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**11 - 13**of**13**### SPARSE LINEAR ALGEBRA in and around the APO-ENSEEIHT-IRIT group

"... We describe the work done in sparse linear algebra, in the "Algorithmique Parall`ele et Optimization" group of the ENSEEIHT-IRIT laboratory. The research activities, described in this paper, result from collaborations with CERFACS, RAL and University of FLorida. These include work on computational ..."

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We describe the work done in sparse linear algebra, in the "Algorithmique Parall`ele et Optimization" group of the ENSEEIHT-IRIT laboratory. The research activities, described in this paper, result from collaborations with CERFACS, RAL and University of FLorida. These include work on computational kernels for linear algebra, the solution of sparse systems by both direct and iterative methods, the study of element-by-element preconditionners. The objective of this paper is to describe the principal research themes explored in these area. We also comment on likely future developments. 1 Introduction We consider the solution of Ax = b; (1) where A is a large sparse matrix. If the matrix A is structured then it may be written as A = p X i=1 A i : (2) Sparse structured linear systems arise in many applications. The elementary matrices A i are usually full or nearly full matrices. Both classes of matrices (structured and unstructured) are being considered in our research studies...

### HSL

"... To compute an orthogonal factorization of a sparse overdetermined matrix A and optionally to solve the least squares problem min ||b−Ax|| 2. Given a sparse matrix A of order m × n, m ≥ n, of full column rank, this subroutine computes the QR factorization A = Q R 0 where Q is an m × m orthogonal matr ..."

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To compute an orthogonal factorization of a sparse overdetermined matrix A and optionally to solve the least squares problem min ||b−Ax|| 2. Given a sparse matrix A of order m × n, m ≥ n, of full column rank, this subroutine computes the QR factorization A = Q R 0 where Q is an m × m orthogonal matrix and R is an n × n upper triangular matrix. Given an m-vector b, this subroutine solves the least squares problem min ||b−Ax|| 2, either computing the solution x using Golub’s method (Numerical methods for solving least squares problems. Numer. Math. 7, 1965, 206-216), R T T T 0 x = Q b, or using the seminormal equations method (R Rx = A b). In the latter case, the Q factor need not be stored. T Given an n-vector b, this subroutine may also compute the minimum 2-norm solution of the linear system A x = b,