TUHH Open Research (TORE)https://tore.tuhh.deTORE captures, stores, indexes, preserves, and distributes digital research material.Mon, 14 Jun 2021 21:30:45 GMT2021-06-14T21:30:45Z50241- Yet more elementary proofs that the determinant of a symplectic matrix is 1http://hdl.handle.net/11420/3836Title: Yet more elementary proofs that the determinant of a symplectic matrix is 1
Authors: Bünger, Florian; Rump, Siegfried M.
Abstract: It seems to be of recurring interest in the literature to give alternative proofs for the fact that the determinant of a symplectic matrix is one. We state four short and elementary proofs for symplectic matrices over general fields. Two of them seem to be new.
Thu, 21 Nov 2019 07:48:56 GMThttp://hdl.handle.net/11420/38362019-11-21T07:48:56Z
- The determinant of a complex matrix and Gershgorin circleshttp://hdl.handle.net/11420/3111Title: The determinant of a complex matrix and Gershgorin circles
Authors: Bünger, Florian; Rump, Siegfried M.
Abstract: Each connected component of the Gershgorin circles of a matrix contains exactly as many eigenvalues as circles are involved. Thus, the Minkowski (set) product of all circles contains the determinant if all circles are disjoint. In [S.M. Rump. Bounds for the determinant by Gershgorin circles. Linear Algebra and its Applications, 563:215219, 2019.], it was proved that statement to be true for real matrices whose circles need not to be disjoint. Moreover, it was asked whether the statement remains true for complex matrices. This note answers that in the affrmative. As a by-product, a parameterization of the outer loop of a Cartesian oval without case distinction is derived.
Fri, 16 Aug 2019 08:45:45 GMThttp://hdl.handle.net/11420/31112019-08-16T08:45:45Z
- Verified solutions of two-point boundary value problems for nonlinear oscillatorshttp://hdl.handle.net/11420/7920Title: Verified solutions of two-point boundary value problems for nonlinear oscillators
Authors: Bünger, Florian
Tue, 24 Nov 2020 09:31:50 GMThttp://hdl.handle.net/11420/79202020-11-24T09:31:50Z
- Lower Bounds for the Smallest Singular Value of Certain Toeplitz-like Triangular Matrices with Linearly Increasing Diagonal Entrieshttp://hdl.handle.net/11420/3366Title: Lower Bounds for the Smallest Singular Value of Certain Toeplitz-like Triangular Matrices with Linearly Increasing Diagonal Entries
Authors: Bünger, Florian; Rump, Siegfried M.
Abstract: Let L be a lower triangular n× n-Toeplitz matrix with first column (μ, α, β, α, β, … ) T, where μ, α, β≥ 0 fulfill α- β∈ [ 0 , 1 ) and α∈ [ 1 , μ+ 3 ]. Furthermore let D be the diagonal matrix with diagonal entries 1 , 2 , … , n. We prove that the smallest singular value of the matrix A: = L+ D is bounded from below by a constant ω= ω(μ, α, β) > 0 which is independent of the dimension n.
Mon, 16 Sep 2019 13:21:27 GMThttp://hdl.handle.net/11420/33662019-09-16T13:21:27Z
- A short note on the ratio between sign-real and sign-complex spectral radius of a real square matrixhttp://hdl.handle.net/11420/3852Title: A short note on the ratio between sign-real and sign-complex spectral radius of a real square matrix
Authors: Bünger, Florian
Abstract: For a real (n×n)-matrix A the sign-real and the sign-complex spectral radius – invented by Rump – are respectively defined as ρR(A):=max|λ|||Ax|=|λx|,λ∈R,x∈Rn{0,ρC(A):=max|λ|||Ax|=|λx|,λ∈C,x∈Cn{0. For n≥2 we prove ρR(A)≥ζn ρC(A) where the constant ζn:=[formula omitted] is best possible.
Fri, 22 Nov 2019 11:25:37 GMThttp://hdl.handle.net/11420/38522019-11-22T11:25:37Z
- Inverses, determinants, eigenvalues, and eigenvectors of real symmetric Toeplitz matrices with linearly increasing entrieshttp://hdl.handle.net/11420/7882Title: Inverses, determinants, eigenvalues, and eigenvectors of real symmetric Toeplitz matrices with linearly increasing entries
Authors: Bünger, Florian
Abstract: We explicitly determine the skew-symmetric eigenvectors and corresponding eigenvalues of the real symmetric Toeplitz matricesT=T(a,b,n):=( a+b|j-k|)1≤j,k≤n of order n≥3 where a, b ∈ ℝ, b ≠0. The matrix T is singular if and only if c := a/b = -n-1/2. In this case we also explicitly determine the symmetric eigenvectors and corresponding eigenvalues of T. If T is regular, we explicitly compute the inverse T- 1, the determinant det T, and the symmetric eigenvectors and corresponding eigenvalues of T are described in terms of the roots of the real self-inversive polynomial pn(δ;z):=(zn+1- δzn-δz+1)/(z+1) if n is even, and pn(δ; z):=zn+1-δzn-δz+1 if n is odd, δ:=1+2/(2c+n-1). © 2014 Elsevier Inc.
Thu, 19 Nov 2020 12:18:41 GMThttp://hdl.handle.net/11420/78822020-11-19T12:18:41Z
- Accelerating interval matrix multiplication by mixed precision arithmetichttp://hdl.handle.net/11420/7823Title: Accelerating interval matrix multiplication by mixed precision arithmetic
Authors: Ozaki, Katsuhisa; Ogita, Takeshi; Bünger, Florian; Oishi, Shin’ichi
Abstract: This paper is concerned with real interval arithmetic. We focus on interval matrix multiplication. Well-known algorithms for this purpose require the evaluation of several point matrix products to compute one interval matrix product. In order to save computing time we propose a method that modifies such known algorithm by partially using low-precision floating-point arithmetic. The modified algorithms work without significant loss of tightness of the computed interval matrix product but are about 30% faster than their corresponding original versions. The negligible loss of accuracy is rigorously estimated.
Fri, 13 Nov 2020 08:11:58 GMThttp://hdl.handle.net/11420/78232020-11-13T08:11:58Z
- A matrix-decomposition theorem for GLn (K)http://hdl.handle.net/11420/8775Title: A matrix-decomposition theorem for GLn (K)
Authors: Bünger, Florian; Nielsen, Klaus
Abstract: Given an arbitrary commutative field K, n ∈ ℕ≥3 and two monic polynomials q and r over K of degree n - 1 and n such that q(0) ≠ 0 ≠ r(0). We prove that any non-scalar invertible n x n matrix M can be written as a product of two matrices A and B, where the minimum polynomial of A is divisible by q and B is cyclic with minimum polynomial r. This result yields that the Thompson conjecture is true for PSLn(F3), n ∈ ℕ≥3, and PSL2n+1(F2), n ∈ ℕ. If G is such a group, then G has a conjugacy class Ω such that G = Ω2. In particular each element of G is a commutator.
Fri, 12 Feb 2021 10:29:17 GMThttp://hdl.handle.net/11420/87752021-02-12T10:29:17Z
- A Taylor model toolbox for solving ODEs implemented in MATLAB/INTLABhttp://hdl.handle.net/11420/3648Title: A Taylor model toolbox for solving ODEs implemented in MATLAB/INTLAB
Authors: Bünger, Florian
Abstract: The new INTLAB release V11 contains two verified ODE solvers. One is a MATLAB implementation of Lohner's classical AWA, the other one follows the so-called Taylor model approach which is the main subject of this article.
Thu, 24 Oct 2019 11:07:39 GMThttp://hdl.handle.net/11420/36482019-10-24T11:07:39Z
- Products of symmetries in unitary groupshttp://hdl.handle.net/11420/8776Title: Products of symmetries in unitary groups
Authors: Bünger, Florian; Knüppel, Frieder; Nielsen, Klaus
Abstract: Given a regular --hermitian form on a finite-dimensional vector space V over a commutative field K of characteristic ≠ 2 such that the norm on K is surjective onto the fixed field of - (this is true whenever K is finite). Call an element σ of the unitary group a symmetry if σ2 = 1 and the negative space of σ is 1-dimensional. If π is unitary and det π ∈ 1, -1, we prove that π is a product of symmetries (with a few exceptions when K = GF 9 and dim V = 2) and we find the minimal number of factors in such a product.
Fri, 12 Feb 2021 10:41:30 GMThttp://hdl.handle.net/11420/87762021-02-12T10:41:30Z
- Products of quasi-involutions in unitary groupshttp://hdl.handle.net/11420/8762Title: Products of quasi-involutions in unitary groups
Authors: Bünger, Florian; Knüppel, Frieder
Abstract: Given a regular - -hermitian form on an n-dimensional vector space V over a commutative field K of characteristic ≠ 2 (n ∈ ℕ). Call an element σ of the unitary group a quasi-involution if σ is a product of commuting quasi-symmetries (a quasi-symmetry is a unitary transformation with a regular (n - 1)-dimensional fixed space). In the special case of an orthogonal group every quasi-involution is an involution. Result: every unitary element is a product of five quasi-involutions. If K is algebraically closed then three quasi-involutions suffice.
Thu, 11 Feb 2021 08:29:03 GMThttp://hdl.handle.net/11420/87622021-02-11T08:29:03Z
- A note on the boundary shape of matrix polytope productshttp://hdl.handle.net/11420/7879Title: A note on the boundary shape of matrix polytope products
Authors: Bünger, Florian
Abstract: Motivated by interval matrix multiplication we consider (matrix) polytopes A ⊆ ℝm,n, B ⊆ ℝn,k, m, n, k ∈ ℕ, and investigate the boundary shape of their pointwise product AB:= AB | A ∈ A,B ∈ B: We prove that AB cannot have outward curved boundary sections while inward curved sections may exist. This is achieved by a simple local extreme point analysis. Results are proved in a more general abstract setting for images of compact sets of (not necessarily finite dimensional) locally convex vector spaces under continuous multilinear mappings. They can be seen as extensions of the Zadeh-Desoer Mapping Theorem which is a fundamental tool in control theory.
Thu, 19 Nov 2020 11:10:35 GMThttp://hdl.handle.net/11420/78792020-11-19T11:10:35Z
- An extended similarity theory applied to heated flows in complex geometrieshttp://hdl.handle.net/11420/8558Title: An extended similarity theory applied to heated flows in complex geometries
Authors: Bünger, Florian; Herwig, Heinz
Abstract: In the traditional similarity theory the influence of temperature- and pressure-dependent fluid properties on the flow field and heat transfer is not described by the basic dimensionless parameters, i.e. Prandtl, Reynolds, Rayleigh,... number. We present an extended similarity theory that not only takes into account the variable material properties but also can handle small variations in other parameters of the physical model like small changes in the (reference) Prandtl number. The method has general applicability that is suitable for a wide variety of fluid dynamic and heat transfer situations in which variable properties with a strong dependence on temperature and pressure play a significant role. It is especially useful in predicting the behaviour of a certain fluid based on the results for a different one. As an example the Nußelt number of a lid driven heated cavity is determined with fluid properties being temperature dependent.
Mon, 25 Jan 2021 09:06:53 GMThttp://hdl.handle.net/11420/85582021-01-25T09:06:53Z
- Improved error bounds for floating-point products and Horner’s schemehttp://hdl.handle.net/11420/5500Title: Improved error bounds for floating-point products and Horner’s scheme
Authors: Rump, Siegfried M.; Bünger, Florian; Jeannerod, Claude Pierre
Abstract: Let (Formula presented.) denote the relative rounding error of some floating-point format. Recently it has been shown that for a number of standard Wilkinson-type bounds the typical factors (Formula presented.) can be improved into (Formula presented.) , and that the bounds are valid without restriction on (Formula presented.). Problems include summation, dot products and thus matrix multiplication, residual bounds for (Formula presented.) - and Cholesky-decomposition, and triangular system solving by substitution. In this note we show a similar result for the product (Formula presented.) of real and/or floating-point numbers (Formula presented.) , for computation in any order, and for any base (Formula presented.). The derived error bounds are valid under a mandatory restriction of (Formula presented.). Moreover, we prove a similar bound for Horner’s polynomial evaluation scheme.
Thu, 26 Mar 2020 09:03:22 GMThttp://hdl.handle.net/11420/55002020-03-26T09:03:22Z
- Products of involutions in unitary groupshttp://hdl.handle.net/11420/8778Title: Products of involutions in unitary groups
Authors: Bünger, Florian
Fri, 12 Feb 2021 12:22:51 GMThttp://hdl.handle.net/11420/87782021-02-12T12:22:51Z
- The product of two quadratic matriceshttp://hdl.handle.net/11420/8761Title: The product of two quadratic matrices
Authors: Bünger, Florian; Knüppel, Frieder; Nielsen, Klaus
Abstract: Let p=(x-β)(x-β-1)∈K[x] where β2≠β-2 and let V be a finite-dimensional vector space over the field K. A linear mapping M:V→V is called quadratic if p(M)=0. We characterize products of two quadratic linear mappings. © 2001 Elsevier Science Inc.
Thu, 11 Feb 2021 08:12:05 GMThttp://hdl.handle.net/11420/87612021-02-11T08:12:05Z
- A short note on the convexity of interval matrix-vector productshttp://hdl.handle.net/11420/7816Title: A short note on the convexity of interval matrix-vector products
Authors: Bünger, Florian; Rump, Siegfried M.
Abstract: We investigate the question under which circumstances the pointwise interval matrix-vector product Ax := fAx j A 2 A; x 2 xg of a real interval matrix A 2 IRm;n and a real interval vector x 2 IRn is convex.
Fri, 13 Nov 2020 07:06:14 GMThttp://hdl.handle.net/11420/78162020-11-13T07:06:14Z
- On the zeros of eigenpolynomials of hermitian Toeplitz matriceshttp://hdl.handle.net/11420/7881Title: On the zeros of eigenpolynomials of hermitian Toeplitz matrices
Authors: Bünger, Florian
Abstract: This article refines a result of Delsarte, Genin, Kamp (Circuits Syst Signal Process 3:207–223, 1984), and Delsarte and Genin (Springer Lect Notes Control Inf Sci 58:194–213, 1984), regarding the number of zeros on the unit circle of eigenpolynomials of complex Hermitian Toeplitz matrices and generalized Caratheodory representations of such matrices. This is achieved by exploring a key observation of Schur (Über einen Satz von C. Carathéodory. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften, pp. 4–15, 1912) stated in his proof of a famous theorem of Carathéodory (Rendiconti del Circolo Matematico di Palermo 32:193–217, 1911). In short, Schur observed that companion matrices corresponding to eigenpolynomials of Hermitian Toeplitz matrices H define isometries with respect to (spectrum shifted) submatrices of H. Looking at possible normal forms of these isometries leads directly to the results. This geometric, conceptual approach can be generalized to Hermitian or symmetric Toeplitz matrices over arbitrary fields. Furthermore, as a byproduct, Iohvidov’s law in the jumps of the ranks and the connection between the Iohvidov parameter and the Witt index are established for such Toeplitz matrices.
Thu, 19 Nov 2020 12:11:06 GMThttp://hdl.handle.net/11420/78812020-11-19T12:11:06Z
- Involutionen als Erzeugende in unitären Gruppenhttp://hdl.handle.net/11420/8777Title: Involutionen als Erzeugende in unitären Gruppen
Authors: Bünger, Florian
Fri, 12 Feb 2021 10:52:43 GMThttp://hdl.handle.net/11420/87772021-02-12T10:52:43Z
- Simple floating-point filters for the two-dimensional orientation problemhttp://hdl.handle.net/11420/5502Title: Simple floating-point filters for the two-dimensional orientation problem
Authors: Ozaki, Katsuhisa; Bünger, Florian; Ogita, Takeshi; Oishi, Shin’ichi; Rump, Siegfried M.
Abstract: This paper is concerned with floating-point filters for a two dimensional orientation problem which is a basic problem in the field of computational geometry. If this problem is only approximately solved by floating-point arithmetic, then an incorrect result may be obtained due to accumulation of rounding errors. A floating-point filter can quickly guarantee the correctness of the computed result if the problem is well-conditioned. In this paper, a simple semi-static floating-point filter which handles floating-point exceptions such as overflow and underflow by only one branch is developed. In addition, an improved fully-static filter is developed.
Thu, 26 Mar 2020 09:07:17 GMThttp://hdl.handle.net/11420/55022020-03-26T09:07:17Z
- Standard integral table algebras with a faithful nonreal element of degree 5http://hdl.handle.net/11420/9429Title: Standard integral table algebras with a faithful nonreal element of degree 5
Authors: Ārād, Ṣevî; Bünger, Florian; Fisman, E.; Muzychuk, Mikhail
Abstract: This chapter deals with the classification of standard integral GT-algebras (A,B) with L(B) = 1 {1} and |b| ≥ 4 for all b ∈ B# which contain a nonreal faithful basis element b of degree 5. Starting from this point using the basic identity λxyz|z|⟨xy,z⟩=⟨x,zy¯⟩=λzy¯x|x|,x,y,z∈B,
one can list all possible representations of bb¯ and b 2 as linear combinations of basis elements (cf. Tables II and III of Subsection 3.3). Assuming that b commutes with b¯ yields the identity ⟨bb¯,bb¯⟩=⟨b2,b2⟩ which reduces the number of these representations (cf. Table III of Subsection 3.3). Then, using various kind of techniques (for example repeated application of the associa- tivity law), each of the reamining cases will be treated separately. In order to state the main result, we introduce the following base of a specific table algebra. © Springer-Verlag Berlin Heidelberg 2002
Tue, 04 May 2021 08:26:58 GMThttp://hdl.handle.net/11420/94292021-05-04T08:26:58Z
- Shrink wrapping for Taylor models revisitedhttp://hdl.handle.net/11420/2878Title: Shrink wrapping for Taylor models revisited
Authors: Bünger, Florian
Abstract: Taylor models have been used successfully to calculate verified inclusions of the solutions of initial value problems for ordinary differential equations. In this context, Makino and Berz introduced an accompanying method called shrink wrapping. This method aims to reduce the wrapping effect which occurs during repeated forward integration of Taylor models. We review shrink wrapping as proposed by Makino and Berz, state examples that point to a flaw in their theorem and concept of proof, and present a new, corrected version of shrink wrapping.
Tue, 02 Jul 2019 15:26:51 GMThttp://hdl.handle.net/11420/28782019-07-02T15:26:51Z
- Complex Disk Products and Cartesian Ovalshttp://hdl.handle.net/11420/3259Title: Complex Disk Products and Cartesian Ovals
Authors: Bünger, Florian; Rump, Siegfried M.
Abstract: Let DR, Dr, DS, Ds be complex disks with common center 1 and radii R, r, S, s, respectively. We consider the Minkowski products A: = DRDr and B: = DSDs and give necessary and sufficient conditions for A being a subset or superset of B. Partially, this extends to n-fold disk products D1… Dn, n> 2. It is well-known that the boundaries of A and B are outer loops of Cartesian ovals. Therefore, our results translate to necessary and sufficient conditions under which such loops encircle each other.
Mon, 02 Sep 2019 13:28:31 GMThttp://hdl.handle.net/11420/32592019-09-02T13:28:31Z
- Minimizing and maximizing the Euclidean norm of the product of two polynomialshttp://hdl.handle.net/11420/7919Title: Minimizing and maximizing the Euclidean norm of the product of two polynomials
Authors: Bünger, Florian
Abstract: We consider the problem of minimizing or maximizing the quotient, where p = p0 + p1x + ... + pmxm, q = q0 + q1x + ... + qnxn ∈ K[x], K ∈ R, C, are non-zero real or complex polynomials of maximum degree m, n ∈ ℕ respectively and double pipepdouble pipe := (pipep0pipe2 + ... + pipepmpipe2)1/2 is simply the Euclidean norm of the polynomial coefficients. Clearly fm,n is bounded and assumes its maximum and minimum values min fm,n = fm,n(pmin, qmin) and max fm,n = f(pmax, qmax). We prove that minimizers pmin, qmin for K = ¢ and maximizers pmax, qmax for arbitrary K fulfill deg(pmin) = m = deg(pmax), deg(qmin) = n = deg(qmax) and all roots of pmin, qmin, pmax, qmax have modulus one and are simple. For K = ℝ we can only prove the existence of minimizers pmin, qmin of full degree m and n respectively having roots of modulus one. These results are obtained by transferring the optimization problem to that of determining extremal eigenvalues and corresponding eigenvectors of autocorrelation Toeplitz matrices. By the way we give lower bounds for min fm,n for real polynomials which are slightly better than the known ones and inclusions for max fm,n. © 2010 Springer Science+Business Media, LLC.
Tue, 24 Nov 2020 09:23:01 GMThttp://hdl.handle.net/11420/79192020-11-24T09:23:01Z