No abstract

realization theory 5.5 Notes 116 6. ISOMETRIC AND INNER OPERATORS 6.1 Realization of inner operators 6.2 External factorization 6.3 State-space properties of isometric systems 6.4 Beurling-Lax like theorem 6.5 Example 7. INNER-OUTER FACTORIZATION AND OPERATOR INVERSION 7.1 Introduction 7.2 Inner-outer factorizations 7.3 Operator inversion 7.4 Examples 7.5 Zero structure and its limiting behavior 7.6 Notes Part II INTERPOLATION AND APPROXIMATION 8. J-UNITARY OPERATORS 8.1 Scattering operators 8.2 Geometry of diagonal J-inner product spaces 8.3 State space properties of J-unitary operators 8.4 Past and future scattering operators 210 8.5 J-unitary external factorization 215 8.6 J-Iossless and J-inner chain scattering operators 218 8.7 The mixed causality case 223 9. ALGEBRAIC INTERPOLATION 233 9.1 Diagonal evaluations or the W-transform 235 9.2 The algebraic Nevanlinna-Pick problem 237 9.3 The tangential Nevanlinna-Pick problem 242 9.4 The Hermite-Fejer interpolation problem 242 9.5 Conjugation of a left interpolation problem 245 9.6 Two sided interpolation 250 9.7 The four block problem 260

Abstract. In this paper we present a fast direct solver for certain classes of dense structured linear systems that works by first converting the given dense system to a larger system of block sparse equations and then uses standard sparse direct solvers. The kind of matrix structures that we consider are induced by numerical low rank in the off-diagonal blocks of the matrix and are related to the structures exploited by the fast multipole method (FMM) of Greengard and Rokhlin. [11], and the introduction of the fast multipole method (FMM) of Greengard and Rokhlin [7], it has become clear that many large matrices that arise in practice have a complex low-rank structure in their submatrices that can be exploited efficiently to speed up matrix algorithms. In particular, such structured matrices arise in the numerical solution of integral equations, as fill-in during Gaussian elimination of sparse matrices that come from the discretization of elliptic PDEs, and in many other applications. In earlier work [2] we introduced techniques to design fast and stable direct solvers for such structured matrices based on an implicit ULV factorization algorithm and a matrix representation that we called hierarchically semiseparable (HSS). In this paper we show that linear systems of equations involving such dense structured matrices can be efficiently converted into a larger sparse system of equations that has an ordering of the unknowns permitting a very efficient direct Gaussian elimination solver to be used. This technique has several advantages. First, it makes it possible to exploit the highly developed sparse direct solver technology to attack dense structured problems. Second, it provides a theoretical tool to study these large dense structured matrices. However, in this paper we just concentrate on showing how this technique can be used to design a fast, stable solver for matrices in HSS form only.

No abstract

An extended sequentially semiseparable (SSS) representation derived from timevarying system theory is used to capture, on the one hand, the low-rank of the off-diagonal blocks of a matrix for the purposes of efficient computations and, on the other, to provide for sufficient descriptive richness to allow for backward stability in the computations. We present (i) a fast algorithm (linear in the number of equations) to solve least squares problems in which the coefficient matrix is in SSS form, (ii) a fast algorithm to find the SSS form of X such that AX = B, where A and B are in SSS form, and (iii) a fast model reduction technique to improve the SSS form.