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INCREMENTAL COMPUTATION OF block triangular matrix EXPONENTIALS WITH APPLICATION TO OPTION PRICING

ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 2017年 47卷 57-72页

作者： Kressner, Daniel Luce, Robert Statti, Francesco Ecole Polytech Fed Lausanne Stn 8 CH-1015 Lausanne Switzerland

We study the problem of computing the matrix exponential of a block triangular matrix in a peculiar way: block column by block column, from left to right. The need for such an evaluation scheme arises naturally in the context of option pricing in polynomial diffusion models. In this setting, a discretization process produces a sequence of nested block triangular matrices, and their exponentials are to be computed at each stage until a dynamically evaluated criterion allows to stop. Our algorithm is based on scaling and squaring. By carefully reusing certain intermediate quantities from one step to the next, we can efficiently compute such a sequence of matrix exponentials.

关键词： matrix exponential block triangular matrix polynomial diffusion models option pricing

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On the Segre characteristic of a block triangular matrix

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LINEAR ALGEBRA AND ITS APPLICATIONS 1999年 303卷 245-263页

作者： Cantó, R Urbano, AM Univ Politecn Valencia Dept Matemat Aplicada E-46071 Valencia Spain

Consider two complex matrices A and B of sizes n x n and m x m respectively, and let lambda be an eigenvalue of both matrices. If alpha = (n(1),n(2)) and beta = (m(1),m(2)) are the Segre characteristics associated with lambda of A and B, respectively, and y = (r(1), r(2),r(3),r(4)) is a nonincreasing sequence of nonnegative integers, then a method for determining when gamma is the Segre characteristic of the block triangular matrix M= [(A)(O) (L)(B)] in terms of alpha and beta, and the structure of matrix L is presented. A completely explicit description of the Segre characteristic of M associated with lambda is obtained. Using similar techniques, general cases when alpha, beta and gamma have more elements and satisfy some size restrictions are considered. (C) 1999 Elsevier Science Inc. All rights reserved.

关键词： Segre characteristics block triangular matrix Carlson problem Jordan canonical form

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Completions of inverse M-matrix patterns

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LINEAR ALGEBRA AND ITS APPLICATIONS 1998年第1-3期282卷 145-160页

作者： Hogben, L Iowa State Univ Sci & Technol Dept Math Ames IA 50011 USA

A list of positions in an n x n real matrix that includes all diagonal positions (a pattern) is said to have inverse M-completion if every partial inverse M-matrix that specifies exactly these positions can be completed to an inverse M-matrix. Johnson and Smith (C.R. Johnson, R.L. Smith, Linear Algebra and its Applications 241-243 (1996) 655-667) characterize the positionally symmetric patterns that have inverse M-completion as those patterns whose graphs are block-clique. In this paper the restriction of positional symmetry is removed: A pattern has inverse M-completion if and only if the digraph G of the pattern has the properties (a) the induced subdigraph of a cycle is a clique, and (b) if G contains both are (i, j) and a path of length >1 between i and j then the induced subdigraph of the path is a clique. Furthermore, any irreducible pattern with inverse M-completion is positionally symmetric and has a block-clique digraph. Any pattern is permutation similar to a block-lower-triangular pattern with irreducible diagonal blocks, and such a pattern has inverse M-completion if and only if (i) the pattern-digraph of each diagonal block is block-clique, (ii) each subdiagonal block contains at most one position, and (iii) the pattern-digraph of the block structure has no alternate path to a single are. (C) 1998 Elsevier Science Inc. All rights reserved.

关键词： M-matrix inverse M-matrix partial matrix completion block-clique block triangular matrix digraph pattern

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block triangular and skew-Hermitian splitting methods for positive-definite linear systems

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2005年第3期26卷 844-863页

作者： Bai, ZZ Golub, GH Lu, LZ Yin, JF Chinese Acad Sci Acad Math & Syst Sci Inst Computat Math & Sci Engn Comp State Key Lab Sci Engn Comp Beijing 100080 Peoples R China Stanford Univ Dept Comp Sci Sci Comp & Computat Math Program Stanford CA 94305 USA Xiamen Univ Dept Math Xiamen 361005 Peoples R China

By further generalizing the concept of Hermitian (or normal) and skew-Hermitian splitting for a non-Hermitian and positive-definite matrix, we introduce a new splitting, called positive-definite and skew-Hermitian splitting (PSS), and then establish a class of PSS methods similar to the Hermitian (or normal) and skew-Hermitian splitting (HSS or NSS) method for iteratively solving the positive-definite systems of linear equations. Theoretical analysis shows that the PSS method converges unconditionally to the exact solution of the linear system, with the upper bound of its convergence factor dependent only on the spectrum of the positive-definite splitting matrix and independent of the spectrum of the skew-Hermitian splitting matrix as well as the eigenvectors of all matrices involved. When we specialize the PSS to block triangular ( or triangular) and skew-Hermitian splitting (BTSS or TSS), the PSS method naturally leads to a BTSS or TSS iteration method, which may be more practical and efficient than the HSS and NSS iteration methods. Applications of the BTSS method to the linear systems of block two-by-two structures are discussed in detail. Numerical experiments further show the effectiveness of our new methods.

关键词： non-Hermitian matrix positive-definite matrix triangular matrix block triangular matrix Hermitian and skew-Hermitian splitting splitting iteration method

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On common α-scalar Lyapunov solutions

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LINEAR ALGEBRA AND ITS APPLICATIONS 2019年 563卷 123-141页

作者： Gumus, Mehmet Xu, Jianhong Auburn Univ Dept Math & Stat Auburn AL 36849 USA Southern Illinois Univ Dept Math Carbondale IL 62901 USA

We present two new characterizations regarding common alpha-scalar solutions for systems of Lyapunov matrix inequalities so as to bridge and broaden relevant existing results. As applications of such results, we further explore the cases involving a set of conformally partitioned block triangular matrices and a set of Z-matrices, respectively, to obtain new results concerning these cases, which are useful in the areas of control and systems theory and engineering. (C) 2018 Elsevier Inc. All rights reserved.

关键词： Stability Lyapunov matrix inequality Diagonal Lyapunov solution Positive semidefinite matrix Hadamard product block triangular matrix Z-matrix

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ON THE INDEX OF block UPPER-triangular MATRICES

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SIAM JOURNAL ON matrix ANALYSIS AND APPLICATIONS 1995年第2期16卷 436-447页

作者： BRU, R CLIMENT, JJ NEUMANN, M UNIV CONNECTICUT DEPT MATHSTORRSCT 06269 UNIV ALACANT DEPT TECNOL INFORMAT & COMPUTACIOE-03071 ALACANTSPAIN

Let M be an upper block triangular matrix with A and B singular diagonal blocks. It is known that max{index(A), index(B)} less than or equal to index(M) less than or equal to index(A) + index(B). Recently, a necessary and sufficient condition has been given so that index(M) = index(A)S-index(B). In this paper we find various characterizations for index(M) to take any specific values between max{ index(il), index(B)} and index(A) + index(B) which generalize previous results.

关键词： INDEX block triangular matrix GENERALIZED EIGENSPACE GENERALIZED EIGENVECTOR DRAZIN INVERSE HEIGHT DEPTH

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On regularity of block triangular fuzzy matrices

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Journal of Applied Mathematics and Computing 2004年第1-2期16卷 207-220页

作者： Meenakshi, Ar. Department of Mathematics Annamalai University Annamalainagar-608002 India

Necessary and sufficient conditions are given for the regularity of block triangular fuzzy matrices. This leads to characterization of idempotency of a class of triangular Toeplitz matrices. As an application, the existence of group inverse of a block triangular fuzzy matrix is discussed. Equivalent conditions for a regular block triangular fuzzy matrix to be expressed as a sum of regular block fuzzy matrices is derived. Further, fuzzy relations equations consistency is studied.

关键词： block triangular matrix Fuzzy matrix Fuzzy relational equation triangular Toeplitz matrix

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block triangular and skew symmetric splitting method for steady state vector of linear system of erogodic block circulant Markov chains

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INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND MATHEMATICS 2017年第4期8卷 384-394页

作者： Donthi, Ranadheer Dasari, Rajaiah Perati, Malla Reddy Medak Coll Engn & Technol Dept Math Math Siddipet Telangana India Kakatiya Univ Dept Math Math Warangal Telangana India Kakatiya Univ Dept Math Warangal Telangana India

In this paper, we determine steady state probability vector p of erogodic block circulant Markov chain using block triangular and skew symmetric method. The homogeneous system pi Q = 0 is transformed to the non homogeneous regularised linear system Ax = b, and proved that the matrix A = Q(T) + epsilon I is positive definite for epsilon > 0. The contraction factor alpha minimises the spectral radius of block iteration matrix of block coefficient matrix A. To improve computing efficiency of the TSS iteration, we employ ITSS iteration. From the numerical results it is clear that the error of TSS iteration method converges rapidly when compared to other existing methods.

关键词： circulant stochastic matrices steady state probability vector block triangular matrix block skew symmetric matrix TSS method convergence analysis

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Centralizing additive maps on rank r block triangular matrices

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ACTA SCIENTIARUM MATHEMATICARUM 2021年第1-2期87卷 63-94页

作者： Chooi, W. L. Mutalib, M. H. A. Tan, L. Y. Univ Malaya Inst Math Sci Kuala Lumpur Malaysia Tunku Abdul Rahman Univ Coll Dept Math Sci Kuala Lumpur Malaysia

Let F be a field and let k, n(1), ..., n(k) be positive integers with n(1) + ...+ n(k) = n >= 2. We denote by Tn(1), ..., n(k) a block triangular matrix algebra over F with unity I-n and center Z(Tn(1), ..., n(k)). Fixing an integer 1 < r <= n with r not equal n when vertical bar F vertical bar = 2, we prove that an additive map psi: Tn(1), ..., n(k) -> Tn(1), ..., n(k )satisfies psi(A)A - A psi(A) is an element of Z(TTn(1), ..., n(k)) for every rank r matrices A is an element of Tn(1), ..., n(k )if and only if there exist an additive map mu: Tn(1), ..., n(k) -> F and scalars lambda, alpha is an element of F, in which alpha not equal 0 only if r = n, n(1) = n(k) = 1 and vertical bar F vertical bar = 3, such that psi(A) = lambda A + mu(A)I-n + alpha(a(11) + a(nn))E-1n for all A = (a(ij)) is an element of T-n1, ..., n(k), where E-ij is an element of Tn(1), ..., n(k) is the matrix unit whose (i, j)th entry is one and zero elsewhere. Using this result, a complete structural characterization of commuting additive maps on rank s > 1 upper triangular matrices over an arbitrary field is addressed.

关键词： centralizing map commuting map block triangular matrix rank functional identity

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Splitting iteration methods for non-Hermitian positive definite systems of linear equations

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Hokkaido Mathematical Journal 2007年第4期36卷 801-814页

作者： Bai, Zhong-Zhi State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100080 P.O. Box 2719 China

For large sparse system of linear equations with a non-Hermitian positive definite coefficient matrix, we review the recently developed Hermitian/skew-Hermitian splitting (HSS) iteration, normal/skew-Hermitian splitting (NSS) iteration, positive-definite/ skew-Hermitian splitting (PSS) iteration, and block triangular/skew-Hermitian splitting (BTSS) iteration. These methods converge unconditionally to the exact solution of the linearsystem, with the upper bounds of their convergence factors being only dependent on the spectrum of the Hermitian (or normal, or positive-definite) splitting matrix, but independent of the spectrum of the skew-Hermitian splitting matrix as well as the eigenvectors of all matrices involved. © 2007 by the University of Notre Dame. All rights reserved.

关键词： block triangular matrix Hermitian and skew-Hermitian splitting Non-Hermitian matrix Normal/Skew-Hermitian splitting Positive definite matrix Positivedefinite/Skew-Hermitian splitting Splitting iteration method

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