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The input to our algorithm is a multivariate polynomial, whose complex rational coefficients are considered imprecise with an unknown error that causes f to be irreducible over the complex numbers C. We seek to perturb the coefficients by a small quantitity such that the resulting polynomial factors over C. Ideally, one would like to minimize the(More)
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 <i>k</i>. In addition, we restrict the deformed coefficients by a given set of linear(More)
We generalize the technique by Peyrl and Parillo [Proc. SNC 2007] to computing lower bound certificates for several well-known factorization problems in hybrid symbolic-numeric 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(More)
We describe the design, implementation and experimental evaluation of new algorithms for computing the approximate factorization of multivariate polynomials with complex coefficients that contain numerical noise. Our algorithms are based on a generalization of the differential forms introduced by W. Ruppert and S. Gao to many variables, and use singular(More)
Algebraic randomization techniques can be applied to hybrid symbolic-numeric algorithms. Here we consider the problem of interpolating a sparse rational function from noisy values. We develop a new hybrid algorithm based on Zippel's original sparse polynomial interpolation technique. We show experimentally that our algorithm can handle sparse polynomials(More)
The black box algorithm for separating the numerator from the denominator of a multivariate rational function can be combined with sparse multivariate polynomial interpolation algorithms to interpolate a sparse rational function. domization and early termination strategies are exploited to minimize the number of black box evaluations. In addition, rational(More)
A series of 3-, 7-, 15-, and 16-methyl-substituted steroid analogs were synthesized via a highly stereoselective 1,6-conjugate addition. Under the catalysis of CuBr, AlMe(3) reacted with four steroid dienone precursors to afford either the corresponding alpha-epimer of C-3 and C-7 methyl-substituted steroids as the major products, and the ratio of(More)
We present a hybrid symbolic-numeric algorithm for certifying a polynomial or rational function with rational coefficients to be non-negative for all real values of the variables by computing a representation for it as a fraction of two polynomial sum-of-squares (SOS) with rational coefficients. Our new approach turns the earlier methods by Peyrl and(More)
The problem of approximating the greatest common divisor(GCD) for polynomials with inexact coefficients can be formulated as a low rank approximation problem with a Sylvester matrix. In this paper, we present an algorithm based on fast Structured Total Least Norm(STLN) for constructing a Sylvester matrix of given lower rank and obtaining the nearest(More)
Background: Tumor necrosis factor alpha (TNFa) has been used to treat certain tumors in clinic trials. However, the curative effect of TNFa has been undermined by the induced-NF-B activation in many types of tumor. Maslinic acid (MA), a pharmacological safe natural product, has been known for its important effects as anti-oxidant, anti-inflammatory, and(More)