J. A. George and M. T. Heath. Solution of Sparse Linear Least Squares Problems Using Givens Rotations. Lin. Alg. and Its Applic., 34:69--83, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....each column subspace. This algorithm, which is called column oriented successive subspace correction (CSSC) in [8] is, in fact, Gauss Seidel iteration on the normal equations. Each subproblem is substantially smaller than the original problem and hence is solved directly using QR factorization [7] and semi normal equations (SNE) 3] on one processor. Each processor computes a correction of the solution vector restricted to the variables associated with the blocks of columns. This research is supported by The Research Council of Norway. Accepted to Euro Par 97, August 26 29, 1997, ....

George, J. A., Heath, M. T.: Solution of Sparse Linear Least Squares Problems Using Givens Rotations. Lin. Alg. Appl. 34 (1980) 69--83

....need much more space and time than the normal equations method. Q is usually much denser than A. Although Q can be discarded, large storage may still be needed for the computation. Several sparse orthogonal factorization algorithms have been suggested [31] 46] 3O] 32] 34] 53] 18] 64] [29] [49] If multiple solutions have to be computed for the same A and multiple b, where not all b are available at the same time, and if Q is not stored, the seminormal equations method can be used. It computes the orthogonal factorization of A for R and solves IT Ix = ATb. 2.2) It would seem ....

....Under mild conditions, a correction step can be added to yield a solution as accurate as the QR method. As with the normal equations, if there are a few dense rows in A, severe fill in occurs in R. Usually, such rows are treated separately by using technique of updating a mR factorization [37] 35][29]. 2.3 An elimination method A method based on an LU factorization can be described as follows: Compute PAP = LU. olv LT Ly LT Pb. olv upT xs . y. where PAP LU is an LU factorization with complete pivoting. P, P are permutation matrices. The method was first proposed by Peters ....

A. George and M. T. Heath. Solution of sparse linear least squares problems using Givens rotations. Linear Algebra and its Applications, 34:69 83, 1980.

....is computed which reduces [A; b] to the form Q ; Q b = c d where R 2 IR n Thetan and c 2 IR . Matrix R can be computed row sequentially using Givens rotations. Givens rotations should be applied simultaneously to b to form Q b. In the implementation by George and Heath [6] the Givens rotations are not stored but discarded after use. Hence, only enough storage to hold the final R and a few extra vectors for the current row and right hand side is needed in the main memory during the factorization phase. Discarding Q creates a problem since we wish to solve ....

J. A. George, M. T. Heath, Solution of Sparse Linear Least Squares Problems Using Givens Rotations, Lin. Alg. Appl. 34, pp. 69--83 (1980).

....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 Norway. have concentrated on finding column orderings that result in as sparse R as possible. George and Heath [12] have observed that a suitable column ordering of A can be found by symmetric fill reducing analysis of A A. Efficient methods, like minimum degree and nested dissection, exist for fill reduction in symmetric factorization. Moreover, it is shown by George and Liu [14] that minimum degree ....

....inner product with column k, takes the sparsity pattern of the union of itself with column k. Proposition 2.2 (Fill in sparse Givens QR factorization) The sparsity patterns of two rows involved in a Givens transformation are replaced by the union of their nonzero patterns. George and Heath [12] show that Givens prediction is always at least as good as symbolic Cholesky factorization. The same is true for symbolic Householder prediction, as shown by Manneback [24] Altogether this can be summarized as follows: struct(R) ae pred(Householder on A) pred(Givens on A) oe ....

[Article contains additional citation context not shown here]

J. A. George and M. T. Heath, Solution of sparse linear least squares problems using Givens rotations, Linear Algebra and Its Applications, 34 (1980), pp. 69--83.

....in iqrd The computation of the QR factorization is not recommended here, since besides being more expensive than a Cholesky factorization, it also destroys the sparsity of the matrix A. Instead, Portugal et al. 116] propose an incomplete QR decomposition of A. Applying Givens rotations [39] to A, using the diagonal elements of D 1 2 T A T , the elements of D 1 2 T A T become null. No fill in is incurred in this factorization. See [116] for an example illustrating this procedure. After the factorization, we have the preconditioner M = FDF ; where F is a matrix ....

A. George and M. Heath. Solution of sparse linear least squares problems using Givens rotations. Linear Algebra and Its Applications, 34:69--83, 1980.

....re nes this upper bound on a supernode s storage dynamically during the LU factorization [17] 3.4 Row and column counts of the upper triangular factor We turn our attention to the upper triangular factor R in the QR factorization of A. This R is the transpose of the Cholesky factor of A T A [9]. Thus we could simply take B = A T A in the Cholesky count algorithm (Section 3.1) The only diculty is that A T A may have many more nonzeros than A, so the running time might be far from linear in jAj. Therefore, instead of taking B = A T A, we choose a B that has at most jAj nonzeros ....

A. George and M. T. Heath. Solution of sparse linear least squares problems using Givens rotations. Linear Algebra Appl., 34:69-83, 1980.

....each of the diagonal blocks. Both classes of 2 matrices have been studied in Puglisi (1993) To illustrate our discussion and for the sake of simplicity we focus, in this paper, on test matrices for which the symbolic factorization of A T A predicts the exact pattern of the actual R factor. George and Heath (1980) were the first exploiting the equality L T = R, but Liu (1986) first introduced the notion of a row merge tree for the sparse orthogonal factorization of a rectangular matrix. The row merging scheme can actually be viewed as the use of the multifrontal method on M = A T A without ever forming ....

....number of operations to be performed, the size of temporary storage and the space required for storing the Householder vectors. It is known that the production of transient fill in during the QR decomposition depends on the row ordering 6 during the factorization (Duff (1974) Gentleman (1976) George and Heath (1980)) It is probably less known that it depends also on the way the orthogonal transformations are applied at each major step. Reid s algorithm (Reid (1967) for banded systems using Householder transformations is an example of reducing transient fill in by taking account of structure when ....

J. A. George and M. T. Heath, (1980), Solution of Sparse Linear Least Squares Problems Using Givens Rotations, Linear Algebra and Its Application, 34, 69--83.

.... situations, but it is not exploited in this paper (because we are interested in obtaining much bigger parallel tasks) There are two versions of the Givens method, classical Givens [16] and GentlemanGivens [12] The classical Givens version was used by George and Heath in the package SPARSPAK B [13, 14, 15]. The Gentleman Givens version (also known as fast Givens) is more economic with regard to floating point operations, but has some overhead and may occasionally require rescaling in an attempt to prevent overflows. It has been applied in the code LLSS by Zlatev [30] The classical Givens version ....

J. A. George and M. T. Heath, Solution of sparse linear least squares problems using Givens rotations, Lin. Alg. Appl., Vol. 34, 1980, pp. 69--73.

....A = QR, where Q is an orthogonal matrix and R is upper triangular with nonnegative diagonal. George and Heath observed that, since this R is the same as the Cholesky factor of A T A, the structure of R can be predicted by forming G (A) and doing structural Cholesky factorization. Theorem 4. 7 ([14]) Let the structure H(A) be given for a rectangular matrix A with at least as many rows as columns. Whatever values A has, if A has full column rank then its orthogonal factorization A = QR satisfies G(R) G (A) The converse of this theorem is false; for example, if A is upper triangular ....

Alan George and Michael T. Heath. Solution of sparse linear least squares problems using Givens rotations. Linear Algebra and its Applications, 34:69--83, 1980.

....m Theta n, with m n. It is often necessary to compute the orthogonal factorization A = QR. Our algorithms can be used also to predict the row counts and column counts of the upper triangular factor R, since the structure of R is always contained in the structure of the Cholesky factor of A T A [12]. Throughout the paper we assume familiarity with graphs, trees, and such basic techniques as depth first search [26] We also assume a basic knowledge of the four steps in solving sparse systems by Cholesky factorization, and with the use of graphs in these algorithms [15] More specifically, we ....

Alan George and Michael T. Heath. Solution of sparse linear least squares problems using Givens rotations. Linear Algebra and its Applications, 34:69--83, 1980.

....each column subspace. This algorithm, which is called column oriented successive subspace correction (CSSC) in [8] is, in fact, Gauss Seidel iteration on the normal equations. Each subproblem is substantially smaller than the original problem and hence is solved directly using QR factorization [7] and semi normal equations (SNE) 3] on one processor. Each processor computes a correction of the solution vector restricted to the variables associated with the blocks of columns. This research is supported by The Research Council of Norway. Accepted to Euro Par 97, August 26 29, 1997, ....

George, J. A., Heath, M. T.: Solution of Sparse Linear Least Squares Problems Using Givens Rotations. Lin. Alg. Appl. 34 (1980) 69--83

....of two specified coordinate axes; a Givens rotation can be chosen to annihilate any single entry of the vector. We consider three algorithms to compute R from A in the sparse setting: row Givens, column Givens, and column Householder . The sparse row Givens algorithm is due to George and Heath [8]. They first predict the nonzero structure of R, and set up a static data structure to hold R. Then they annihilate nonzeros from one row of A at a time, processing each row until either it becomes completely zero, or its structure fits into an empty row of the static data structure. This approach ....

....weaker bound is tight, and hence in that case the two bounds are the same. If A = QR then A T A = R T Q T QR = R T R. Thus (the upper triangular part of) R is equal to the Cholesky factor of the normal equations matrix A T A (which is symmetric and positive definite) George and Heath [8] used this fact in their implementation of sparse orthogonal factorization by Givens rotations. They predict the structure of A T A to be the column intersection graph G (A) which has a nonzero in position (i; j) whenever columns i and j of A have a common nonzero row; then they predict the ....

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Alan George and Michael T. Heath. Solution of sparse linear least squares problems using Givens rotations. Linear Algebra and its Applications, 34:69--83, 1980.

....has good separators. Section 3 gives upper bounds on jQj, jH j, and jRj in the case where the only restriction on A is that its column intersection graph has good separators. These upper bounds follow in a straightforward manner from results in George, Heath, Lipton, Liu, Ng, Rose, and Tarjan [8, 9, 11, 22]. In Section 4 we show that the upper bounds given in Section 3 are the best possible, at least for a large class of matrices. The argument combines some new analysis of separators in the intersection graph of A with 3 a sequence of results of Coleman, Edenbrandt, Gilbert, Hare, Johnson, Ng, ....

....an example of a filled column intersection graph. The filled column intersection graph represents the nonzero structure of the Cholesky factor of the normal equations matrix A T A. Since that Cholesky factor is equal to R in the QR factorization, R R T G (A) 2) George and Heath [8] use this containment to design data structures for sparse QR factorization. The conditions under which we have equality in (2) are subtle; several of the references study this question, and we return to it in Section 4. In cases where equality in (2) occurs with probability one, we again call ....

Alan George and Michael T. Heath. Solution of sparse linear least squares problems using Givens rotations. Linear Algebra and its Applications, 34:69--83, 1980.

....fill in and Q can be stored implicitly as the Householder matrices H k of steps k = 1; q. It is elementary to verify that the number of nonzero elements in A, R, and H are nnz(A) mw, nnz(R) nw Gamma 1 2 w(w Gamma 1) and P nnz(H k ) mw, respectively. George and Heath [12] showed that for general sparse matrices the problem with intermediate fill in could be avoided by using Givens rotations, processing the matrix A row by row, and gradually building up the matrix R. Although they mention the possibility of storing the sequence of Givens rotations on an external ....

....to operations and intermediate fill in is to sort the rows of A after increasing order with respect to minimum column subscript. This will also give an efficient representation of Q in terms of Householder matrices of the factorizations of sub blocks. In the Givens method of George and Heath [12] the heuristic ordering used is to sort the rows into increasing order with respect to the maximum column subscript. They compare this good ordering with a bad row ordering (the reverse of the good ordering) They give examples where using the bad ordering the number of operations for the ....

J. A. George and M. T. Heath. Solution of sparse linear least squares problems using Givens rotations. Linear Algebra Appl., 34:69--83, 1980.

....a solution to the linear system. When the matrix A is sparse, the numeric computations are preceded by a symbolic step to order the columns so that the factor suffers low fill, i.e. few of the zeroes of A become nonzero. The column ordering can be induced by applying a fill reducing heuristic [14, 16] to G (A) a column intersection graph of A which represents the structure of the matrix A T A under assumption of no numeric cancellation. A fully parallel solution requires parallelization of both symbolic and numeric steps. Computing an ordering in parallel is a very hard problem but ....

....elimination and to R for orthogonal factorization. We also use NCND to denote CND for nonsymmetric matrices and P to denote the total number of processors. 1.1. Related research. Our work is based on methods developed earlier by several researchers for solving nonsymmetric systems. A central idea [14, 16] is that of formulating the symbolic steps in terms of those for computing the Cholesky factor of A T A. Serial algorithms to compute a fill reducing ordering for a nonsymmetric sparse matrix A and to predict the structure of the factor using an elimination tree have been developed by several ....

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J. A. George and M. T. Heath, Solution of sparse linear least squares problems using Givens rotations, Linear Alg. Appl., 88 (1980), pp. 223--238.

....overly expensive in storage and has the benefit of good stability but within a subsequent static data structure. There are, of course, cases when the storage penalty is high. Methods using a sparse QR factorization can be used for general unsymmetric systems. These are usually based on work of George and Heath (1980) and first obtain the structure of R through a symbolic factorization of the structure of the normal equations matrix A T A. It is common not to keep Q but to solve the system using the semi normal equations R T Rx = A T b; with iterative refinement usually used to avoid numerical ....

George, A. and Heath, M. T. (1980), `Solution of sparse linear least squares problems using Givens rotations', Linear Algebra and its Applications 34, 69--83.

....There are several algorithms for finding this decomposition: Householder reflections or transformations, Givens rotations and Modified Gram Schmidt algorithm. Different results obtained for dense matrices can be found in [4, 18, 16, 17] on distributed memory machines; or for sparse matrices in [15, 13, 9, 14]. The algorithm we will implement is the one based on Householder transformations [10, chapter 5] with column pivoting in order to contemplate those cases in which the rank of matrix M is not maximum (B) Pivoting will also be used in order to provide numerical stability. This algorithm obtains a ....

J.A. George and M.T. Heath, Solution of Sparse Linear Least Squares Problems using Givens Rotations, Linear Algebra Appl., Vol.34, pp.69-83, 1980.

....edges in the graph. This ability to identify independent rows will be needed in x 4. Also, the rigidity matrix is quite sparse, having only 2d nonzeros in each row. To save time and space, sparse techniques could be used for large problems. There are sparse QR algorithms, but none for SVD [8, 16, 17]. There are also efficient parallel algorithms for finding the rank of a matrix. Ibarra, Moran and Rosier [23] discovered an algorithm that runs in O(log 2 m) time on O(m 4 ) processors. This means that rigidity testing is in random NC for any dimension. The class NC is the set of problems ....

A. George and M. T. Heath, Solution of sparse linear least squares problems using Givens rotations, Linear Algebra Appl., 34 (1980), pp. 69--83.

....need much more space and time than the normal equations method. Q is usually much denser than A. Although Q can be discarded, large storage may still be needed for the computation. Several sparse orthogonal factorization algorithms have been suggested [31] 46] 30] 32] 34] 53] 18] 64] [29] [49] If multiple solutions have to be computed for the same A and multiple b, where not all b are available at the same time, and if Q is not stored, the seminormal equations method can be used. It computes the orthogonal factorization of A for R and solves R T Rx = A T b: 2:2) It would ....

....Under mild conditions, a correction step can be added to yield a solution as accurate as the QR method. As with the normal equations, if there are a few dense rows in A, severe fill in occurs in R. Usually, such rows are treated separately by using technique of updating a QR factorization [37] 35][29]. 2.3 An elimination method A method based on an LU factorization can be described as follows: Algorithm: begin Compute P 1 AP 2 = LU . Solve L T Ly = L T P 1 b. Solve UP T 2 xLS = y. end where P 1 AP 2 = LU is an LU factorization with complete pivoting. P 1 , P 2 are permutation ....

A. George and M. T. Heath. Solution of sparse linear least squares problems using Givens rotations. Linear Algebra and its Applications, 34:69--83, 1980.

....reduces [A; b] to the form Q T A = R 0 ; Q T b = c d ; where R 2 IR n Thetan and c 2 IR n . Matrix R can be computed row sequentially using Givens rotations. Givens rotations should be applied simultaneously to b to form Q T b. In the implementation by George and Heath [6] the Givens rotations are not stored but discarded after use. Hence, only enough storage to hold the final R and a few extra vectors for the current row and right hand side is needed in the main memory during the factorization phase. Discarding Q creates a problem since we wish to solve additional ....

J. A. George, M. T. Heath, Solution of Sparse Linear Least Squares Problems Using Givens Rotations, Lin. Alg. Appl. 34, pp. 69--83 (1980).

....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 Norway. have concentrated on finding column orderings that result in as sparse R as possible. George and Heath [12] have observed that a suitable column ordering of A can be found by symmetric fill reducing analysis of A T A. Efficient methods, like minimum degree and nested dissection, exist for fill reduction in symmetric factorization. Moreover, it is shown by George and Liu [14] that minimum degree ....

....inner product with column k, takes the sparsity pattern of the union of itself with column k. Proposition 2.2 (Fill in sparse Givens QR factorization) The sparsity patterns of two rows involved in a Givens transformation are replaced by the union of their nonzero patterns. George and Heath [12] show that Givens prediction is always at least as good as symbolic Cholesky factorization. The same is true for symbolic Householder prediction, as shown by Manneback [24] Altogether this can be summarized as follows: struct(R) ae pred(Householder on A) pred(Givens on A) oe ....

[Article contains additional citation context not shown here]

J. A. George and M. T. Heath, Solution of sparse linear least squares problems using Givens rotations, Linear Algebra and Its Applications, 34 (1980), pp. 69--83.

....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 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 ....

....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 time we estimate the condition of the triangular portion of the matrix computed thus far. A rank deficiency is signaled by the condition estimate exceeding a tolerance, in which case we remove the column in the computed factor that ....

J.A. George and M.T. Heath, Solution of sparse linear least squares problems using Givens rotations, Linear Alg. and Appl., 34, 1980, pp. 69--83.

....be applied to A T A to obtain a sparse triangular factor R. Similarly, the symbolic factorization algorithm for sparse Cholesky factorization described in [5] can be applied to A T A to predict the symbolic structure of R prior to numeric factorization. This approach is initially proposed in [4] and is now widely used for implementing the symbolic phase of the overall process for solving sparse linear least squares problems. In practice, a sparse linear least squares problem frequently contains relatively dense rows. Therefore, the corresponding triangular factor R becomes nearly or ....

....containing relatively dense rows. Experimental results are discussed in x4. Concluding remarks are contained in x5. Finally, a MATLAB implementation of our overall approach is provided in the Appendix. 2 A Solution Updating Algorithm In this section, we briefly describe an algorithm proposed in [4] for updating the solution to a sparse linear least squares problem. Let A 1 be an m 1 Theta n sparse matrix with full column rank, and A 2 an m 2 Theta n matrix. Suppose that the solution to the following sparse linear least squares problem min x kA 1 x Gamma b 1 k 2 (4) has already been ....

[Article contains additional citation context not shown here]

J. A. George and M. T. Heath, Solution of sparse linear least squares problems using Givens rotations, Linear Algebra and its Appl., 34 (1980), pp. 69--83.

....than one would expect or deserve, a phenomenon that is discussed in more detail by Shanno and Simantiraki (1996) A further problem with using the normal equations is that a single dense row in A would result in the normal equations being completely dense. This problem was recognized some time ago (George and Heath 1980, George, Heath and Ng 1983) and is avoided by partitioning the system and treating the dense rows by an updating scheme. The selection of which rows to include in the dense part is still an open question (for example, Sun 1995) There is difficulty in selecting rows so that the remaining problem ....

....R T R = A T b: Moreover, analysis by Bjorck (1987) has shown that, in most cases, numerically satisfactory results can be obtained by using the corrected semi normal equations (CSNE) where one step of iterative refinement is used. Sparse QR factorization uses the observation exploited by George and Heath (1980) that the factor R is the same as the Cholesky factor of the normal equations matrix. Although this is true in exact arithmetic, the difficulty in recognizing numerical cancellation means that the computed structure of the Cholesky factor can overestimate the structure of R. However, in many ....

George, A. and Heath, M. T. (1980), `Solution of sparse linear least squares problems using Givens rotations', Linear Algebra and its Applications 34, 69--83.

....matrix A is non symmetric and sparse. Let A have M N rows and rank N . Then, direct methods for the solution require either an LU or QR factorization. For either factorization, a fill reducing permutation for the symmetric matrix A T A can be used induce a sparsity preserving column order of A [2, 5]. Observe that a row of A forms a clique in the graph of A T A. When the columns correspond to vertices embedded in R d , this observation can be used along with our Cartesian nested dissection algorithm to compute a fill reducing ordering without forming A T A. We assume that we are given ....

J. A. George and M. T. Heath, Solution of sparse linear least squares problems using givens rotations, Linear Alg. Appl., 88 (1980), pp. 223--238.

....Q # QR = R # R. The computation of the QR factorization is not recommended here, since besides being more expensive than a Cholesky factorization, it also destroys the sparsity of the matrix A. Instead, Portugal et al. 116] propose an incomplete QR decomposition of A. Applying Givens rotations [39] to A, using the diagonal elements of D 1 2 T A # T , the elements of D 1 2 T A # T become null. No fill in is incurred in this factorization. See [116] for an example illustrating this procedure. After the factorization, we have the preconditioner M = FDF # , where F is a matrix ....

A. George and M. Heath. Solution of sparse linear least squares problems using Givens rotations. Linear Algebra and Its Applications, 34:69--83, 1980.

....is justified since the rows of a sparse matrix with full column rank can be reordered to make the diagonal zero free. It is also assumed that the matrix A is permuted according to a a column ordering selected to reduce fill; typically such a column ordering is obtained by an ordering for A T A [7, 9]. We use the letter F to denote the factor; F corresponds to U for Gaussian elimination and to R for orthogonal factorization. 2. Graph Model of Factorization. We assume the reader is familiar with basic graph theoretic concepts; a good reference is [15] The graph theoretic model of ....

....6 16,386 114,690 same as above Table 1 Description of Test Problems, Square A. tree of A T A; the structure of rows of the factor are determined using symbolic factorization on A T A, and rows of A are are transformed by diagonal pivoting against rows of the factor; the reader is referred to [7] for more details. In our approach, the structure is estimated using the separator tree; unless the separators are minimal this could lead to an overestimate of the nonzero structure and arithmetic work. Furthermore, for overdetermined systems, the processing of rows greatly affects the amount of ....

J. A. George and M. T. Heath, Solution of sparse linear least squares problems using Givens rotations, Linear Alg. Appl., 88 (1980), pp. 223--238.

....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 factorization of an n Theta n sparse symmetric positive definite matrix and the QR factorization of an m Theta n sparse ....

....a multifrontal method to implement our sparse QR decomposition. We make use of the observation that the Cholesky factor L T of the normal equation matrix A T A is identical to the R matrix of the QR factorization, except for possible sign differences in the rows. This observation was used by George and Heath (1980) to develop an algorithm for solving sparse least squares problems and it is essentially the same approach that we use here. We form the normal equations matrix, M = A T A symbolically and then order M using a minimum degree ordering, which gives us the pattern of the factors of L T and hence ....

[Article contains additional citation context not shown here]

J. A. George and M. T. Heath, (1980), Solution of Sparse Linear Least Squares Problems Using Givens Rotations, Linear Algebra and Its Application, 34, 69--83.

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J. A. George and M. T. Heath. Solution of Sparse Linear Least Squares Problems Using Givens Rotations. Lin. Alg. and Its Applic., 34:69--83, 1980.

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A. George, M. T. Heath, 1980. Solution of sparse linear least squares problems using Givens rotations. Lin. Alg. Appl. 34: 69-83, 1980.

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A. George and M. T. Heath. Solution of sparse linear least squares problems using givens rotations. Linear Algebra and its Applications, 34:69#83, 1980.

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A. George and M. T. Heath, Solution of sparse linear least squares problems using Givens rotations, Linear Algebra and its Applications, 34(1980), pp. 69--83.

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