For solving the total least squares problems, minE;f k(E; f)kF subject to (A+E)x = b+f , where A is large and sparse or structured Björck suggested a method based on Rayleigh quotient iteration. This method reduces the problem to the solution of a sequence of symmetric, positive definite linear systems of the form (A T A \Gamma ¯ oe 2 I)z = g, where ¯ oe is an approximation to the smallest singular value of (A; b). A preconditioned conjugate gradient method, using a sparse, possibly incomplete, Cholesky factor of A T A can be used for solving these systems. In this paper the method is further developed. The choice of initial approximation and termination criteria are discussed. Numerical results confirm that the method achieves rapid convergence and good accuracy for problems which are not too ill-conditioned.

. This paper studies the solution of the linear least squares problem for a large and sparse m by n matrix A with m n by QR factorization of A and transformation of the righthand side vector b to Q T b. A multifrontal-based method for computing Q T b using Householder factorization is presented. A theoretical operation count for the K by K unbordered grid model problem and problems defined on graphs with p n-separators shows that the proposed method requires O(NR ) storage and multiplications to compute Q T b, where NR = O(n log n) is the number of nonzeros of the upper triangular factor R of A. In order to introduce BLAS-2 operations, Schreiber and Van Loan's Storage-Efficient-WY Representation [SIAM J. Sci. Stat. Computing, 10(1989),pp. 55-57] is applied for the orthogonal factor Q i of each frontal matrix F i . If this technique is used, the bound on storage increases to O(n(logn) 2 ). Some numerical results for the grid model problems as well as Harwell-Boeing problems...

Sparse linear least squares problems containing a few relatively dense rows occur frequently in practice. Straightforward solution of these problems could cause catastrophic fill and delivers extremely poor performance. This paper studies a scheme for solving such problems efficiently by handling dense rows and sparse rows separately. How a sparse matrix is partitioned into dense rows and sparse rows determines the efficiency of the overall solution process. A new algorithm is proposed to find a partition of a sparse matrix which leads to satisfactory or even optimal performance. Extensive numerical experiments are performed to demonstrate the effectiveness of the proposed scheme. A MATLAB implementation is included. 1 This work was supported in part by the Cornell Theory Center which receives funding from members of its Corporate Research Institute, the National Science Foundation (NSF), the Advanced Research Projects Agency (ARPA), the National Institutes of Health (NIH), New York S...

The minimal 2-norm solution to an underdetermined system Ax = b of full rank can be computed using a QR factorization of A T in two di erent ways. One requires storage and re-use of the orthogo-nal matrix Q while the method of semi-normal equations does not. Existing error analyses show that both methods produce computed solutions whose normwise relative error is bounded to rst order by c 2(A)u, where c is a constant depending on the dimensions of A, 2(A) = kA + k2kAk2 is the 2-norm condition number, and u is the unit roundo. We show that these error bounds can be strength-ened by replacing 2(A) by the potentially much smaller quantity cond2(A) = kjA + j jAjk2, which isinvariant under row scaling of A. We also show that cond2(A) re ects the sensitivity of the minimum norm solution x to row-wise relative perturbations in the data A and b. For square linear systems Ax = b row equilibration is shown to endow

. We survey the numerical stability of some fast algorithms for solving systems of linear equations and linear least squares problems with a low displacement-rank structure. For example, the matrices involved may be Toeplitz or Hankel. We consider algorithms which incorporate pivoting without destroying the structure, and describe some recent results on the stability of these algorithms. We also compare these results with the corresponding stability results for the well known algorithms of Schur/Bareiss and Levinson, and for algorithms based on the semi-normal equations. Key words. Bareiss algorithm, Levinson algorithm, Schur algorithm, Toeplitz matrices, displacement rank, generalized Schur algorithm, numerical stability. AMS subject classifications. 65F05, 65G05, 47B35, 65F30 1. Motivation. The standard direct method for solving dense n \Theta n systems of linear equations is Gaussian elimination with partial pivoting. The usual implementation requires of order n 3 arithmetic op...

. We describe the issues involved in the design and implementation of efficient parallel algorithms for solving sparse linear least squares problems on distributed-memory multiprocessors. We consider both the QR factorization method due to Golub and the method of corrected semi-normal equations due to Bj¨orck. The major tasks involved are sparse QR factorization, sparse triangular solution and sparse matrix-vector multiplication. The sparse QR factorization is accomplished by a parallel multifrontal scheme recently introduced. New parallel algorithms for solving the related sparse triangular systems and for performing sparse matrix-vector multiplications are proposed. The arithmetic and communication complexities of our algorithms on regular grid problems are presented. Experimental results on an Intel iPSC/860 machine are described. Key words. parallel algorithms, sparse matrix, orthogonal factorization, multifrontal method, least squares problems, triangular solution, distributed-me...