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  • Articles  (35)
  • CR: 5.14  (35)
  • 1980-1984  (35)
  • Mathematics  (35)
  • Technology
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  • Articles  (35)
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  • Mathematics  (35)
  • Technology
  • 1
    Electronic Resource
    Electronic Resource
    Calculation of Gauss quadratures with multiple free and fixed knots (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 147-163 
    Details
    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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  • 2
    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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  • 3
    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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  • 4
    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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  • 5
    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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  • 6
    Electronic Resource
    Electronic Resource
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01459075
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  • 7
    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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  • 8
    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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  • 9
    Electronic Resource
    Electronic Resource
    Computing theCS decomposition of a partitioned orthonormal matrix (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 297-306 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396447
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  • 10
    Electronic Resource
    Electronic Resource
    Strong underrelaxation in Kaczmarz's method for inconsistent systems (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 83-92 
    Details
    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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