P. MATSTOMS. Sparse QR factorization in MATLAB. ACM Trans. Math. Software, 20:136--159, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....usually determined empirically. The e#ect of smoothing is demonstrated in Chapter 5. In practice most of the singular values do not diminish to zero, and due to the unique smoothing constraints for each table entry, col(B) usually has a full rank. Thus, we may consider a sparse QR factorization [18] of the matrix B instead of the SVD. The QR factorization may be faster than the SVD, if the algorithm takes advantage of the sparsity. The sparsity may also allow for more compact representation of the factorized matrix. Factorization As Change of Basis Finding the BRDF approximation from the ....

Pontus Matstoms. Sparse QR Factorization in MATLAB. In Transactions on Mathematical Software, 1994.

....i ) from the same class of matrices that Gilbert, et al. considered. They obtained the following bound on the number of nonzeros by storing the Householder vectors from the frontal matrices, nnz(Y ) O(n log n) In last two column of Table 3. 2 we have used a modified version of sqr, by Matstoms [48], to compute nnz(Y ) As could be expected we see a dramatic reduction of the storage needed for a representation of Q. The number of nonzero for the Q matrix obtained from Matlab is unnecessary large, only the n first columns are needed in general. 3.4.2 Sparse Cholesky factorization Since the ....

P. Matstoms. Sparse QR factorization in MATLAB. ACM Trans. Math. Software, 20:136--159, 1994.

....a priori. The algorithms are based on the elimination directed acyclic graph (dag) and its reductions, which are similar to the assembly dag presented in this paper. Moreover, we also indicate how our graphs can be applied to the case where the pivot ordering is not known a priori. Also, Matstoms [28] has recently developed a multifrontal QR factorization algorithm. 5 1.2 Outline The following sections present the unsymmetric pattern multifrontal method and three sequential algorithms based on the method. Section 2 describes LU factorization in terms of general frontal matrices. Section 3 ....

P. Matstoms. Sparse QR factorization in MATLAB. Technical Report LiTH-MAT-R1992 -05, Dept. of Mathematics, Linkoping Univ., Linkoping, Sweden, March 1992.

....0 Geodesy problem 5 ash958 958 292 1916 6:9 Delta 10 0 Geodesy problem 6 wl1033 1033 320 4732 1:7 Delta 10 2 Gravity meter obs. 7 wl1850 1850 712 8758 1:1 Delta 10 2 Gravity meter obs. In all numerical experiments we used the sparse QR factorization code for Matlab described by Matstoms [10]. Each full row was split into k = 1, 2, 4, 128 rows. A minimum degree column ordering with tight setting was used in all the factorizations. Note that it is very important to perform the minimum degree ordering after the rows have been split and not on the original matrix A. Figure 5.1 ....

P. Matstoms, Sparse QR factorization in MATLAB, ACM Trans. Math. Software, 20 (1994), pp. 136--159.

....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 factor of A T A, ....

....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 elimination tree, T (M) of a symmetric ....

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P. Matstoms. Sparse QR factorization in MATLAB. ACM Trans. Math. Software, 20:136--159, 1994.

....correct ordering; our programs of course perform this ordering automatically, using the multiple minimum degree ordering genmmd as used in Sparsepak [5] Note that R is the transposed Cholesky factor of B. Alternatively, one can obtain R from a sparse QR factorization of A, see, e.g. Matstoms [39]. To take care of dependent (or nearly dependent) linear equations in the model formulation, we replace in the factorization small pivots B ii by 1. The choice = macheps) 2=3 , where macheps is the machine accuracy, proved to be suitable. The exponent is less than 1 to allow for some ....

P. Matstoms. Sparse QR factorization in MATLAB. ACM Trans. Math. Software 20 (1994), 136--159.

....of the oscillations of these quantities. The resulting rough parameters are then fitted to the data using a least squares approach. The state of the art of numerical methods for least squares calculations is surveyed in Bj orck [22, 23] for the linear case. In addition, recent work by Matstoms [201] on multifrontal orthogonal factorizations for large and sparse least squares problems is relevant. The nonlinear case is reduced to the linear case, most commonly by means of damped Gauss Newton steps, see, e.g. Dennis Schnabel [74] Fletcher [97] For a survey of parameter fitting procedures ....

P. Matstoms, Sparse QR Factorization in MATLAB, ACM Trans. Math. Software 20 (1994), pp. 136--159.

....assumption of a symmetric nonzero pattern, and so has a poor performance on matrices whose patterns are very unsymmetric. None of the previous parallel methods for unsymmetric patterned matrices use dense matrix kernels [2, 10, 11] with the exception of a multifrontal QR factorization algorithm [15] (which will be compared later on with the algorithms we develop) available via anonymous ftp to cis.ufl.edu as cis tech reports tr92 tr92 014.ps.Z This paper presents a new unsymmetric pattern multifrontal method that takes full advantage of dense matrix kernels, maintains a purely ....

P. Matstoms. Sparse QR factorization in MATLAB. Technical Report LiTH-MAT-R-1992-05, Dept. of Mathematics, Linkoping Univ., Linkoping, Sweden, March 1992.

....matrices. Gilbert et al. 1992) have introduced a sparse matrix structure and some sparse algorithms into MATLAB. Their aim has been ease of use and functionality rather than efficiency, although increasingly researchers are making codes available to MATLAB users through M files (for example, Matstoms 1994). Saad (1994b) has developed, over many years, a tool kit called SPARSKIT for sparse matrix computations, Gupta and Rothberg (1994) have proposed an environment for handling sparse matrices on distributed memory machines, and Alvarado (1989) has designed an integrated package as a teaching and ....

Matstoms, P. (1994), `Sparse QR factorization in MATLAB', ACM Trans. Math. Softw. 20, 136--159.

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P. Matstoms, Sparse QR factorization in MATLAB, ACM Trans. Math. Software, 20 (1994), pp. 136--159.

....squares problems. The coefficient matrix is then assumed to be large with relatively few nonzero entries. The introduction of multifrontal methods have made direct methods based on sparse QR factorization attractive and competitive to previously recommended alternatives, as explained by Matstoms [26]. In this paper, we consider the QR factorization, A = Q of a large and sparse Strong Hall matrix A 2 R of full column rank, where m n. The column ordering on A is decisive for the efficiency of the factorization. Researchers This research is partly supported by The Research Council of ....

....for sparse least squares, the main alternatives are the normal equations, the augmented system method and different variants of QR factorization. Methods based on QR factorization have, during the last decade, become highly interesting and recommended for sparse least squares problems (Matstoms [26]) We should keep in mind that, until the early 80s, QR factorization was rejected for general sparse problems (Duff and Reid [6] The main explanation for why QR factorization has become useful, is the multifrontal technique. This technique makes it possible to use Householder transformations ....

[Article contains additional citation context not shown here]

P. Matstoms, Sparse QR factorization in MATLAB, ACM Trans. Math. Software, 20 (1994), pp. 136--159.

No context found.

P. Matstoms, Sparse QR factorization in MATLAB, ACM Trans. Math. Software, 20 (1994), pp. 136--159.

....squares problems. The coefficient matrix is then assumed to be large with relatively few nonzero entries. The introduction of multifrontal methods have made direct methods based on sparse QR factorization attractive and competitive to previously recommended alternatives, as explained by Matstoms [26]. In this paper, we consider the QR factorization, A = Q R 0 ; of a large and sparse Strong Hall matrix A 2 R m Thetan of full column rank, where m n. The column ordering on A is decisive for the efficiency of the factorization. Researchers This research is partly supported by The ....

....for sparse least squares, the main alternatives are the normal equations, the augmented system method and different variants of QR factorization. Methods based on QR factorization have, during the last decade, become highly interesting and recommended for sparse least squares problems (Matstoms [26]) We should keep in mind that, until the early 80s, QR factorization was rejected for general sparse problems (Duff and Reid [6] The main explanation for why QR factorization has become useful, is the multifrontal technique. This technique makes it possible to use Householder transformations ....

[Article contains additional citation context not shown here]

P. Matstoms, Sparse QR factorization in MATLAB, ACM Trans. Math. Software, 20 (1994), pp. 136--159.

No context found.

P. MATSTOMS. Sparse QR factorization in MATLAB. ACM Trans. Math. Software, 20:136--159, 1994.

No context found.

P. MATSTOMS. Sparse QR factorization in MATLAB. ACM Trans. Math. Software, 20:136-159, 1994.

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