J. G. Lewis, D. J. Pierce, and D. K. Wah, Multifrontal Householder QR factorization, Tech. Rep. ECA-TR-127-Revised, Boeing Computer Services, Seattle, WA, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....proposed efficient scheme for sparse QR factorization by Givens rotations, makes the paper by George and Heath an important starting point for today s algorithms. References on the way from George and Heath s paper to modern multifrontal methods include Liu [22] George and Liu [13] Lewis et al. [20], Puglisi [29] and Matstoms [26] There are essentially three important features characterizing the multifrontal method for sparse QR factorization. First, flexibility in the elimination order of the columns is provided by the underlying elimination tree. This tree gives information about ....

J. G. Lewis, D. J. Pierce, and D. K. Wah, Multifrontal Householder QR factorization, Tech. Rep. ECA-TR-127-Revised, Boeing Computer Services, Seattle, WA, 1989.

....(necessarily dynamic) structures were allowed to exist simultaneously. Givens rotations are also used in Liu s scheme. George and Liu (1987) presented a modified version of Liu s algorithm, consisting in the use of a row oriented version of the Householder reflections instead of Givens rotations. Lewis, Pierce, and Wah (1989) developed a multifrontal QR based on Householder transformations which differs from the modified version of the row merging scheme of Liu essentially in one aspect. Liu restricts himself to merging either a triangle or a collection of rows at a time whereas Lewis et al. gather all such triangles ....

....126736 90x90 grid 335440 Nfac100 39204 10000 156816 100x100 grid 477500 large 28254 17264 75018 animal science 200938 medium2 18794 12238 75039 animal science 535670 large2 56508 34528 225054 animal science 2380788 Table 1. Test matrices. The matrices called Nfacxxx in Table 1 were used by Lewis, Pierce, and Wah (1989) and arise in the natural formulation of the finite element method (Argyris and Bronlund (1975) Consider a regular k by k grid with (k Gamma 1) 2 elements. With each element there is an associated set of s equations involving the four variables corresponding to the corners. The assembling of ....

J. G. Lewis, D. J. Pierce, and D. C. Wah, (1989), A Multifrontal Householder QR Factorization, Technical Report ECA-TR-127, Engineering and Scientific Services Division, Boeing Computer Services.

....90x90 grid with 4 eqns elements Nfac100 39204 10000 156816 100x100 grid with 4 eqns elements large 28254 17264 75018 animal science medium2 18794 12238 75039 animal science large2 56508 34528 225054 animal science Table 6: Test matrices. The matrices called Nfacxxx in Table 6 were used by Lewis, Pierce, and Wah (1989) and arise in the natural formulation of the finite element method (Argyris and Bronlund (1975) The other test problems in Table 6 are rectangular matrices arising in animal breeding science for the estimation of breeding values. The data was used for pig breeding in Switzerland and was supplied ....

J. G. Lewis, D. J. Pierce, and D. C. Wah, (1989), A Multifrontal Householder QR Factorization, Technical Report ECA-TR-127, Engineering and Scientific Services Division, Boeing Computer Services.

....up the matrix R. Although they mention the possibility of storing the sequence of Givens rotations on an external file, they simply discard the orthogonal transformations which make up Q. A similar approach is taken in later multifrontal sparse QR algorithms, e.g. by Lewis, Pierce and Wah [24], Matstoms [28, 29] and Sun [38] The multifrontal method, which will be discussed further below, can be considered as a generalization of Reid s method for banded matrices. 2.2 Row and column orderings The R factor from the QR factorization can also be interpreted as the (unique) Cholesky ....

....of rows of each A[i] is bounded by a constant. Then Lu and Barlow show that their method requires only O(n log n) storage if A 2 IR m Thetan is defined on a p n separable graph 3 The multifrontal QR method The multifrontal method was adapted for the QR factorization by Liu [25] Lewis et al. [24], Puglisi [34] Matstoms [27, 28] and Sun [37] For a more complete treatment of the multifrontal QR method we refer to Matstoms [29] A useful tool for analyzing the multifrontal QR factorization of a sparse matrix A is the elimination tree of A T A. Definition 3.1 (Elimination tree) The ....

J. G. Lewis, D. J. Pierce, and D. K. Wah. Multifrontal Householder QR factorization. Technical Report ECA-TR-127-Revised, Boeing Computer Services, 1989.

....remarks are contained in x7. 2. Parallel multifrontal sparse QR factorization. Typically, the first task in a direct method for solving sparse linear least squares problems is to compute a sparse QR factorization. The multifrontal method has proved to be effective for sparse QR factorization [6, 9, 12, 14]. Parallel implementations of multifrontal sparse QR factorization have been discussed in [3, 15, 19] Sparse QR factorization involves the following steps: 1. Find a permutation matrix P such that AP has a sparse upper triangular factor R. 2. Determine the symbolic structure of R. 3. Perform ....

J. G. Lewis, D. J. Pierce, and D. K. Wah, Multifrontal Householder QR factorization, Tech. Report ECATR -127, Boeing Computer Services, Seattle, WA, November 1989.

....remarks are contained in x7. 2. Parallel multifrontal sparse QR factorization. Typically, the first step in a direct method for solving a sparse linear least squares problem is to compute a sparse QR factorization. Multifrontal method is shown to be effective for sparse QR factorization in [6, 11, 14, 15]. Parallel implementation of the multifrontal method is extensively discussed in literature(e.g. 2, 16, 20] The parallel multifrontal sparse QR factorization algorithm proposed in [20] is used in this work to perform parallel sparse QR factorization. The parallel multifrontal sparse QR ....

J. G. Lewis, D. J. Pierce, and D. K. Wah, Multifrontal Householder QR factorization, Tech. Report ECATR -127, Boeing Computer Services, Seattle, WA, November 1989.

....if m is much larger than n. Other results on the nonzero structures of the Householder matrix H and the orthogonal factor Q for a sparse matrix using G (A) are, for example, given in [14, 26] In this paper, we study the computation of orthogonal factor using the multifrontal QR factorization [20, 24]. Associated with each row of the upper triangular factor R , is a frontal matrix F i . Likewise for each F i , there is a frontal Householder matrix Y i . Note that Y i is the H matrix for F i . Figure 1 is a small sample matrix A and its column intersection graph. Figure 2 is the Householder ....

....the p n separator problems, the multifrontal Householder QR factorization, and the application of supernodes. Section 3 proposes a multifrontal based method for computing Q T b. Section 4 proves an upper bound on the nonzero counts of all Y i s. This section builds on the work of Lewis et.al. [20] for K by K grid problem. We extend their result to the p n separator problem. Section 5 introduces BLAS 2 operations in computing Q T b by using the YTY representation of Schreiber and Van Loan [29] for the orthogonal factor Q i of each frontal matrix F i . The upper 6 S. M. LU AND J. L. ....

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J. G. Lewis, D. J. Pierce, and D. K. Wah, Multifrontal Householder QR factorization, Technical Report ECA-TR-127, Boeing Computer Services, Seattle WA, Nov. (1989).

....proposed efficient scheme for sparse QR factorization by Givens rotations, makes the paper by George and Heath an important starting point for today s algorithms. References on the way from George and Heath s paper to modern multifrontal methods include Liu [22] George and Liu [13] Lewis et al. [20], Puglisi [29] and Matstoms [26] There are essentially three important features characterizing the multifrontal method for sparse QR factorization. First, flexibility in the elimination order of the columns is provided by the underlying elimination tree. This tree gives information about ....

J. G. Lewis, D. J. Pierce, and D. K. Wah, Multifrontal Householder QR factorization, Tech. Rep. ECA-TR-127-Revised, Boeing Computer Services, Seattle, WA, 1989.

....elimination tree. The third and last step also driven by the elimination tree, is the solver phase, where the solution is computed. In fact our analysis phase generates a tree with multiple eliminations at each node so, strictly speaking, we use an assembly tree rather than an elimination tree. Lewis, Pierce, and Wah (1989) have developed a multifrontal QR factorization algorithm based on Householder transformations. Following George and Heath (1980) they build the elimination tree by performing a symbolic Cholesky factorization on the matrix A T A. In fact, there is a close relationship between the Cholesky ....

....the actual R factor. In Section 2, we describe the multifrontal QR factorization showing how the assembly tree built during the symbolic Cholesky factorization of the matrix A T A, can drive the QR factorization of the matrix A. In other implementations of the multifrontal QR factorization (Lewis, Pierce, and Wah (1989) and Matstoms (1994) the orthogonal transformations are discarded as soon as they have been used. We store the factors Q and thus we are also interested in the storage required for the Q matrix and in the computational work for performing the sparse matrix vector product Q T b. It is well known ....

[Article contains additional citation context not shown here]

J. G. Lewis, D. J. Pierce, and D. C. Wah, (1989), A Multifrontal Householder QR Factorization, Technical Report ECA-TR-127, Engineering and Scientific Services Division, Boeing Computer Services.

....but we must pay the cost of dynamic data structures for the factor R. Our algorithm uses the multifrontal QR algorithm as the basis of the factorization. The reader is assumed to have some familiarity with this sparse QR algorithm, first introduced by Liu [27] other references include [18] [25], 30] The multifrontal algorithm was chosen for efficiency. It has a low operation count, compared to the classical work of George and Heath [16] and it organizes the computations to take advantage of dense matrix computational kernels [11] 25] 27] We are able to exploit the multifrontal ....

....by Liu [27] other references include [18] 25] 30] The multifrontal algorithm was chosen for efficiency. It has a low operation count, compared to the classical work of George and Heath [16] and it organizes the computations to take advantage of dense matrix computational kernels [11] [25], 27] We are able to exploit the multifrontal paradigm even as we allow dynamic changes in the the factor R. The columns of A are initially reordered to preserve sparsity, using the structure of A T A, as first proposed in [16] The QR factorization is computed one row at a time. At the same ....

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J.G. Lewis, D.J. Pierce and D.K. Wah, Multifrontal Householder QR factorization, Technical Report ECA-TR-127, Boeing Computer Services, 1989.

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