Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for solving this type of systems. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.

Abstract not found

this paper is the modified version of Gram-Schmidt orthogonalization with a rejection test applied right after the formation of the off-diagonal elements of the factor R. For a given rejection parameter 0 / 1, the rejection test is: if r ij ! /= k a

Abstract. A new preconditioner for symmetric positive definite systems is proposed, analyzed, and tested. The preconditioner, compressed incomplete modified Gram–Schmidt (CIMGS), is based on an incomplete orthogonal factorization. CIMGS is robust both theoretically and empirically, existing (in exact arithmetic) for any full rank matrix. Numerically it is more robust than an incomplete Cholesky factorization preconditioner (IC) and a complete Cholesky factorization of the normal equations. Theoretical results show that the CIMGS factorization has better backward error properties than complete Cholesky factorization. For symmetric positive definite M-matrices, CIMGS induces a regular splitting and better estimates the complete Cholesky factor as the set of dropped positions gets smaller. CIMGS lies between complete Cholesky factorization and incomplete Cholesky factorization in its approximation properties. These theoretical properties usually hold numerically, even when the matrix is not an M-matrix. When the drop set satisfies a mild and easily verified (or enforced) property, the upper triangular factor CIMGS generates is the same as that generated by incomplete Cholesky factorization. This allows the existence of the IC factorization to be guaranteed, based solely on the target sparsity pattern.

. The performance of CGLS, a basic iterative method whose main idea is to organize the computation of conjugate gradient method applied to normal equations for solving least squares problems. On modern architecture is always limited because of the global communication required for inner products. Inner products often therefore present a bottleneck, and it is desirable to reduce or even eliminate all the inner products. Following a note of B. Fischer and R. Freund [11], an inner product-free conjugate gradient-like algorithm is presented that simulates the standard conjugate gradient by approximating the conjugate gradient orthogonal polynomial by suitable chosen orthogonal polynomial from Bernstein-Szego class. We also apply this kind of algorithm into normal equations as CGLS to solve the least squares problems and compare the performance with the standard and modified approaches. 1 Introduction Many scientific and engineering applications such as linear programming [4], augmented La...

In the factorization A = QR of a matrix A, the orthogonal matrix Q can be represented either explicitly (as a matrix) or implicitly (as a matrix H of Householder vectors). We derive both upper and lower bounds on the number of nonzeros in H and the number of nonzeros in Q, in the case where the graph of A T A has "good" separators and A need not be square. We also derive an upper bound on the number of nonzeros in the null-basis part of Q in the case where A is the edge-vertex incidence matrix of a planar graph. The significance of these results is that they both illuminate and amplify a folk theorem of sparse QR factorization, which holds that the matrix H of Householder vectors represents the orthogonal factor of A much more compactly than Q itself. To facilitate discussion of this and related issues, we review several related results which have appeared previously. Keywords: Sparse matrix algorithms, QR factorization, separators, column intersection graph, strong Hall...

. In this paper we study the parallelization of CGLS, a basic iterative method for large and sparse least squares problems whose main idea is to organize the computation of conjugate gradient method to normal equations. A performance model called isoefficiency concept is used to analyze the behavior of this method implemented on massively parallel distributed memory computers with two dimensional mesh communication scheme. Two different mappings of data to processors, namely simple stripe and cyclic stripe partitionings are compared by putting these communication times into the isoefficiency concept which models scalability aspects. Theoretically, the cyclic stripe partitioning is shown to be asymptotically more scalable. 1 Introduction Many scientific and engineering applications such as linear programming [4], augmented Lagrangian method for CFD [10], and the natural factor method in structure engineering [1, 13] give rise to the least squares problems min x kAx \Gamma...

In this report a new version of the multifrontal sparse QR factorization routine sqr, originally by Matstoms, for general sparse matrices is described and evaluated. In the previous version the orthogonal factor Q is discarded due to storage considerations. The new version provides Q and uses the multifrontal structure to store this orthogonal factor in a compact way. A new data class with overloaded operators is implemented in Matlab to provide an easy usage of the compact orthogonal factors. This implicit way of storing the orthogonal factor also results in faster computation and application of Q and Q T . Examples are given, where the new version is up to four times faster when computing only R and up to 1000 times faster when computing both Q and R, than the built-in function qr in Matlab. The sqr package is available at URL: http://www.mai.liu.se/~milun/sls/. Key words: QR factorization, sparse problems, multifrontal method, orthogonal factorization. 1 Introduction. Let A 2 IR...

On modern architecture the performance of CGLS, a basic iterative method whose main idea is to organize the computation of conjugate gradient method applied to normal equations for solving sparse least squares problems is always limited because of the global communication required for inner products. Inner products often therefore present a bottleneck, and it is desirable to reduce or even eliminate all the inner products. Here we present an inner productfree conjugate gradient-like algorithm, that simulates the standard conjugate gradient by approximating the conjugate gradient orthogonal polynomial by suitable chosen orthogonal polynomial from Bernstein-Szego class, with Chebyshev polynomial preconditioner which is constructed based on the observation that good estimates for the eigenvalue distribution can be surprisingly derived after only a few steps of the conjugate gradient-type method. For the inner product-free algorithm, the parallel performance has been compared with the stan...