Structured Perturbations Part I: Normwise Distances

Rump, Siegfried M.

doi:10.1137/s0895479802405732

Cited by 66 publications

(75 citation statements)

References 34 publications

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“…It characterizes the sensitivity of the solution x with respect to infinitely small Toeplitz structured perturbations of the matrix A and perturbations of b. It has been proved in [37] that the ratio between the Toeplitz and the unstructured condition number satisfies…”

Section: Problem and Mathematical Backgroundmentioning

confidence: 99%

Exact certification of global optimality of approximate factorizations via rationalizing sums-of-squares with floating point scalars

Kaltofen

Yang

et al. 2008

Proceedings of the Twenty-First International Symposium on Symbolic and Algebraic Computation

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We generalize the technique by Peyrl and Parillo [Proc. SNC 2007] to computing lower bound certificates for several well-known factorization problems in hybrid symbolicnumeric computation. The idea is to transform a numerical sum-of-squares (SOS) representation of a positive polynomial into an exact rational identity. Our algorithms successfully certify accurate rational lower bounds near the irrational global optima for benchmark approximate polynomial greatest common divisors and multivariate polynomial irreducibility radii from the literature, and factor coefficient bounds in the setting of a model problem by Rump (up to n = 14, factor degree = 13).The numeric SOSes produced by the current fixed precision semi-definite programming (SDP) packages (SeDuMi, SOSTOOLS, YALMIP) are usually too coarse to allow successful projection to exact SOSes via Maple 11's exact linear algebra. Therefore, before projection we refine the SOSes by rank-preserving Newton iteration. For smaller problems the starting SOSes for Newton can be guessed without SDP ("SDP-free SOS"), but for larger inputs we additionally appeal to sparsity techniques in our SDP formulation.

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Section: Problem and Mathematical Backgroundmentioning

confidence: 99%

Exact certification of global optimality of approximate factorizations via rationalizing sums-of-squares with floating point scalars

Kaltofen

Yang

et al. 2008

Proceedings of the Twenty-First International Symposium on Symbolic and Algebraic Computation

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“…Therefore [14, p. 775]. Such a difference between structured and unstructured distances to singularity is impossible for Toeplitz matrices [29]. 2…”

Section: Algorithmic Details and Experimentsmentioning

confidence: 99%

Approximate greatest common divisors of several polynomials with linearly constrained coefficients and singular polynomials

Kaltofen

Yang

Zhi

2006

Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation

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We consider the problem of computing minimal real or complex deformations to the coefficients in a list of relatively prime real or complex multivariate polynomials such that the deformed polynomials have a greatest common divisor (GCD) of at least a given degree k. In addition, we restrict the deformed coefficients by a given set of linear constraints, thus introducing the linearly constrained approximate GCD problem. We present an algorithm based on a version of the structured total least norm (STLN) method and demonstrate, on a diverse set of benchmark polynomials, that the algorithm in practice computes globally minimal approximations. As an application of the linearly constrained approximate GCD problem, we present an STLN-based method that computes for a real or complex polynomial the nearest real or complex polynomial that has a root of multiplicity at least k. We demonstrate that the algorithm in practice computes, on the benchmark polynomials given in the literature, the known globally optimal nearest singular polynomials. Our algorithms can handle, via randomized preconditioning, the difficult case when the nearest solution to a list of real input polynomials actually has non-real complex coefficients.

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“…Condition numbers relative to application-specific, structured perturbations [7,8,25,26,33] should capture many of the successful cases our cautious settings forgo. The rounding errors δr and δx affect the algorithm adversely only for ill-conditioned systems.…”

Section: Limitations Of Refinement and Our Boundsmentioning

confidence: 99%

Error bounds from extra-precise iterative refinement

Demmel

Hida

Kahan

et al. 2006

ACM Trans. Math. Softw.

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We present the design and testing of an algorithm for iterative refinement of the solution of linear equations, where the residual is computed with extra precision. This algorithm was originally proposed in the 1960s [6,22] as a means to compute very accurate solutions to all but the most ill-conditioned linear systems of equations. However two obstacles have until now prevented its adoption in standard subroutine libraries like LAPACK: (1) There was no standard way to access the higher precision arithmetic needed to compute residuals, and (2) it was unclear how to compute a reliable error bound for the computed solution. The completion of the new BLAS Technical Forum Standard [5] has recently removed the first obstacle. To overcome the second obstacle, we show how a single application of iterative refinement can be used to compute an error bound in any norm at small cost, and use this to compute both an error bound in the usual infinity norm, and a componentwise relative error bound.We report extensive test results on over 6.2 million matrices of dimension 5, 10, 100, and 1000. As long as a normwise (resp. componentwise) condition number computed by the algorithm is less than 1 /max{10, √ n}εw, the computed normwise (resp. componentwise) error bound is at most 2 max{10, √ n} · ε w , and indeed bounds the true error. Here, n is the matrix dimension and ε w is single precision roundoff error. For worse conditioned problems, we get similarly small correct error bounds in over 89.4% of cases.

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Structured Perturbations Part I: Normwise Distances

Cited by 66 publications

References 34 publications

Exact certification of global optimality of approximate factorizations via rationalizing sums-of-squares with floating point scalars

Exact certification of global optimality of approximate factorizations via rationalizing sums-of-squares with floating point scalars

Approximate greatest common divisors of several polynomials with linearly constrained coefficients and singular polynomials

Error bounds from extra-precise iterative refinement

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