Matrix Computations and Semiseparable Matrices: Linear Systems

Notation

Notation Throughout the book we use the following notation, unless stated otherwise. A = (ai,j)i,j A general (or a quasiseparable) matrix A with elements ai,j. A(i : j, k : l) Submatrix of A, consisting of rows i up to and including j, and columns k up to and including l. A(α; β) . . . . Submatrix of A, consisting of indices out of the set α and β. α × β . . . . . . α × β denotes the product set {(i, j)|i ∈ α, j ∈ β}, with α and β sets. B . . . . . . . . . . A band matrix. D. . . . . . . . . . A diagonal matrix. D. . . . . . . . . . The class of diagonal matrices. det (A) . . . . The determinant of a matrix A. diag (d). . . . Denotes a diagonal matrix, with as diagonal elements, the elements from the vector d. G. . . . . . . . . . A Givens transformation. H . . . . . . . . . A (generalized) Hessenberg matrix. Ik . . . . . . . . . The identity matrix of size k × k. inv . . . . . . . . E.g. Ainv: the subclass of invertible matrices from the class A. L . . . . . . . . . . A lower triangular matrix (with regard to the LUfactorization ). M . . . . . . . . . An elementary Gaussian transformation (with regard to the LU-factorization). P . . . . . . . . . . A permutation matrix (with regard to the LUfactorization ). r(Σ; A) . . . . The structured rank of the matrix A, with structure Σ. rA . . . . . . . . A representation map for the class A. rank (A) . . The rank of a matrix A. Σ . . . . . . . . . . Defines a structure. sA . . . . . . . . A map for retrieving the representation of a class A, in some sense the inverse of rA. S . . . . . . . . . . A semiseparable matrix. xvii xviii Notation S(u, v). . . . . . . . . . . . . A symmetric generator representable semiseparable matrix with generators u and v. S(u, v, p, q) . . . . . . . . An unsymmetric generator representable semiseparable matrix with generators u, v, p and q. S . . . . . . . . . . . . . . . . . . The class of semiseparable matrices. S(d) . . . . . . . . . . . . . . . . The class of semiseparable plus diagonal matrices. S(g) . . . . . . . . . . . . . . . . The class of generator representable semiseparable matrices . S(g,d) . . . . . . . . . . . . . . The class of generator representable semiseparable plus diagonal matrices. S(s) . . . . . . . . . . . . . . . . The class of diagonal-subdiagonal representable semiseparable matrices. sym. . . . . . . . . . . . . . . . E.g. Asym: the subclass of symmetric matrices from the class A. ti:j . . . . . . . . . . . . . . . . . An abbreviation for ti:j = titi−1 . . . tj if i ≥ j. An abbreviation for ti:j = titi+1 . . . tj if i ≤ j. T . . . . . . . . . . . . . . . . . . A tridiagonal or a Toeplitz matrix. T . . . . . . . . . . . . . . . . . . The class of tridiagonal matrices. T (i) . . . . . . . . . . . . . . . . The class of irreducible tridiagonal matrices. tril(A, p) . . . . . . . . . . Denotes the lower triangular part of the matrix A, below and including subdiagonal p. triu(A, p). . . . . . . . . . . Denotes the upper triangular part of the matrix A, above and including superdiagonal p. Q . . . . . . . . . . . . . . . . . . A unitary(orthogonal) matrix (with regard to the QRfactorization ). Q . . . . . . . . . . . . . . . . . . The class of quasiseparable matrices. Q(g) . . . . . . . . . . . . . . . . The class of generator representable quasiseparable matrices . Q(s) . . . . . . . . . . . . . . . . The class of quasiseparable matrices with a symmetric rank structure. R . . . . . . . . . . . . . . . . . . An upper triangular matrix (with regard to the QRfactorization ). U . . . . . . . . . . . . . . . . . . An upper triangular matrix (with regard to the LUfactorization ). uT = [u1, u2, . . . , un] A column vector u, with entries ui. All vectors are column vectors and denoted in bold. u(i : j) . . . . . . . . . . . . . A subvector of u, consisting of the elements i up to and including j. Z . . . . . . . . . . . . . . . . . . A Hessenberg-like matrix. ...