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  • 1
    Electronic Resource
    Electronic Resource
    Remarks on band matrices (1967)
    Springer
    Numerische Mathematik 10 (1967), S. 153-161 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this note we consider band- or tridiagonal-matrices of orderk whose elements above, on, and below the diagonal are denoted byb i ,a i,c i . In the periodic case, i.e.b i+m =b i etc., we derive fork=nm andk=nm−1 formulas for the characteristic polynomial and the eigenvectors under the assumption that $$\mathop \prod \limits_{i = 1}^m c_i b_i 〉 0$$ i=1 In the latter case it is shown that the characteristic polynomial is divisible by them−1-th minor, as was already observed byRósa. We also give estimations for the number of real roots and an application to Fibonacci numbers.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02174148
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  • 2
    Electronic Resource
    Electronic Resource
    Bemerkung zu einer Arbeit von Marshall und Olkin (1969)
    Springer
    Numerische Mathematik 13 (1969), S. 245-246 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02167555
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  • 3
    Electronic Resource
    Electronic Resource
    Convergence of sequential and asynchronous nonlinear paracontractions (1992)
    Springer
    Numerische Mathematik 62 (1992), S. 305-319 
    Details
    ISSN: 0945-3245
    Keywords: MSC 1991 ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We establish the convergence of sequential and asynchronous iteration schemes for nonlinear paracontracting operators acting in finite dimensional spaces. Applications to the solution of linear systems of equations with convex constraints are outlined. A first generalization of one of our convergence results to an infinite pool of asymptotically paracontracting operators is also presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396232
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  • 4
    Electronic Resource
    Electronic Resource
    Convergence properties of ART and SOR algorithms (1991)
    Springer
    Numerische Mathematik 59 (1991), S. 91-106 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) 65F10 ; 15A06 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary ART algorithms with relaxation parameters are studied for general (consistent or inconsistent) linear algebraic systemsRx=f, and a general convergence theorem is formulated. The advantage of severe underrelaxation is reexamined and clarified. The relationship to solutions obtained by applying SOR methods to the equationRR T y=f is investigated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385772
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  • 5
    Electronic Resource
    Electronic Resource
    A note on optimal block-scaling of matrices (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 127-128 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F35 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary After pointing out that two recent results on optimal blockscaling are equivalent, a new short and simple proof of both results is given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389760
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  • 6
    Electronic Resource
    Electronic Resource
    Minimizing the condition number of a positive definite matrix by completion (1994)
    Springer
    Numerische Mathematik 69 (1994), S. 17-23 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65F35, 15A12
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. We consider the problem of minimizing the spectral condition number of a positive definite matrix by completion: $$\min\left\{ {\rm cond}\left(\mat{cc} A & B^{\rm H} \\ B & X \rix\right): \mat{cc} A & B^{\rm H} \\ B & X \rix \mbox {\rm positive definite} \right\},$$ \noindent where $A$ is an $n\times n$ Hermitian positive definite matrix, $B$ a $p\times n$ matrix and $X$ is a free $p\times p$ Hermitian matrix. We reduce this problem to an optimization problem for a convex function in one variable. Using the minimal solution of this problem we characterize the complete set of matrices that give the minimum condition number.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050077
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  • 7
    Electronic Resource
    Electronic Resource
    Verfahren zur Berechnung des Spektralradius nichtnegativer irreduzibler Matrizen (1971)
    Springer
    Computing 8 (1971), S. 32-39 
    Details
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Summary Several iterative methods for determining the spectral radius ϱ of a nonnegative matrixA are given. Convergence is proved under the assumption of irreducibility. Contrary to the power method, these methods are insensitive to the presence of other eigenvalues with modulus ϱ.
    Notes: Zusammenfassung Es werden mehrere Iterationsverfahren zur Bestimmung des Spektralradius ϱ einer nichtnegativen MatrixA angegeben. Ihre Konvergenz wird unter der Voraussetzung bewiesen, daßA irreduzibel ist. Die Verfahren sind, im Gegensatz zur Potenzmethode, unempfindlich gegen das Vorhandensein mehrerer Eigenwerte vom Betrage ϱ.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02234041
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  • 8
    Electronic Resource
    Electronic Resource
    Verfahren zur Berechnung des Spektralradius nichtnegativer irreduzibler Matrizen II (1972)
    Springer
    Computing 9 (1972), S. 69-73 
    Details
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Summary Supplementary to a former paper with the same title additional iterative methods for computing the spectralradius and the positive eigenvector of a nonnegative irreducible matrix are given and convergence is proved. Here the smallest quotients of the approximating vectors are monotously converging to the spectral radius. Inclusions for the positive eigenvectory are given, from which the convergence of the approximations toy is deduced.
    Notes: Zusammenfassung In Ergänzung einer früheren gleichlautenden Arbeit werden weitere iterative Verfahren zur Berechnung des Spektralradius und des zugehörigen positiven Eigenvektors einer nichtnegativen irreduziblen Matrix angegeben und die Konvergenz bewiesen. Dabei streben die kleinsten Quotienten der Näherungsvektoren monoton gegen den Spektralradius. Es werden Einschließungsaussagen für den positiven Eigenvektory bewiesen und daraus die Konvergenz der Näherungen, gegeny hergeleitet.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02236379
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  • 9
    Electronic Resource
    Electronic Resource
    The limit of the spectral radius of block Toeplitz matrices with nonnegative entries (2000)
    Springer
    Integral equations and operator theory 36 (2000), S. 193-200 
    Details
    ISSN: 1420-8989
    Keywords: Primary 15A42 ; 15A48 ; 47B35 ; 47B65
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A bi-infinite sequence ...,t −2,t −1,t 0,t 1,t 2,... of nonnegativep×p matrices defines a sequence of block Toeplitz matricesT n =(t ik ),n=1,2,...,, wheret ik =t k−i ,i,k=1,...,n. Under certain irreducibility assumptions, we show that the limit of the spectral radius ofT n , asn tends to infinity, is given by inf{σ(ξ)∶ξ∈[0,∞]}, where σ(ξ) is the spectral radius of Σ j∈z t j ξ j .
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01202094
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  • 10
    Electronic Resource
    Electronic Resource
    On matrices that admit unitary reduction to band form (1998)
    Springer
    Mathematical notes 64 (1998), S. 753-760 
    Details
    ISSN: 1573-8876
    Keywords: Hermitean matrix ; unitary matrix ; band form of a matrix ; unitary reduction ; Krylov sequence ; Lanczos procedure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It is proved in the paper that any low rank perturbation of a Hermitian matrix is unitarily reducible to band form. Moreover, if a normal matrix is unitarily reducible to band form, then any of its rank one perturbations is unitarily reducible as well.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02313033
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