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  • 1
    Electronic Resource
    Electronic Resource
    Numerical computation of an analytic singular value decomposition of a matrix valued function (1991)
    Springer
    Numerische Mathematik 60 (1991), S. 1-39 
    Details
    ISSN: 0945-3245
    Keywords: 65F20 ; 65F25 ; 65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper extends the singular value decomposition to a path of matricesE(t). An analytic singular value decomposition of a path of matricesE(t) is an analytic path of factorizationsE(t)=X(t)S(t)Y(t) T whereX(t) andY(t) are orthogonal andS(t) is diagonal. To maintain differentiability the diagonal entries ofS(t) are allowed to be either positive or negative and to appear in any order. This paper investigates existence and uniqueness of analytic SVD's and develops an algorithm for computing them. We show that a real analytic pathE(t) always admits a real analytic SVD, a full-rank, smooth pathE(t) with distinct singular values admits a smooth SVD. We derive a differential equation for the left factor, develop Euler-like and extrapolated Euler-like numerical methods for approximating an analytic SVD and prove that the Euler-like method converges.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385712
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils (1998)
    Springer
    Numerische Mathematik 78 (1998), S. 329-358 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65F15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. A new method is presented for the numerical computation of the generalized eigenvalues of real Hamiltonian or symplectic pencils and matrices. The method is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order $\sqrt{\varepsilon}$ , where $\varepsilon $ is the machine precision, the new method computes the eigenvalues to full possible accuracy.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050315
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Regular solutions of nonlinear differential-algebraic equations and their numerical determination (1998)
    Springer
    Numerische Mathematik 79 (1998), S. 581-600 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65L99, 34A09
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. For a general class of nonlinear (possibly higher index) differential-algebraic equations we show existence and uniqueness of solutions. These solutions are regular in the sense that Newton's method will converge locally and quadratically. On the basis of the presented theoretical results, numerical methods for the determination of consistent initial values and for the computation of regular solutions are developed. Several numerical examples are included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050353
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Convergence of block iterative methods for linear systems arising in the numerical solution of Euler equations (1991)
    Springer
    Numerische Mathematik 59 (1991), S. 541-559 
    Details
    ISSN: 0945-3245
    Keywords: 65F10 ; 65N22 ; 15A48
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We discuss block matrices of the formA=[A ij ], whereA ij is ak×k symmetric matrix,A ij is positive definite andA ij is negative semidefinite. These matrices are natural block-generalizations of Z-matrices and M-matrices. Matrices of this type arise in the numerical solution of Euler equations in fluid flow computations. We discuss properties of these matrices, in particular we prove convergence of block iterative methods for linear systems with such system matrices.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385795
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    A quaternion QR algorithm (1989)
    Springer
    Numerische Mathematik 55 (1989), S. 83-95 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F 15 ; 65-04 ; 15-04 ; 15A 33 ; 81-04 ; CR: G 1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper extends the Francis QR algorithm to quaternion and antiquaternion matrices. It calculates a quaternion version of the Schur decomposition using quaternion unitary similarity transformations. Following a finite step reduction to a Hessenberg-like condensed form, a sequence of implicit QR steps reduces the matrix to triangular form. Eigenvalues may be read off the diagonal. Eigenvectors may be obtained from simple back substitutions. For serial computation, the algorithm uses only half the work and storage of the unstructured Francis QR iteration. By preserving quaternion structure, the algorithm calculates the eigenvalues of a nearby quaternion matrix despite rounding errors.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01395873
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  • 6
    Electronic Resource
    Electronic Resource
    Smooth factorizations of matrix valued functions and their derivatives (1991)
    Springer
    Numerische Mathematik 60 (1991), S. 115-131 
    Details
    ISSN: 0945-3245
    Keywords: 65L02 ; 65F25 ; 93C15 ; 93B40
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we study the numerical factorization of matrix valued functions in order to apply them in the numerical solution of differential algebraic equations with time varying coefficients. The main difficulty is to obtain smoothness of the factors and a numerically accessible form of their derivatives. We show how this can be achieved without numerical differentiation if the derivative of the given matrix valued function is known. These results are then applied in the numerical solution of differential algebraic Riccati equations. For this a numerical algorithm is given and its properties are demonstrated by a numerical example.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385717
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    The linear quadratic optimal control problem for linear descriptor systems with variable coefficients (1997)
    Springer
    Mathematics of control, signals, and systems 10 (1997), S. 247-264 
    Details
    ISSN: 1435-568X
    Keywords: Descriptor systems ; Differential algebraic equations ; Optimal control ; Strangeness index ; Riccati differential-algebraic equations ; Euler-Lagrange equations ; Linear feedback
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology , Mathematics , Technology
    Notes: Abstract We study linear quadratic optimal control problems for linear variable coefficient descriptor systems. Generalization from the case of standard control problems leads to several difficulties. We discuss a behavioral approach that solves some of these difficulties. Furthermore, an analysis of general rectangular systems is given and generalized Euler-Lagrange equations and Riccati differential algebraic equations are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01211506
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    Paper (German National Licenses)
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  • 8
    Electronic Resource
    Electronic Resource
    Minimization of the norm, the norm of the inverse and the condition number of a matrix by completion (1995)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 2 (1995), S. 155-171 
    Details
    ISSN: 1070-5325
    Keywords: condition number ; norm of a matrix ; matrix completion ; dilation theory ; robust regularization of descriptor systems ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We study the problem of minimizing the norm, the norm of the inverse and the condition number with respect to the spectral norm, when a submatrix of a matrix can be chosen arbitrarily. For the norm minimization problem we give a different proof than that given by Davis/Kahan/Weinberger. This new approach can then also be used to characterize the completions that minimize the norm of the inverse. For the problem of optimizing the condition number we give a partial result.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nla.1680020207
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  • 9
    Book
    Book
    Differential-algebraic equations : analysis and numerical solutions (2006)
    Zürich :European Mathematical Soc./Publ. House,
    Details
    Title: Differential-algebraic equations : analysis and numerical solutions
    Author: Kunkel, Peter
    Contributer: Mehrmann, Volker
    Publisher: Zürich :European Mathematical Soc./Publ. House,
    Year of publication: 2006
    Pages: VIII, 377 S.
    Series Statement: EMS textbooks in mathematics
    ISBN: 3-03719-017-5
    Type of Medium: Book
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  • 10
    Book
    Book
    Production factor mathematics / (2010)
    Berlin ; London :Springer,
    Details
    Title: Production factor mathematics /
    Contributer: Grötschel, Martin , Lucas, Klaus , Mehrmann, Volker
    Uniform Title: Produktionsfaktor Mathematik, engl.
    Publisher: Berlin ; London :Springer,
    Year of publication: 2010
    Pages: XIV, 402 S. ; , 24 cm
    ISBN: 978-3-642-11247-8 , 3-642-11247-1
    Type of Medium: Book
    Language: English
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