple rank-1 update/downdate of a sparse LL T factorization that does not change the nonzero pattern of L is discussed in detail in [6]; that algorithm is a mere 35 lines of C. The rank-1 update/... |

10 |
A Cholesky up- and downdating algorithm for systolic and SIMD architectures
- Bischof, Pan, et al.
- 1993
(Show Context)
Citation Context ... [Davis and Hager 1999].This includes the time required to modify the nonzero pattern of L, if the patternneeds to change. For additional background on the update/downdate problem, see (for example) [=-=Bischof et al. 1993-=-; Carlson 1973; Gill et al. 1974; Stewart 1979;1998; Pan 1990]. A simple rank-1 update/downdate of a sparse LLT factorizationthat does not change the nonzero pattern of L is discussed in detail in [Da... |

5 | Experiences of sparse direct symmetric solvers - Scott, Hu - 2007 |

4 |
W.W.: Dual multilevel optimization
- Davis, Hager
- 2008
(Show Context)
Citation Context ...= AFAT F + βI was factorized, and then a rank-128 update was selected at random from the 13scolumns in A but not in AF. This procedure mimics the use of CHOLMOD in a linear programming solver, LPDASA =-=[10, 9]-=-. The method cannot be compared with CSparse, since the pattern of L is changing. Matrix name: Qaplib/lp nug15 source: linear programming problem n: 6330 |S|, lower triangular part: 129 × 10 3 orderin... |

3 |
Fast triangular factorization of the square root filter
- Carlson
- 1973
(Show Context)
Citation Context ...9].This includes the time required to modify the nonzero pattern of L, if the patternneeds to change. For additional background on the update/downdate problem, see (for example) [Bischof et al. 1993; =-=Carlson 1973-=-; Gill et al. 1974; Stewart 1979;1998; Pan 1990]. A simple rank-1 update/downdate of a sparse LLT factorizationthat does not change the nonzero pattern of L is discussed in detail in [Davis 2006];that... |

1 |
Dynamic supernodal update/downdate and triangular solve * 17
- Gill, Golub, et al.
- 1974
(Show Context)
Citation Context ...es the time required to modify the nonzero pattern of L, if the patternneeds to change. For additional background on the update/downdate problem, see (for example) [Bischof et al. 1993; Carlson 1973; =-=Gill et al. 1974-=-; Stewart 1979;1998; Pan 1990]. A simple rank-1 update/downdate of a sparse LLT factorizationthat does not change the nonzero pattern of L is discussed in detail in [Davis 2006];that algorithm is a me... |

1 |
A sparse proximal implementation
- Davis, Hager
- 2006
(Show Context)
Citation Context ... only added. During a downdate, A = A − ww T , entries could be dropped (but not added) if A = CC T and w is one of the columns of C. The driving motivation is an active-set linear programming method =-=[10, 9]-=-, where C is the matrix comprised of the columns corresponding to the active variables. Dropping entries requires the nonzero pattern of L to be held as a multi-set, with multiplicities for each entry... |

1 | Article 27, Pub. date: February 2009. Supernodes in Cholesky Up/Downdate and Triangular Solves - DAVIS, HAGER |