M MATRIX

In this paper we explore discrete monitored barrier options in the Black-Scholes framework. The discontinuity arising at each monitoring data requires a careful numerical method to avoid spurious oscillations when low volatility is assumed. Here a technique mixing the Laplace Transform and the finite difference method to solve Black-Scholes PDE is considered. Equivalence between the Post-Widder inversion formula joint with finite difference and the standard finite difference technique is proved. The mixed method is positivity-preserving, satisfies the discrete maximum principle according to financial meaning of the involved PDE and converges to the underlying solution. In presence of low volatility, equivalence between methods allows some physical interpretations.

In this article a new preconditioner from class of (I+S)-type based on the Modified Accelerated
Overrelaxation(MAOR) iterative method has been introduced and convergence properties of the proposed
method have been analyzed and compared with the some other preconditioners. Moreover, comparisons
between different splittings are derived. Numerical example is also given to illustrate our results.

This work deals with the H 1 condition numbers and the distribution of the~ N;Msingular values of the preconditioned operators f~ ?1 N;M W N;MÂN;M g. N;M is the matrix representation of the Legendre Spectral Collocation discretization of the elliptic operator A de ned by Au := ? u + a 1 u x + a 2 u y + a 0 u in (the unit square) with boundary conditions: u = 0 on ? 0 ; @u @ A = u on ? 1 .~ N;M is the sti ness matrix associated with the nite element discretization of the positive de nite elliptic operator B de ned by Bv := ? v + b 0 v in with boundary conditions v = 0 on ? 0 ; @v @ B = v on ? 1 . The nite element space is either the space of continuous functions which are bilinear on the rectangles determined by the Legendre-Gauss-Lobatto (LGL) points or the space of continuous functions which are linear on a triangulation of determined by the LGL points. W N;M is the matrix of quadrature weights. When A = B we obtain results on the eigenvalues of~ ?1 N;M W N;MBN;M . We show that there is an integer N 0 and constants ; with 0 < < , such that: if min(N; M) N 0 , then all the~ N;M -singular values of~ ?1 N;M W N;MÂN;M lie in the interval ; ]. Moreover, there is a smaller interval, 0 ; 0 ], independent of the operator A, such that: if min(N; M) N 0 , then all but a xed nite number of the~ N;M -singular value lie in 0 ; 0 ]. These results are related to results of Manteu el and Parter MP] Parter and Wong PW] and Wong W1], W2] for nite element discretizations. 2 10

Diagonal tensor flux approximations are commonly used in fluid dynamics. This approximation introduces an O(1) error in flux whenever the coordinate system is nonaligned with the principal axes of the tensor which is particularly common when employing curvilinear gridding. In general a consistent full tensor flux approximation leads to a significant increase in support and consequent size of the Jacobian matrix. After decomposition of a general full tensor flux into a diagonal tensor flux together with cross terms, time-split semi-implicit, stable, full tensor flux approximations are introduced with in a general finite volume formalism, enabling the standard diagonal tensor Jacobian matrix structure to be retained for single phase flow, IMPES, and standard block fully implicit formulations while ensuring spatial consistency of the discretization for both structured and unstructured grids. Stability of the scheme is proven for constant elliptic coefficients. The results presented demonstrate the benefits of the method for multiphase flow within a fully implicit framework on structured and unstructured grids.

A brief but concise review of methods to generate P-matrices (i.e., matrices having positive principal minors) is provided and motivated by open problems on P-matrices and the desire to develop and test efficient methods for the detection of P-matrices. Also discussed are operations that leave the class of P-matrices invariant. Some new results and extensions of results regarding P-matrices are included.

The preconditioner for solving the linear system Ax = b introduced in [D.J. Evans, M.M. Martins, M.E. Trigo, The AOR iterative method for new preconditioned linear systems, J. Comput. Appl. Math. 132 -466] is generalized. Results obtained in this paper show that the convergence rate of Jacobi and Gauss-Seidel type methods can be increased by using the preconditioned method when A is an M-matrix.

For the Hadamard product A • A −1 of an M-matrix A and its inverse A −1 , we give new lower bounds for the minimum eigenvalue of A • A −1 . These bounds are strong enough to prove the conjecture of Fiedler and Markham [An inequality for the Hadamard product of an M-matrix and inverse M-matrix, Linear Algebra Appl. 101 (1988) 1-8].

A well-known property of an M-matrix M is that the inverse is element-wise non-negative, which we write as M −1 0. In this paper, we consider element-wise perturbations of non-symmetric tridiagonal M-matrices and obtain bounds on the perturbations so that the non-negative inverse persists. Sufficient bounds are written in terms of decay estimates which characterize the decay of the elements of the inverse of the unperturbed matrix. Results for general symmetric matrices and symmetric Toeplitz matrices are obtained as special cases and compared with known results.

The linear systems with M and H-matrices as coefficient matrix many often arise in science and engineering. In this article we consider some preconditioners of [12] for solving a linear system with the above coefficient matrix. Also we use preconditioned AOR method for the solution of the systems. The convergence theorems for the preconditioners also presented. A numerical example is given to test the methods.

The problem of optimally fitting controllers to data is examined for the identification of a controller from a given class of MIMO controllers used in model reference adaptive control. The problem of identifying a MIMO controller from this class is formulated. This formulation leads to an optimization problem where a best 2nm × m matrix of parameters is looked for. It is proved that the solution to this problem can be given in terms of the solution of an optimization problem where a best 2nm 2 × 1 vector of parameters is looked for, which has already been solved for the SISO case. Simulations are provided for an example which has appeared a few times in the model reference adaptive control literature. The formulation and solution of this problem illustrates a unifying link between the design of model reference adaptive control for SISO and MIMO linear systems.

In this paper, we give some new centrosymmetric splittings of a centrosymmetric matrix A and obtain corresponding iterative methods for a linear system Ax = b. These iterative schemes are particularly based on the centrosymmetric property of A and aimed at reduction of the cost of computation and storage.

It is well known that each tree metric M has a unique realization as a tree, and that this realization minimizes the total length of the edges among all other realizations of M . We extend this result to the class of symmetric matrices M with zero diagonal, positive entries, and such that mij + m kl ≤ max{m ik + m jl , m il + m jk } for all distinct i, j, k, l.

In this paper, we obtain an inverse M-matrix completion, with zeros in the inverse completion, of a noncombinatorially symmetric partial inverse M-matrix, when the associated graph is acyclic without specified paths or, in the other case, when the subgraph induced by the vertices of any cycle or specified path is a clique.

In this note, we present sharp bounds for the componentwise perturbation of matrix inverse and linear systems, especially for nonsingular M-matrix. We also use matrix derivatives to deduce the matrix componentwise condition number and present some numerical tests to show our results.

The higher rank numerical range is useful for constructing quantum error correction code for a noisy quantum channel. It is known that if a normal matrix A ∈ Mn has eigenvalues a1, . . . , an, then its rank-k numerical range Λ k (A) is the intersection of convex polygons with vertices aj 1 , . . . , aj n−k+1 , where 1 ≤ j1 < · · · < j n−k+1 ≤ n. In this paper, it is shown that the higher rank numerical range of a normal matrix with m distinct vertices can be written as the intersection of no more than m closed half planes. In addition, given a convex polygon P a construction is given for a normal matrix A ∈ Mn with minimum n such that Λ k (A) = P. In particular, if P has p vertices, with p ≥ 3, there is a normal matrix A ∈ Mn with

In two recent works the condition of the diagonal entries of the group inverse of a singular and irreducible M-matrix being uniform (constant) has arisen: in resistive electrical circuits and in the eect upon the Perron root of certain diagonal perturbation of a nonnegative matrix. In this paper we ®rst show a negative result that the group inverse of the Laplacian matrix of an undirected weighted graph G on n vertices with a cutpoint cannot have uniform diagonal. This includes the case when G is a tree. We characterize, however, all weighted n-cycles G whose Laplacian has a group inverse with a uniform diagonal. Finally, we consider the mean ®rst passage matrix M of an ergodic Markov chain with a doubly stochastic transition matrix T. We show that if the group inverse of s À has a uniform diagonal, then the group inverse of the M-matrix qws À w is again an M-matrix. Ó 1 NSERC Grant No. OGP0138251. 0024-3795/99/$ ± see front matter Ó 1999 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 -3 7 9 5 ( 9 9 ) 0 0 1 2 7 -5 Linear Algebra and its Applications 296 (1999) 153±170

Recently, two distinct directions have been taken in an attempt to generalize the definition of an M-matrix. Even for nonsingular matrices, these two generalizations are not equivalent. The role of these and other classes of recently defined matrices is indicated showing their usefulness in various applications.

The concept of global conformal invariance (GCI) opens the way of applying algebraic techniques, developed in the context of two-dimensional chiral conformal field theory, to a higher (even) dimensional spacetime. In particular, a system of GCI scalar fields of conformal dimension two gives rise to a Lie algebra of harmonic bilocal fields, VM(x, y), where the M span a finite dimensional real matrix algebra \MA closed under transposition. The associative algebra \MA is irreducible iff its commutant \MA^{\prime} coincides with one of the three real division rings. The Lie algebra of (the modes of) the bilocal fields is in each case an infinite-dimensional Lie algebra: a central extension of {sp}(\infty,{\bb R}) corresponding to the field {{\bb R}} of reals, of u(∞, ∞) associated with the field {{\bb C}} of complex numbers, and of so*(4∞) related to the algebra {{\bb H}} of quaternions. They give rise to quantum field theory models with superselection sectors governed by the (global) gauge groups O(N), U(N) and U(N,{{\bb H}})=Sp(2N) , respectively. Lecture at the workshops 'Lie Theory and Its Applications in Physics', 18-24 June 2007, Varna, Bulgaria; 'Infinite- Dimensional Algebras and Quantum Integrable Systems', 23-27 July, 2007, Faro, Portugal; and 'Supersymmetries and Quantum Symmetries', 30 July-4 August, 2007, Dubna, Russia.

If one were to talk about special families of matrices on which Dr. Fiedler has worked, one could focus on families determined by a single letter: F-matrices, K-matrices, M-matrices, N-matrices, Pmatrices, Z-matrices. One could focus on families determined by a letter and a subscript: K 0-matrices, M 0-matrices, N 0-matrices. One could focus on families determined by two letters, PM-matrices, MMmatrices, ZM-matrices. One could even focus on families determined by three letters, MMA-matrices, ZME-matrices, ZMO-matrices, ZMA-matrices.

In this paper we characterize all nonnegative matrices whose inverses are M-matrices with unipathic digraphs. A digraph is called unipathic if there is at most one simple path from any vertex j to any other vertex k. The set of unipathic digraphs on n vertices includes the simple n-cycle and all digraphs whose underlying undirected graphs are trees (or forests). Our results facilitate the construction of nonnegative matrices whose inverses are M-matrices with unipathic digraphs. We highlight this procedure for inverses of tridiagonal M-matrices and of M-matrices whose digraphs are simple n-cycles with loops. AMS subject classifications. 15A09, 15A57