and Applied Analysis 3 Instead of solving for y directly from the triangular linear system (15), Paige and Saunders [1] factorize the matrix Tn into a lower triangular matrix with bandwidth three (resulting in the SYMMLQmethod). Also, we have TnQn−1 = ̂ Ln = [ [ [ [ [ [ [ [ [ [ γ0 δ1 γ1 ε2 δ2 γ2 d d d εn−3 δn−3 γn−3 εn−2 δn−2 γn−2 εn−1 δn−1 γn−1 ] ] ] ] ] ] ] ] ]

The Jacobian-free Newton-Krylov method is widely used in solving nonlinear equations arising in many applications. However, an effective preconditioner is required each iteration and determining such may be hard or expensive. In this paper, we propose an efficient two-sided bicoloring method to determine the lower triangular half of the sparse Jacobian matrix via automatic differentiation. Then...

A singular matrix A may have more than one LU factorizations. In this work the set of all LU factorizations of A is explicitly described when the lower triangular matrix L is nonsingular. To this purpose, a canonical form of A under left multiplication by unit lower triangular matrices is introduced. This canonical form allows us to characterize the matrices that have an LU factorization and to...

In order to estimate the condition number of the preconditioned matrix proposed in [F.R. Lin, W.K. Ching, Inverse Toeplitz preconditioners for Hermitian Toeplitz systems, Numer. Linear Algebra Appl. 12 (2005) 221–229], we study the inverse of band triangular Toeplitz matrix. We derive an explicit formula for the entries of the inverse of band lower triangular Toeplitz matrix by means of divided...

Let T be a block lower triangular contraction, i.e., a contractive operator T = (ti,j) ∞ i,j=−∞ acting from a doubly infinite Hilbert space direct sum ⊕∞ j=−∞Kj into a doubly infinite Hilbert space direct sum ⊕∞ j=−∞ Lj . The operator ti,j , which maps Kj into Li, is the (i, j)-th entry in the operator matrix representation of T relative to the natural Hilbert space direct sum decompositions. I...

The lgrade of a n × n matrix A is the largest rank of any subdiagonal block of a symmetric partition of A. A number of algebraic results on lgrade are given. When A has lgrade d, it can be be approximately decomposed as A = U + V , where U is an upper triangular matrix and V has rank d. If A satisfies GA = N with G and N have lower bandwidths dG and dN , then the decomposition is exact: A = U +...

Let L n be a lower triangular matrix of dimension n each of whose nonzero entries is an independent N(0; 1) variable, i.e., a random normal variable of mean 0 and variance 1. It is shown that n , the 2-norm condition number of L n , satisses n p n ! 2 almost surely as n ! 1. This exponential growth of n with n is in striking contrast to the linear growth of the condition numbers of random dense...

It is well known that the group of all nonsingular lower block -triangular pxp matrices acts transitively on the cone p* of all positive definite pxp matrices. This result has been applied to obtain several major reSUlts in mUltivariate statistical distribution theory and decision theory. Here a converse is established: if a matrix group acts transitively on P*, then its group algebra must be (...

In this work an algorithm for solving triangular systems of equations for multiple right hand sides is presented. The algorithm for solving triangular systems for multiple right hand sides, commonly referred to as the TRSM problem, is a very important in dense linear algebra as it is a subroutine for most decompositions of matrices as LU or QR. To improve performance over the standard iterative...