Optimally scaled matrices

Bauer, Friedrich L.

doi:10.1007/bf01385880

Cited by 195 publications

(67 citation statements)

References 6 publications

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“…The two computations give rise to identically the same sequence of significands; only the exponents may differ. Bauer [2] made an analogous observation with respect to the scaling of linear equation problems. In order to have a fair basis for comparison with Clenshaw's algorithm, we shall assume the problem has been normalized so that |a| Si 1.…”

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confidence: 85%

Error Analysis for Polynomial Evaluation

Newbery¹

1974

Mathematics of Computation

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Abstract.A floating-point error analysis is given for the evaluation of a real polynomial at a real argument by Horner's scheme. A computable error bound is derived. It is observed that when a polynomial has coefficients of constant sign or of strictly alternating sign, one cannot expect better accuracy by reformulating the problem in terms of Chebyshev polynomials.Given that the real polynomial P{x) = 22 A*' is to be evaluated at a real argument a under conditions of normalized floating-point arithmetic, we wish to study the bounds for accumulated round-off effects. Two algorithms for evaluation will be compared: (i) the standard Homer scheme and (ii) Clenshaw's algorithm [1] applied to the Chebyshev form.First, we note that for practical purposes there is no loss of generality in assuming that |a| Si 1. For instance, on a binary machine one could define H{x) = P{2kx), and, in the absence of overflow/underflow, the coefficients of H are exactly determined in terms of those of P. The problem of evaluating P(a) can be replaced by that of evaluating H{x) at x = 2~ka, where |2~*a| Si 1. The two computations give rise to identically the same sequence of significands; only the exponents may differ. Bauer [2] made an analogous observation with respect to the scaling of linear equation problems. In order to have a fair basis for comparison with Clenshaw's algorithm, we shall assume the problem has been normalized so that |a| Si 1.According to the Horner scheme, we have (1) P{a) = q0, where qn = p", and q, = pr + aqr+x, r = n -1, n -2, ..., 0.Computationally, since the result of each arithmetical operation in (1) is subject to a relative error in the range ±e, we shall generate a sequence {

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confidence: 85%

Error Analysis for Polynomial Evaluation

Newbery¹

1974

Mathematics of Computation

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“…This idea was explored in [2,9] as a method for finding a diagonal scaling such that κ(DAE) κ(A). By applying the SK algorithm to the matrix whose (i, j)th element is |a p i j |, it is easily seen that the problem is essentially identical for 1 < p < ∞.…”

Section: The Sinkhorn-knopp Algorithmmentioning

confidence: 99%

The Sinkhorn–Knopp Algorithm: Convergence and Applications

Knight¹

2008

SIAM J. Matrix Anal. & Appl.

219

186

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Abstract. As long as a square nonnegative matrix A contains sufficient nonzero elements, then the Sinkhorn-Knopp algorithm can be used to balance the matrix, that is, to find a diagonal scaling of A that is doubly stochastic. It is known that the convergence is linear and an upper bound has been given for the rate of convergence for positive matrices. In this paper we give an explicit expression for the rate of convergence for fully indecomposable matrices.We describe how balancing algorithms can be used to give a measure of web page significance. We compare the measure with some well known alternatives, including PageRank. We show that with an appropriate modification, the Sinkhorn-Knopp algorithm is a natural candidate for computing the measure on enormous data sets.

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“…As the matrix is symmetric and positive definite the elements of D 1 and D 2 are positive. These matrices are known to be scaled best if the diagonal matrices D 1 and D 2 are equal to the identity matrix (see [2,1,5]). This can be achieved rather easily by applying a diagonal scaling of the matrix.…”

Section: Introductionmentioning

confidence: 99%

A small note on the scaling of symmetric positive definite semiseparable matrices

Vandebril¹,

Golub²,

Barel³

2006

Numer Algor

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In this paper we will adapt a known method for diagonal scaling of symmetric positive definite tridiagonal matrices towards the semiseparable case. Based on the fact that a symmetric, positive definite tridiagonal matrix T satisfies property A, one can easily construct a diagonal matrix (D) over cap such that (D) over capT (D) over cap has the lowest condition number over all matrices DTD, for any choice of diagonal matrix D. Knowing that semiseparable matrices are the inverses of tridiagonal matrices, one can derive similar properties for semiseparable matrices. Here, we will construct the optimal diagonal scaling of a semiseparable matrix, based on a new inversion formula for semiseparable matrices. Some numerical experiments are performed. In a first experiment we compare the condition numbers of the semiseparable matrices before and after the scaling. In a second numerical experiment we compare the scalability of matrices coming from the reduction to semiseparable form and matrices coming from the reduction to tridiagonal form. Article information• Vandebril, Raf; Golub, G.; Van Barel, Marc. A small note on the scaling of symmetric positive definite semiseparable matrices, Numerical Algorithms, volume 41, issue 3, pages 319-326, 2006.• The content of this article is identical to the content of the published paper, but without the final typesetting by the publisher.• Journal's homepage: http://link.springer.com/journal/11075 Abstract In this paper we will adapt a known method for diagonal scaling of symmetric positive definite tridiagonal matrices towards the semiseparable case. Based on the fact that a symmetric, positive definite tridiagonal matrix T satisfies property A, one can easily construct a diagonal matrixD such thatDTD has the lowest condition number over all matrices DT D, for any choice of diagonal matrix D. Knowing that semiseparable matrices are the inverses of tridiagonal matrices, one can derive similar properties for semiseparable matrices. Here, we will construct the optimal diagonal scaling of a semiseparable matrix, based on a new inversion formula for semiseparable matrices.Some numerical experiments are performed. In a first experiment we compare the condition numbers of the semiseparable matrices before and after the scaling. In a second numerical experiment we compare the scalability of matrices coming from the reduction to semiseparable form and matrices coming from the reduction to tridiagonal form.

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Optimally scaled matrices

Cited by 195 publications

References 6 publications

Error Analysis for Polynomial Evaluation

Error Analysis for Polynomial Evaluation

The Sinkhorn–Knopp Algorithm: Convergence and Applications

A small note on the scaling of symmetric positive definite semiseparable matrices

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