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Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms
Get PDFTheoretical and computational aspects of matrix inverse trigonometric and inverse hyperbolic functions are studied. Conditions for existence are given, all possible values are characterized, and principal values \acos, \asin, \acosh, and \asinh are defined and shown to be unique primary matrix functions. Various functional identities are derived, some of which are new even in the scalar case, with care taken to specify precisely the choices of signs and branches. New results include a ``round trip'' formula that relates to and similar formulas for the other inverse functions. Key tools used in the derivations are the matrix unwinding function and the matrix sign function. A new inverse scaling and squaring type algorithm employing a Schur decomposition and variable-degree Pad\'e approximation is derived for computing acos, and it is shown how it can also be used to compute asin, acosh, and asinh. In numerical experiments the algorithm is found to behave in a forward stable fashion and to be superior to computing these functions via logarithmic formulas
A Catalogue of Software for Matrix Functions. Version 2.0
Get PDFA catalogue of software for computing matrix functions and their Fr\'echet derivatives is presented. For a wide variety of languages and for software ranging from commercial products to open source packages we describe what matrix function codes are available and which algorithms they implement
Engel-type subgroups and length parameters of finite groups
Get PDFLet be an element of a finite group . For a positive integer , let be the subgroup generated by all commutators over , where is repeated times. By Baer's theorem, if , then belongs to the Fitting subgroup . We generalize this theorem in terms of certain length parameters of . For soluble we prove that if, for some , the Fitting height of is equal to , then belongs to the th Fitting subgroup . For nonsoluble the results are in terms of nonsoluble length and generalized Fitting height. The generalized Fitting height of a finite group is the least number such that , where , and is the inverse image of the generalized Fitting subgroup . Let be the number of prime factors of counting multiplicities. It is proved that if, for some , the generalized Fitting height of is equal to , then belongs to , where depends only on and . The nonsoluble length~ of a finite group~ is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if , then belongs to a normal subgroup whose nonsoluble length is bounded in terms of and . We also state conjectures of stronger results independent of and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups
Less is More II: an optimistic view of piecewise smooth bifurcation theory
Get PDFThe analysis of piecewise smooth bifurcations reveals an alarming proliferation of cases as the dimension of phase space increases. Rather than attempt the derivation of exhaustive lists of possibilities we describe ways of giving less detailed, but possibly more useful, results
An inequality concerning the expected values of row-sum and column-sum products in Boolean matrices
Get PDFWe give a proof of a matrix inequality and indicate how it can be applied to establish a natural analogy principle in Pure Inductive Logic
Odd Order Products of Conjugate Involutions in Linear Groups over GF(2^a)
Get PDFLet be isomorphic to , , or , where . If is an involution lying in a -conjugacy class , then for arbitrary we show that as becomes large, the proportion of elements of which have odd-order product with tends to . Furthermore, for at most we give formulae for the number of elements in which have odd-order product with , in terms of
A Catalogue of Software for Matrix Functions. Version 2.0
Get PDFA catalogue of software for computing matrix functions and their Fr\'echet derivatives is presented. For a wide variety of languages and for software ranging from commercial products to open source packages we describe what matrix function codes are available and which algorithms they implement
A Catalogue of Software for Matrix Functions. Version 2.0
Get PDFA catalogue of software for computing matrix functions and their Fr\'echet derivatives is presented. For a wide variety of languages and for software ranging from commercial products to open source packages we describe what matrix function codes are available and which algorithms they implement
Computing the weighted geometric mean of two large-scale matrices and its inverse times a vector
Get PDFWe investigate different approaches for the computation of the action of the weighted geometric mean of two large-scale positive definite matrices on a vector. We derive several algorithms, based on numerical quadrature and the Krylov subspace, and compare them in terms of convergence speed and execution time. By exploiting an algebraic relation between the weighted geometric mean and its inverse, we show how these methods can be used for the solution of large linear system whose coefficient matrix is a weighted geometric mean. We derive two novel algorithms, based on Gauss�Jacobi quadrature, and tailor an existing technique based on contour integration. On the other hand, we adapt several existing Krylov subspace techniques to the computation of the weighted geometric mean. According to our experiments, both classes of algorithms perform well on some problems but there is no clear winner, while some problem-dependent recommendations are provided