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  • Articles  (52)
  • CR: 5.14  (52)
  • Springer  (52)
  • Cambridge University Press
  • Frontiers Media
  • National Academy of Sciences
  • 2020-2022
  • 1980-1984  (35)
  • 1975-1979  (17)
  • 1970-1974
  • Mathematics  (52)
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  • Articles  (52)
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  • Springer  (52)
  • Cambridge University Press
  • Frontiers Media
  • National Academy of Sciences
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  • 2020-2022
  • 1980-1984  (35)
  • 1975-1979  (17)
  • 1970-1974
  • 1985-1989  (1)
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  • Mathematics  (52)
  • 1
    Electronic Resource
    Electronic Resource
    An alternative givens ordering (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 83-90 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F05 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A new Givens ordering is shown, empirically and by an approximate theoretical analysis, to take appreciably fewer stages than the standard scheme. Sharper error bounds than Gentleman's ensue, and the scheme is better suited to parallel computation. Other schemes, less efficient but more easily analysed, are discussed. The effect of a possible limit in practice on the number of simultaneous computations is considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389639
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  • 2
    Electronic Resource
    Electronic Resource
    An accelerated bisection method for the calculation of eigenvalues of a symmetric tridiagonal matrix (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 153-160 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 15A18 ; 15-04 ; 65F10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We present a method for the determination of eigenvalues of a symmetric tridiagonal matrix which combines Givens' Sturm bisection [4, 5] with interpolation, to accelerate convergence in high precision cases. By using an appropriate root of the absolute value of the determinant to derive the interpolation weight, results are obtained which compare favorably with the Barth, Martin, Wilkinson algorithm [1].
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389644
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  • 3
    Electronic Resource
    Electronic Resource
    Fast Givens rotations for orthogonal similarity transformations (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 47-56 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65F30 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01459074
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  • 4
    Electronic Resource
    Electronic Resource
    Exact solution of linear equations usingP-adic expansions (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 137-141 
    Details
    ISSN: 0945-3245
    Keywords: (MR 1980) AMS(MOS) ; 65F05, 15A06, 10M10, 10A30 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01459082
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  • 5
    Electronic Resource
    Electronic Resource
    Error bounds for computed eigenvalues and eigenvectors. II (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 201-206 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F05, 65F15, 65H10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01400539
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  • 6
    Electronic Resource
    Electronic Resource
    The perturbation bounds for eigenspaces of a definite matrix-pair (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 321-343 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F15 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary LetA andB be Hermitian matrices. The matrix-pair (A, B) is called “definite pair” and the corresponding eigenvalue problem,Ax=λBx is definite if $$c(A,B) \equiv \mathop {\min }\limits_{\left\| x \right\| = 1} \{ |x^H (A + iB)x|\} 〉 0$$ . The perturbation bounds for eigenspaces of a definite pair on every unitary-invariant matrix norm were obtained by imposing additional restrictions on the location of the generalized eigenvalues. Thus it gives a positive answer for an open question proposed by Stewart [7]. The famous Davis-Kahan sin θ theorems and sin 2θ Theorem [2] can also be deduced from the present results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01418329
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  • 7
    Electronic Resource
    Electronic Resource
    Rank-one modification of the symmetric eigenproblem (1978)
    Springer
    Numerische Mathematik 31 (1978), S. 31-48 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F15 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary An algorithm is presented for computing the eigensystem of the rank-one modification of a symmetric matrix with known eigensystem. The explicit computation of the updated eigenvectors and the treatment of multiple eigenvalues are discussed. The sensitivity of the computed eigenvectors to errors in the updated eigenvalues is shown by a perturbation analysis.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396012
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  • 8
    Electronic Resource
    Electronic Resource
    Updating the singular value decomposition (1978)
    Springer
    Numerische Mathematik 31 (1978), S. 111-129 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65F30 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary LetA be anm×n matrix with known singular value decomposition. The computation of the singular value decomposition of a matrixà is considered, whereà is obtained by appending a row or a column toA whenm≧n or by deleting a row or a column fromA whenm〉n. An algorithm is also presented for solving the updated least squares problemà y−b≈, obtained from the least squares problemAx−b by appending an equation, deleting an equation, appending an unknown, or deleting an unknown.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01397471
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  • 9
    Electronic Resource
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    Convergent nonnegative matrices and iterative methods for consistent linear systems (1978)
    Springer
    Numerische Mathematik 31 (1978), S. 265-279 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 F 10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we study linear stationary iterative methods with nonnegative iteration matrices for solving singular and consistent systems of linear equationsAx=b. The iteration matrices for the schemes are obtained via regular and weak regular splittings of the coefficients matrixA. In certain cases when only some necessary, but not sufficient, conditions for the convergence of the iterations schemes exist, we consider a transformation on the iteration matrices and obtain new iterative schemes which ensure convergence to a solution toAx=b. This transformation is parameter-dependent, and in the case where all the eigenvalues of the iteration matrix are real, we show how to choose this parameter so that the asymptotic convergence rate of the new schemes is optimal. Finally, some applications to the problem of computing the stationary distribution vector for a finite homogeneous ergodic Markov chain are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01397879
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  • 10
    Electronic Resource
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    Reguläre Zerlegungen und Berechnung des Spektralradius nichtnegativer Matrizen (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 201-216 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396060
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  • 11
    Electronic Resource
    Electronic Resource
    Realistic error bounds for a simple eigenvalue and its associated eigenvector (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 113-126 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65F15 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396310
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  • 12
    Electronic Resource
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    Characterizations of dominant and dominated solutions of linear recursions (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 421-442 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65Q05, 65F99, 39A10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01399009
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  • 13
    Electronic Resource
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    Stability of fast algorithms for matrix multiplication (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 63-72 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 15A63 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01395989
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  • 14
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    Least squares with a quadratic constraint (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 291-307 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F20 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
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    URL: http://dx.doi.org/10.1007/BF01396656
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  • 15
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    Matrices related to interpolatory quadratures (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 309-318 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65F15 ; 65D30 ; CR: 5.14 ; 5.16
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
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    URL: http://dx.doi.org/10.1007/BF01396657
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  • 16
    Electronic Resource
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    LU decompositions of generalized diagonally dominant matrices (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 57-69 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F05 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
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    URL: http://dx.doi.org/10.1007/BF01459075
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  • 17
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    Computing theCS decomposition of a partitioned orthonormal matrix (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 297-306 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65F25 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper describes an algorithm for simultaneously diagonalizing by orthogonal transformations the blocks of a partitioned matrix having orthonormal columns.
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    URL: http://dx.doi.org/10.1007/BF01396447
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  • 18
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    Strong underrelaxation in Kaczmarz's method for inconsistent systems (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 83-92 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65 F 10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We investigate the behavior of Kaczmarz's method with relaxation for inconsistent systems. We show that when the relaxation parameter goes to zero, the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system. This point minimizes the sum of the squares of the Euclidean distances to the hyperplanes of the system. If the starting point is chosen properly, then the limits approach the minimum norm weighted least squares solution. The proof is given for a block-Kaczmarz method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396307
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  • 19
    Electronic Resource
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    Calculation of Gauss quadratures with multiple free and fixed knots (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 147-163 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65F15 ; 65D30 ; CR: 5.14 ; 5.16
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Algorithms are derived for the evaluation of Gauss knots in the presence of fixed knots by modification of the Jacobi matrix for the weight function of the integral. Simple Gauss knots are obtained as eigenvalues of symmetric tridiagonal matrices and a rapidly converging simple iterative process, based on the merging of free and fixed knots, of quadratic convergence is presented for multiple Gauss knots. The procedures also allow for the evaluation of the weights of the quadrature corresponding to the simple Gauss knots. A new characterization of simple Gauss knots as a solution of a partial inverse eigenvalue problem is derived.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390210
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  • 20
    Electronic Resource
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    A new method for the solution ofAx=b (1978)
    Springer
    Numerische Mathematik 29 (1978), S. 287-289 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65FO5 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary An alternative to Gauss elimination is developed to solveAx=b. The method enables one to exploit zeros in the lower triangle ofA, so that the number of multiplications needed to perform the algorithm can be substantially reduced.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389213
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  • 21
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    Adaptive procedure for estimating parameters for the nonsymmetric Tchebychev iteration (1978)
    Springer
    Numerische Mathematik 31 (1978), S. 183-208 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary An iteration based upon the Tchebychev polynomials in the complex plane can be used to solve large sparse nonsymmetric linear systems whose eigenvalues lie in the right half plane. The iteration depends upon two parameters which can be chosen from knowledge of the convex hull of the spectrum of the linear operator. This paper deals with a procedure based upon the power method for dynamically estimating the convex hull of the spectrum. The stability of the procedure is discussed in terms of the field of values of the operator. Results show the adaptive procedure to be an effective method of determining parameters. The Tchebychev iteration compares favorably with several competing iterative methods.
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    URL: http://dx.doi.org/10.1007/BF01397475
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  • 22
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    Alternating methods for sets of linear equations (1979)
    Springer
    Numerische Mathematik 32 (1979), S. 105-108 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A generalization of alternating methods for sets of linear equations is described and the number of operations calculated. It is shown that the lowest number of arithmetic operations is achieved in the SSOR algorithm.
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    URL: http://dx.doi.org/10.1007/BF01397654
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  • 23
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    A factorization method for the solution of constrained linear least squares problems allowing subsequent data changes (1978)
    Springer
    Numerische Mathematik 31 (1978), S. 431-463 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65F20, 90C05 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we describe how to use Gram-Schmidt factorizations of Daniel et al. [1] to realize the method of [8] for solving linearly constrained linear least squares problems. The main advantage of using these factorizations is that it is relatively easy to take data changes into account, if necessary. The algorithm is compared numerically with two other codes, one of them published by Lawson and Hanson [3]. Further computational tests show the efficiency of the proposed methods, if a few data of the original problem are changed subsequently.
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    URL: http://dx.doi.org/10.1007/BF01404569
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  • 24
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    On a new class of elementary matrices (1979)
    Springer
    Numerische Mathematik 33 (1979), S. 173-180 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65F15 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A new class of elementary matrices, a block-generalisation of plane rotations, is presented and the application in constructing quadratically convergent block diagonalisation algorithms of Jacobi type is discussed.
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    URL: http://dx.doi.org/10.1007/BF01399552
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  • 25
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    On a class of Jacobi-like procedures for diagonalising arbitrary real matrices (1979)
    Springer
    Numerische Mathematik 33 (1979), S. 157-172 
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    ISSN: 0945-3245
    Keywords: AMS (MOS): 65F15 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A new class of elementary matrices is presented which are convenient in Jacobi-like diagonalisation methods for arbitrary real matrices. It is shown that the presented transformations possess the normreducing property and that they produce an ultimate quadratic convergence even in the case of complex eigenvalues. Finally, a quadratically convergent Jacobi-like algorithm for real matrices with complex eigenvalues is presented.
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    URL: http://dx.doi.org/10.1007/BF01399551
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  • 26
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    A quadratically convergent Jacobi-like method for real matrices with complex eigenvalues (1979)
    Springer
    Numerische Mathematik 33 (1979), S. 425-435 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F15 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A real Jacobi-like algorithm for diagonalizing arbitrary real matrices with complex eigenvalues is described and its applicability discussed. Numerical results are given and compared with those of the well-known real and complex algorithm of Eberlein.
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    URL: http://dx.doi.org/10.1007/BF01399324
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  • 27
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    An algorithm for a linear diophantine equation and a problem of Frobenius (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 349-352 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 10B05 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
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    URL: http://dx.doi.org/10.1007/BF01403673
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  • 28
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    A generalization of the Stein-Rosenberg theorem to Banach spaces (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 403-409 
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    ISSN: 0945-3245
    Keywords: AMS(MOS):65F 10, 47B 55 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
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    URL: http://dx.doi.org/10.1007/BF01403677
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  • 29
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    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 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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].
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    AB-Algorithm and its modifications for the spectral problems of linear pencils of matrices (1984)
    Springer
    Numerische Mathematik 43 (1984), S. 329-342 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F15 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper describes and algorithm and its modifications for solving spectral problems for linear pencils of matrices both regular as well as singular.
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    Applications of shortest path algorithms to matrix scalings (1984)
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    Numerische Mathematik 44 (1984), S. 111-126 
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    Keywords: AMS(MOS): 65F35 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A symmetric scaling of a nonnegative, square matrixA is a matrixXAX −1, whereX is a nonsingular, nonnegative diagonal matrix. By associating a family of weighted directed graphs with the matrixA we are able to adapt the shortest path algorithms to compute an optimal scaling ofA, where we call a symmetric scalingA′ ofA optimal if it minimizes the maximum of the ratio of non-zero elements.
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    Zur Kondition des linearen Ausgleichsproblems mit linearen Gleichungen als Nebenbedingungen (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 139-152 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F20 ; CR: 5.14
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    Topics: Mathematics
    Notes: Summary F.L. Bauer has treated in several papers [1, 3, 4] the condition related to the solution of linear equations and to the algebraic eigenvalue problem. We study the condition for the linear least squares problem with linear equality constraints (problem LSE). A perturbation theory of problem LSE is presented and three condition numbers are defined. Problem LSE includes the linear least squares problem (problem LS). There are examples with identical solution of problem LSE and of problem LS. Sometimes the condition of problem LSE is better and sometimes the condition of problem LS is better. Several numerical tests illustrate the theory.
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    The rate of convergence of an approximate matrix factorization semi-direct method (1980)
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    Numerische Mathematik 36 (1980), S. 237-251 
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    ISSN: 0945-3245
    Keywords: CR: 5.14 ; 5.17 ; AMS(MOS): 65F10 ; 65N10
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    Topics: Mathematics
    Notes: 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.
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    The Tchebychev iteration for nonsymmetric linear systems (1977)
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    Numerische Mathematik 28 (1977), S. 307-327 
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    Keywords: AMS (Mos): 65 F 10 ; CR: 5.14
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    Topics: Mathematics
    Notes: Summary In this paper an iterative method for solving nonsymmetric linear systems based on the Tchebychev polynomials in the complex plane is discussed. The iteration is shown to converge whenever the eigenvalues of the linear system lie in the open right half complex plane. An algorithm is developed for finding optimal iteration parameters as a function of the convex hull of the spectrum.
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    On fixed point linear equations (1982)
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    Numerische Mathematik 38 (1982), S. 53-59 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; 47B55 ; CR: 5.14
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    Topics: Mathematics
    Notes: 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].
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    On theLU factorization ofM-matrices (1981)
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    Numerische Mathematik 38 (1981), S. 179-192 
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    Keywords: AMS(MOS) 15 A 23 ; CR: 5.14
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    Topics: Mathematics
    Notes: Summary In this paper, we give in Theorem 1 a characterization, based on graph theory, of when anM-matrixA admits anLU factorization intoM-matrices, whereL is a nonsingular lower triangularM-matrix andU is an upper triangularM-matrix. This result generalizes earlier factorization results of Fiedler and Pták (1962) and Kuo (1977). As a consequence of Theorem 1, we show in Theorem 3 that the conditionx T A≧0 T for somex〉0, for anM-matrixA, is both necessary and sufficient forPAP T to admit such anLU factorization for everyn×n permutation matrixP. This latter result extends recent work of Funderlic and Plemmons (1981). Finally, Theorem 1 is extended in Theorem 5 to give a characterization, similarly based on graph theory, of when anM-matrixA admits anLU factorization intoM-matrices.
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    Ein linear konvergentes zyklisches jacobiähnliches Verfahren für beliebige reelle Matrizen (1981)
    Springer
    Numerische Mathematik 38 (1981), S. 245-253 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65 F 15 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A modification of Eberleins jacobi-like algorithm for solving the eigenproblem of arbitrary real matrices is described. The algorithm is always convergent. The convergence is linear for diagonalizable matrices.
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    A modified bisection algorithm for the determination of the eigenvalues of a symmetric tridiagonal matrix (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 417-419 
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    Keywords: AMS(MOS): 65F10 ; CR: 5.14
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    Topics: Mathematics
    Notes: 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.
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    A contribution to the theory of condition (1982)
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    Numerische Mathematik 39 (1982), S. 85-96 
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    Keywords: AMS65F05 ; 65F 15 ; 65F 20 ; 65F 35 ; CR: 5.14
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    Topics: Mathematics
    Notes: 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.
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    Round-off error analysis of iterations for large linear systems (1978)
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    Numerische Mathematik 30 (1978), S. 301-314 
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    Keywords: AMS(MOS): 65F10 ; CR: 5.14
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    Topics: Mathematics
    Notes: Summary We deal with the rounding error analysis of successive approximation iterations for the solution of large linear systemsA x =b. We prove that Jacobi, Richardson, Gauss-Seidel and SOR iterations arenumerically stable wheneverA=A *〉0 andA has PropertyA. This means that the computed resultx k approximates the exact solution α with relative error of order ζ ‖A‖·‖A −1‖ where ζ is the relative computer precision. However with the exception of Gauss-Seidel iteration the residual vector ‖Ax k −b‖ is of order ζ ‖A‖2 ‖A −1‖ ‖α‖ and hence the remaining three iterations arenot well-behaved.
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    Numerical solution of eigentuple-eigenvector problems in Hilbert spaces by a gradient method (1978)
    Springer
    Numerische Mathematik 31 (1978), S. 231-246 
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    Keywords: AMS(MOS): 65F15 ; CR: 5.14
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    Topics: Mathematics
    Notes: Summary A gradient technique previously developed for computing the eigenvalues and eigenvectors of the general eigenproblemAx=λBx is generalized to the eigentuple-eigenvector problem $$Ax = \sum\limits_{i = 1}^p {\lambda _i B_i x} $$ . Among the applications of the latter are (1) the determination of complex (λ,x) forAx=λBx using only real arithmetic, (2) a 2-parameter Sturm-Liouville equation and (3) λ-matrices. The use of complex arithmetic in the gradient method is also discussed. Computational results are presented.
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    On inverses of Vandermonde and confluent Vandermonde matrices III (1978)
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    Numerische Mathematik 29 (1978), S. 445-450 
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    Keywords: AMS(MOS): 15A12, 65F35 ; CR: 5.14
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    Topics: Mathematics
    Notes: Summary We derive lower bounds for the norm of the inverse Vandermonde matrix and the norm of certain inverse confluent Vandermonde matrices. They supplement upper bounds which were obtained in previous papers.
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    Comparison of splittings used with the conjugate gradient algorithm (1979)
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    Numerische Mathematik 33 (1979), S. 181-193 
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    Keywords: AMS(MOS): 65F10 ; CR: 5.14
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    Topics: Mathematics
    Notes: Abstract We consider several possible criteria for comparing splittings used with the conjugate gradient algorithm. We present sharp upper bounds for the error at each step of the algorithm and compare several widely used splittings with respect to their effect on these sharp upper bounds. We then consider a more stringent comparison test, and present necessary and sufficient conditions for the error at each step with one splitting to be smaller than that with another, for all pairs of corresponding initial guesses.
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    Einschließung inverser Elemente durch Fixpunktverfahren (1978)
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    Numerische Mathematik 31 (1978), S. 313-320 
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    Keywords: AMS(MOS): 65F05 ; CR: 5.14
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    Topics: Mathematics
    Notes: Summary In a partially ring, linearly convergent methods are constructed in order to enclose the inverse of a given element monotonously. For this purpose by means of a first approximation the original problem is transformed into a fixed-point equation with a contractive operator. The approach is describted if the element to inverse is known exactly or approximatively, too.
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    Factorization iterative methods,M-operators andH-operators (1978)
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    Numerische Mathematik 31 (1978), S. 335-357 
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    Keywords: AMS(MOS): 65F10 ; CR: 5.14
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    Topics: Mathematics
    Notes: Summary We set up here a general formalism for describing factorization iterative methods of the first order and we use it to review various methods that have been proposed in the literature; next we introduce the notions ofM- andH-operators which generalize those of block-M- and block-H-matrices; finally we discuss the properties of factorization iterative methods in relation with characteristic properties ofM- andH-operators.
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    Eigenvector matrices of symmetric tridiagonals (1984)
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    Numerische Mathematik 44 (1984), S. 103-110 
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    Keywords: AMS(MOS) 6540 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A simple test is given for determining whether a given matrix is the eigenvector matrix of an (unknown) unreduced symmetric tridiagonal matrix. A list of known necessary conditions is also provided. A lower bound on the separation between eigenvalues of tridiagonals follows from our Theorem 3.
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    On neighbouring matrices with quadratic elementary divisors (1984)
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    Numerische Mathematik 44 (1984), S. 1-21 
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    Keywords: AMS(MOS): 65F 15 ; 65F20 ; CR: 5.14
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    Topics: Mathematics
    Notes: Summary This paper presents new results on matrices with quadratic elementary divisors neighbouring a given matrixA and relates them to those given by Kakan. It shows that all results of this kind may give severe overestimates of the distance to the nearest defective matrix and that matrices with elementary divisors of degrees higher than two are often at a much greater distance than those with quadratic divisors.
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    Comparisons of regular splittings of matrices (1984)
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    Numerische Mathematik 44 (1984), S. 23-35 
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    Keywords: AMS(MOS): 65F10 ; CR: 5.14
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    Topics: Mathematics
    Notes: Summary In this article, new comparison theorems for regular splittings of matrices are derived. In so doing, the initial results of Varga in 1960 on regular splittings of matrices, and the subsequent unpublished results of Woźnicki in 1973 on regular splittings of matrices, will be seen to be special cases of these new comparison theorems.
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    A Newton iteration process for inverse eigenvalue problems (1981)
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    Numerische Mathematik 37 (1981), S. 349-354 
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    Keywords: AMS(MOS): 65F15 ; CR: 5.14
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    Topics: Mathematics
    Notes: Summary We suppose an inverse eigenvalue problem which includes the classical additive and multiplicative inverse eigenvalue problems as special cases. For the numerical solution of this problem we propose a Newton iteration process and compare it with a known method. Finally we apply it to a numerical example.
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    Inversion of tridiagonal matrices (1982)
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    Numerische Mathematik 38 (1982), S. 333-345 
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    Keywords: AMS(MOS): 15 A 09, 15-04 ; CR: 5.14
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    Topics: Mathematics
    Notes: 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.
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    On the convergence of the symmetric SOR method for matrices with red-black ordering (1982)
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    Numerische Mathematik 39 (1982), S. 113-117 
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    Keywords: AMS(MOS): 65F10 ; CR: 5.14
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    Topics: Mathematics
    Notes: 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.
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    Zur Konvergenz eines Verfahrens von D.J. Evans zur iterativen Verbesserung einer Näherung für die inverse Matrix (1982)
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    Numerische Mathematik 39 (1982), S. 163-173 
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    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: 5.14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
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