L. V. Foster. Rank and null space calculations using matrix decomposition without column interchanges. Linear Algebra and Its Applications, 74:47-- 71, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....So although in practice the diagonal elements of R are rough approximations to the singular values, this need not always be the case. Indeed, the Kahan matrix provides a well known counterexample. Kah66] This is one consideration that motivates the rank revealing QR factorization (RRQR) [Fos86, Cha87]. If R is partitioned as A ; an RRQR algorithm tries to maximize the smallest singular value of R 11 and or minimize the largest singular value of R 22 [CI94] This essentially means making, respectively, kR 11 k as large as possible and kR 22 k as small as possible. From the interlacing ....

L. V. Foster. \Rank and null space calculations using matrix decomposition without column interchanges." Linear Algebra and Its Applications, 74:47-71, 1986. 83

....matrices have been investigated. For QR factorization we try to choose a permutation matrix Pi so that A Pi = QR is a rank revealing QR factorization. The standard column pivoting strategy [69, x5.4.1] tends to be rank revealing, but it can completely fail to reveal near rank deficiency. Foster [63] and Chan [25] develop iterative algorithms for computing rankrevealing QR factorizations; for both algorithms the bounds that show by how much, in the worst case, the factorizations may fail to reveal the rank contain factors exponential in the rank deficiency. Both methods need a condition ....

Leslie V. Foster. Rank and null space calculations using matrix decomposition without column interchanges. Linear Algebra and Appl., 74:47--71, 1986.

....the initial triangular factorization requires O(mn 2 ) ops, while the rank revealing step only requires O( n k)n 2 ) ops if k n, and O(kn 2 ) ops if k n. The updating can always be done in O(n 2 ) ops, when implemented properly. We refer to the original papers [9] 10] 16] [18], 19] 23] 31] 32] for details about the algorithms. For structured matrices (e.g. Hankel and Toeplitz matrices) the initial triangular factorization in the RRQR and UTV algorithms has the same complexity as the rank revealing step, namely, O(mn) ops; see [7, x8.4.2] for signal processing ....

L. V. Foster, Rank and null space calculations using matrix decomposition without column interchanges, Lin. Alg. Appl., 74 (1986), pp. 47-71.

....on parallel computers is normally more efficient when the orthogonal factorization is carried out without pivoting for numerical stability. Column interchanges are also (explicitly or implicitly) needed if one wants to compute a rank revealing QR factorization; see, for example, Chan [5] Foster [10], or Pierce and Lewis [25] Developing a parallel rank revealing code for sparse matrices will be one of the topics for future work. Both column and row interchanges must be used in order to preserve better the sparsity of matrix A during the orthogonal decomposition. Finding a good pivoting 2 ....

L. V. Foster, Rank and null space calculations using matrix decomposition without column interchanges, Lin. Alg. Appl., Vol. 74 1986, pp. 47-71.

....In the past few years, a number of other methods have been developed to alleviate the computational burden of the SVD, yet retaining important information such as rank and principal subspaces. Some of these techniques are the URV decomposition [2 ] and the rank revealing QR decomposition (RRQR) [3 , 4, 5]. Recently, there has been an increased interest in updating techniques for the SVD and URV decomposition, which converge to the exact SVD or URV under certain stationarity conditions [6 , 7] It should be noted that all these decompositions require O(am 2 n) operations, for an m n matrix, ....

L.V. Foster, "Rank and null space calculations using matrix decomposition without column interchanges," Lin. Alg. Appl., vol. 74, pp. 47--71, 1986.

....n with m n and with numerical rank k n, then the initial triangular factorization requires O(mn 2 ) flops, while the rank revealing step only requires O( n Gamma k)n 2 ) flops. The updating can be done in O(n 2 ) flops, when implemented properly. We refer to the original papers [7] [11], 20] and [21] for details about the algorithms. For structured matrices (e.g. Hankel and Toeplitz matrices) RRQR and UTV decompositions are also alternatives to the SVD, but the situation is more complicated here where the initial triangular factorization algorithm is of complexity O(mn) ....

L. V. Foster, Rank and null space calculations using matrix decomposition without column interchanges, Lin. Alg. Appl., 74 (1986), pp. 47--71.

.... demanding and difficult to update for either dense [5] or sparse matrices [1,19] This can be a drawback for recursive procedures which require simple matrix updates (e.g. appending or deleting a row or column) Alternatively, rank revealing QR (RRQR) algorithms such as those by Foster [15], Chan [6] and modifications [4] can be used to obtain subspace information from matrices [7] 8] RRQR decompositions, however, yield subspaces whose accuracies depend on the gap in the singular values [13] in the sense that a large gap is required to produce good approximations to the singular ....

L. Foster. Rank and Null Space Calculations Using Matrix Decomposition Without Column Interchanges. Linear Algebra and Its Applications, 74:47--71, 1986.

....that (oe min (W 2 ) Gamma1 = kW Gamma1 2 k 2 is not large, we are guaranteed that the elements of the bottom right p Theta p block of R will be small. Algorithms based on computing well conditioned nullspace bases for A include these by Golub, Klema, and Stewart [1976] Chan [1987] and Foster [1986]. Other algorithms addressing task 2 are these by Stewart [1984] and Gragg and Stewart [1976] Algorithms addressing task 1 include those of Chan and Hansen [1994] and Golub, Klema, and Stewart [1976] In fact, the latter achieves both task 1 and task 2 and, therefore, reveals the rank, but it is ....

Foster, L. V. 1986 . Rank and null space calculations using matrix decomposition without column interchanges. Linear Algebra and Its Applications 74, 47--71.

....the computational requirements, yet retain important information such as numerical rank and principal subspaces. Some of these techniques are the URV decomposition [1] which is a rank revealing form of a complete orthogonal decomposition [2] and the rank revealing QR decomposition (RRQR) [3 8], see [8] for a review. The URV algorithm is iterative and requires estimates of the conditioning of certain submatrices at every step of the iteration. This is a global and data dependent operation: not a very attractive feature. The SVD and URV decomposition can be updated [9, 1] which is still ....

L.V. FOSTER, Rank and null space calculations using matrix decomposition without column interchanges, Lin. Alg. Appl., 74 (1986), pp. 47--71.

....of the matrix S T # into the factor R to insure that kTk 2 is small enough. The first pass of our algorithm is similar to a Type I method, in the terminology of [9] However, our method allows columns that were placed into the R factor to be removed later, similar to Foster s Algorithm 2 [15]. The second phase of the algorithm, which ensures that kTk 2 is small, is similar to a Type II algorithm in [9] Iterating this process is a method that might produce an RRQR factorization satisfying either or both Problem I and Problem II. However, iterating would dramatically affect the ....

....triangular submatrix to form an approximate right singular vector for the smallest singular value. The index of the component of largest absolute value in the approximate right singular vector designates the column k to be removed from the matrix. This is identical to the method proposed in Foster [15] and Golub, Klema and Stewart [21] except that we use an approximate singular vector. The redundant column k occurs in supernode I, either the current supernode or one of its descendants. We apply the permutation (1; 2; k Gamma 1; k 1; r; k) symmetrically to the rows and ....

L.V. Foster, Rank and null space calculations using matrix decompositions without column interchanges, Linear Algebra and its Applications, 74, 1986, pp. 47--71.

....(oe min (W 2 ) Gamma1 = kW Gamma1 2 k 2 is not large, then we are guaranteed that the elements of the bottom right p Theta p block of R will be small. Algorithms based on computing well conditioned nullspace bases for A include these by Golub, Klema, and Stewart [1976] Chan [1987] and Foster [1986]. Other algorithms addressing task 2 are these by Stewart [1984] and Gragg and Stewart [1976] Algorithms addressing task 1 include those of Chan and Hansen [1994] and Golub, Klema, and Stewart [1976] In fact, the latter achieves both task 1 and task 2 and, therefore, reveals the rank, but it is ....

Foster, L. V. 1986 . Rank and null space calculations using matrix decomposition without column interchanges. Linear Algebra and Its Applications 74, 47--71.

....1 ) Its dimension is equal to the number of singular values of X that are larger than g, d say. A number of alternative decompositions have been developed to estimate R(U 1 ) in a computationally more efficient way. Examples are the URV decomposition [1] and the Rank Revealing QR decomposition [2, 3]. The complexity of these algorithms can be reduced further by considering updating techniques [4, 5] although at this point norm bounds on the estimation error hold only asymptotically. Furthermore, a condition estimation is usually required for determining d, which might not be very accurate or ....

....R(A) which is usually taken to be the d principal left singular vectors of X, i.e. U S = U 1 . We will compare this with the Schur based subspace estimates (U S = U SSE1 and U S = U SSE2 ) and estimates based on the complete orthogonal decomposition (COD) 9] and Rank Revealing QR (RRQR) [2,3] Once the signal subspaces are estimated, the DOAs are obtained from a certain eigenvalue decomposition based on these subspaces. The quality of subspace estimates can be measured by computing the distance between the estimated and the true subspaces, as defined in [9] We conducted three sets of ....

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L. V. Foster, "Rank and null space calculations using matrix decomposition without column interchanges, " Lin. Alg. Appl., vol. 74, pp. 47--71, 1986.

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L. V. Foster. Rank and null space calculations using matrix decomposition without column interchanges. Linear Algebra and Its Applications, 74:47-- 71, 1986.

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L.V. FOSTER, Rank and null space calculations using matrix decomposition without column interchanges. Lin. Alg. Appl. 74 (1986), pp. 47-71.

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L. V. Foster. Rank and null space calculations using matrix decomposition without column interchanges. Linear Algebra and Its Applications, 74:47{ 71, 1986.

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L. V. Foster. Rank and null space calculations using matrix decomposition without column interchanges. Linear Algebra and Its Applications, 74:47-71, 1986.

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L. V. Foster. Rank and null space calculations using matrix decomposition without column interchanges. Linear Algebra and Its Applications, 74:47-- 71, 1986.

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