Abstract not found

Abstract not found

Let ¯ X = [X1,..., Xn] and f ∈ R [ ¯ X]. We consider the problem of computing the global infimum of f when f is bounded below. For A ∈ GLn(C), we denote by f A the polynomial f(A ¯ X). Fix a number M ∈ R greater than infx∈Rn f(x). We prove that there exists a Zariski-closed subset A � GLn(C) such that for all A ∈ GLn(Q) \ A, we have f A ≥ 0 on R n if and only if for all ɛ> 0, there exist sums of squares of polynomials s and t in R [ ¯ X] and polynomials φi ∈ R [ ¯ X] such that f A + ɛ = s + t ( M − f A) + ∑ A ∂f 1≤i≤n−1 φi. ∂Xi Hence we can formulate the original optimization problems as semidefinite programs which can be solved efficiently in Matlab. Some numerical experiments are given. We also discuss how to exploit the sparsity of SDP problems to overcome the ill-conditionedness of SDP problems when the infimum is not attained.

Abstract not found

We present certificates for the positive semidefiniteness of an n × n matrix A, whose entries are integers of binary length log ‖A‖, that can be verified in O(n 2+ǫ (log ‖A‖) 1+ǫ) binary operations for any ǫ> 0. The question arises in Hilbert/Artin-based rational sum-of-squares certificates, i.e., proofs, for polynomial inequalities with rational coefficients. We allow certificates that are validated by Monte Carlo randomized algorithms, as in Rusins M. Freivalds’s famous 1979 quadratic time certification for the matrix product. Our certificates occupy O(n 3+ǫ (log ‖A‖) 1+ǫ) bits, from which the verification algorithm randomly samples a quadratic amount. In addition, we give certificates of the same space and randomized validation time complexity for the Frobenius form and the characteristic and minimal polynomials. For determinant and rank we have certificates of essentially-quadratic binary space and time complexity via Storjohann’s algorithms.

We present two new algorithms FastAccSum and FastPrecSum, one to compute a faithful rounding of the sum of floating-point numbers and the other for a result “as if” computed in K-fold precision. Faithful rounding means the computed result either is one of the immediate floating-point neighbors of the exact result or is equal to the exact sum if this is a floating-point number. The algorithms are based on our previous algorithms AccSum and PrecSum and improve them by up to 25%. The first algorithm adapts to the condition number of the sum; i.e., the computing time is proportional to the difficulty of the problem. The second algorithm does not need extra memory, and the computing time depends only on the number of summands and K. Both algorithms are the fastest known in terms of flops. They allow good instruction-level parallelism so that they are also fast in terms of measured computing time. The algorithms require only standard floating-point addition, subtraction, and multiplication in one working precision, for example, double precision.

JHD’s notes taken at the time. Note that there were parallel sessions, so not every talk is even mentioned. JHD has also not yet proof-read them, or checked