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  • 1
    Digitale Medien
    Digitale Medien
    Fast Givens rotations for orthogonal similarity transformations (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 47-56 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65F30 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Fast Givens rotations with half as many multiplications are proposed for orthogonal similarity transformations and a matrix notation is introduced to describe them more easily. Applications are proposed and numerical results are examined for the Jacobi method, the reduction to Hessenberg form and the QR-algorithm for Hessenberg matrices. It can be seen that in general fast Givens rotations are competitive with Householder reflexions and offer distinct advantages for sparse matrices.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01459074
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  • 2
    Digitale Medien
    Digitale Medien
    Exact solution of linear equations usingP-adic expansions (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 137-141 
    Details
    ISSN: 0945-3245
    Schlagwort(e): (MR 1980) AMS(MOS) ; 65F05, 15A06, 10M10, 10A30 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A method is described for computing the exact rational solution to a regular systemAx=b of linear equations with integer coefficients. The method involves: (i) computing the inverse (modp) ofA for some primep; (ii) using successive refinements to compute an integer vector $$\bar x$$ such that $$A\bar x \equiv b$$ (modp m ) for a suitably large integerm; and (iii) deducing the rational solutionx from thep-adic approximation $$\bar x$$ . For matricesA andb with entries of bounded size and dimensionsn×n andn×1, this method can be implemented in timeO(n 3(logn)2) which is better than methods previously used.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01459082
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  • 3
    Digitale Medien
    Digitale Medien
    Error bounds for computed eigenvalues and eigenvectors. II (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 201-206 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F05, 65F15, 65H10 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this paper, motivated by Symm-Wilkinson's paper [5], we describe a method which finds the rigorous error bounds for a computed eigenvalue λ(0) and a computed eigenvectorx (0) of any matrix A. The assumption in a previous paper [6] that λ(0),x (0) andA are real is not necessary in this paper. In connection with this method, Symm-Wilkinson's procedure is discussed, too.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400539
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  • 4
    Digitale Medien
    Digitale Medien
    Reguläre Zerlegungen und Berechnung des Spektralradius nichtnegativer Matrizen (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 201-216 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F10 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary On the basis of a Rayleigh Quotient Iteration method in [10] and a Maximal Quotient Iteration method in [5, 8] two algorithms for solving special eigenvalue problems are developed. The characteristic properties of these methods lie in the application of iterative linear methods to solving systems of linear equations. The convergence properties are investigated. We apply the algorithms to the computation of the spectralradius of a nonnegative irreducible matrix.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396060
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  • 5
    Digitale Medien
    Digitale Medien
    Realistic error bounds for a simple eigenvalue and its associated eigenvector (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 113-126 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65F15 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary An algorithm is described which, given an approximate simple eigenvalue and a corresponding approximate eigenvector, provides rigorous error bounds for improved versions of them. No information is required on the rest of the eigenvalues, which may indeed correspond to non-linear elementary divisors. A second algorithm is described which gives more accurate improved versions than the first but provides only error estimates rather than rigorous bounds. Both algorithms extend immediately to the generalized eigenvalue problem.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396310
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  • 6
    Digitale Medien
    Digitale Medien
    Characterizations of dominant and dominated solutions of linear recursions (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 421-442 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65Q05, 65F99, 39A10 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Dominated solutions of a linear recursion (i.e. solutions which are outgrown by other ones) cannot be computed in a stable way by forward recursion. We analyze this dominance phenomenon more closely and give practically significant characterizations for dominated and dominant solutions. For a dominated solution, in particular, this leads to a stable computational method.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01399009
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  • 7
    Digitale Medien
    Digitale Medien
    Stability of fast algorithms for matrix multiplication (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 63-72 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 15A63 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Non commutative fast algorithms to compute 2×2 matrix product are classified with regard to stability. An analysis of the rounding error propagation is presented for then×n matrix multiplication algorithms obtained by recursive partitioning.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395989
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  • 8
    Digitale Medien
    Digitale Medien
    Least squares with a quadratic constraint (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 291-307 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F20 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We present the theory of the linear least squares problem with a quadratic constraint. New theorems characterizing properties of the solutions are given. A numerical application is discussed.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396656
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  • 9
    Digitale Medien
    Digitale Medien
    Matrices related to interpolatory quadratures (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 309-318 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65F15 ; 65D30 ; CR: 5.14 ; 5.16
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary An explicit formula for the coefficients of interpolatory quadratures with knots of arbitrary multiplicity which uses the eigenvectors and principal vectors of certain matrices is derived. The construction and properties of such matrices are discussed, and applications for the evaluation of Gauss and generalized Gauss-Lobatto quadratures are indicated.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396657
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  • 10
    Digitale Medien
    Digitale Medien
    LU decompositions of generalized diagonally dominant matrices (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 57-69 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F05 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Using the simple vehicle ofM-matrices, the existence and stability ofLU decompositions of matricesA which can be scaled to diagonally dominant (possibly singular) matrices are investigated. Bounds on the growth factor for Gaussian elimination onA are derived. Motivation for this study is provided in part by applications to solving homogeneous systems of linear equationsAx=0, arising in Markov queuing networks, input-output models in economics and compartmental systems, whereA or −A is an irreducible, singularM-matrix. This paper extends earlier work by Funderlic and Plemmons and by Varga and Cai.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01459075
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  • 11
    Digitale Medien
    Digitale Medien
    Computing theCS decomposition of a partitioned orthonormal matrix (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 297-306 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65F25 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary This paper describes an algorithm for simultaneously diagonalizing by orthogonal transformations the blocks of a partitioned matrix having orthonormal columns.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396447
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  • 12
    Digitale Medien
    Digitale Medien
    An algorithm for a linear diophantine equation and a problem of Frobenius (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 349-352 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 10B05 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We solve the diophantine equation $$\sum\limits_{j = 1}^n {a_j x_j } = L$$ for nonnegative variablesx j , wherea j andL are positive integers. We characterize both the values ofL that lead to solutions and those that do not lead to solutions. We solve the Frobenius problem of finding the largest value ofL for which no solution exists.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01403673
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  • 13
    Digitale Medien
    Digitale Medien
    A generalization of the Stein-Rosenberg theorem to Banach spaces (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 403-409 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS):65F 10, 47B 55 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In the first part of this note we prove a generalization of the Stein-Rosenberg theorem; the context is that of real Banach spaces with a normal reproducing cone and the operators involved are positive and completely continuous. Our generalization of the Stein-Rosenberg theorem improves the modern version of it as stated by F. Robert in [5, §2]. In the second part, we discuss briefly how our results are related to other versions of the Stein-Rosenberg theorem. In the last section we describe a situation to which the results in the first part can be applied.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01403677
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  • 14
    Digitale Medien
    Digitale Medien
    On the sharpness of some upper bounds for the spectral radii of S.O.R. iteration matrices (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 69-79 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F10 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Sharpness is shown for three upper bounds for the spectral radii of point S.O.R. iteration matrices resulting from the splitting (i) of a nonsingularH-matrixA into the ‘usual’D−L−U, and (ii) of an hermitian positive definite matrixA intoD−L−U, whereD is hermitian positive definite andL=1/2(A−D+S) withS some skew-hermitian matrix. The first upper bound (which is related to the splitting in (i)) is due to Kahan [6], Apostolatos and Kulisch [1] and Kulisch [7], while the remaining upper bounds (which are related to the splitting in (ii)) are due to Varga [11]. The considerations regarding the first bound yield an answer to a question which, in essence, was recently posed by Professor Ridgway Scott: What is the largest interval in ω, ω≧0, for which the point S.O.R. iterative method is convergent for all strictly diagonally dominant matrices of arbitrary order? The answer is, precisely, the interval (0, 1].
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396371
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  • 15
    Digitale Medien
    Digitale Medien
    The rate of convergence of an approximate matrix factorization semi-direct method (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 237-251 
    Details
    ISSN: 0945-3245
    Schlagwort(e): CR: 5.14 ; 5.17 ; AMS(MOS): 65F10 ; 65N10
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The definition of acceleration parameters for the convergence of a sparseLU factorization semi-direct method is shown to be based on lower and upper bounds of the extreme eigevalues of the iteration matrix. Optimum values of these parameters are established when the eigenvalues of the iteration matrix are either real or complex. Estimates for the computational work required to reduce theL 2 norm of the error by a specified factor ɛ are also given.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396653
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  • 16
    Digitale Medien
    Digitale Medien
    On fixed point linear equations (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 53-59 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F10 ; 47B55 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary By means of successive partial substitutions it is possible to obtain new fixed point linear equations from old ones and it is interesting to determine how the spectral radius of the corresponding matrices varies. We prove that, when the original matrix is nonnegative, this variation is decreasing or increasing, depending on whether the original matrix has its spectral radius smaller or greater than 1. We answer in this way a question posed by F. Robert in [5].
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395808
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  • 17
    Digitale Medien
    Digitale Medien
    A modified bisection algorithm for the determination of the eigenvalues of a symmetric tridiagonal matrix (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 417-419 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F10 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A modification to the well known bisection algorithm [1] when used to determine the eigenvalues of a real symmetric matrix is presented. In the new strategy the terms in the Sturm sequence are computed only as long as relevant information on the required eigenvalues is obtained. The resulting algorithm usingincomplete Sturm sequences can be shown to minimise the computational work required especially when only a few eigenvalues are required. The technique is also applicable to other computational methods which use the bisection process.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396441
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  • 18
    Digitale Medien
    Digitale Medien
    A contribution to the theory of condition (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 85-96 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS65F05 ; 65F 15 ; 65F 20 ; 65F 35 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this paper the use of the condition number of a problem, as defined by Rice in 1966, is discussed. For the eigenvalue, eigenvector, and linear least squares problems either condition numbers according to various norms are determined or lower and upper bounds for them are derived.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01399313
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  • 19
    Digitale Medien
    Digitale Medien
    Inversion of tridiagonal matrices (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 333-345 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 15 A 09, 15-04 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary This paper presents a simple algorithm for inverting nonsymmetric tridiagonal matrices that leads immediately to closed forms when they exist. Ukita's theorem is extended to characterize the class of matrices that have tridiagonal inverses.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396436
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  • 20
    Digitale Medien
    Digitale Medien
    On the convergence of the symmetric SOR method for matrices with red-black ordering (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 113-117 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F10 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We prove that if the matrixA has the structure which results from the so-called “red-black” ordering and ifA is anH-matrix then the symmetric SOR method (called the SSOR method) is convergent for 0〈ω〈2. In the special case thatA is even anM-matrix we show that the symmetric single-step method cannot be accelerated by the SSOR method. Symmetry of the matrixA is not assumed.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01399315
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  • 21
    Digitale Medien
    Digitale Medien
    Zur Konvergenz eines Verfahrens von D.J. Evans zur iterativen Verbesserung einer Näherung für die inverse Matrix (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 163-173 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F10 ; CR: 5.14
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Recently D.J. Evans introduced an implicit matrix inversion process showing asymptotic behaviour which is superior to that of the well known Schulz-method. In this paper we give sufficient conditions for convergence, prove some error bounds and show that under certain conditions the iterates are converging monotonously.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01408691
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