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COMPUTING RATIONAL POINTS IN CONVEX SEMIALGEBRAIC SETS AND SOS DECOMPOSITIONS
"... Abstract. Let P = {h1,..., hs} ⊂ Z[Y1,..., Yk], D ≥ deg(hi) for 1 ≤ i ≤ s, σ bounding the bit length of the coefficients of the hi’s, and Φ be a quantifierfree Pformula defining a convex semialgebraic set. We design an algorithm returning a rational point in S if and only if S ∩ Q ̸ = ∅. It requ ..."
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Abstract. Let P = {h1,..., hs} ⊂ Z[Y1,..., Yk], D ≥ deg(hi) for 1 ≤ i ≤ s, σ bounding the bit length of the coefficients of the hi’s, and Φ be a quantifierfree Pformula defining a convex semialgebraic set. We design an algorithm returning a rational point in S if and only if S ∩ Q ̸ = ∅. It requires σO(1) DO(k3) bit operations. If a rational point is outputted its coordinates have bit length dominated by σDO(k3). Using this result, we obtain a procedure deciding if a polynomial f ∈ Z[X1,..., Xn] is a sum of squares of polynomials in Q[X1,..., Xn]. Denote by d the degree of f, τ the maximum bit length of the coefficients in f, D = `n+d ´ `n+2d ´ and k ≤ D(D + 1) −. This n n procedure requires τ O(1) DO(k3) bit operations and the coefficients of the outputted polynomials have bit length dominated by τDO(k3). Key words. rational sum of squares, semidefinite programming, convex, complexity. 1. Introduction. Motivation and problem statement. Suppose f ∈ R[x1,..., xn], then f is a sum of squares (SOS) in R[x1,..., xn] if and only if it can be written in the form f = v T · M · v, (1.1)
ULTIMATELY FAST ACCURATE SUMMATION

, 2009
"... We present two new algorithms FastAccSum and FastPrecSum, one to compute a faithful rounding of the sum of floatingpoint numbers and the other for a result “as if” computed in Kfold precision. Faithful rounding means the computed result either is one of the immediate floatingpoint neighbors of th ..."
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We present two new algorithms FastAccSum and FastPrecSum, one to compute a faithful rounding of the sum of floatingpoint numbers and the other for a result “as if” computed in Kfold precision. Faithful rounding means the computed result either is one of the immediate floatingpoint neighbors of the exact result or is equal to the exact sum if this is a floatingpoint 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 instructionlevel parallelism so that they are also fast in terms of measured computing time. The algorithms require only standard floatingpoint addition, subtraction, and multiplication in one working precision, for example, double precision.
Global Optimization of Polynomials Using Generalized Critical Values and Sums of Squares
, 2010
"... 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 Zariskiclosed subset A � GLn(C) such t ..."
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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 Zariskiclosed 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 illconditionedness of SDP problems when the infimum is not attained.
Computing the radius of positive semidefiniteness of a multivariate real polynomial via a dual of Seidenberg’s method
, 2010
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QuadraticTime Certificates in Linear Algebra*
"... 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/Artinbased rational sumofsquares certificates, i.e., p ..."
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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/Artinbased rational sumofsquares 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 essentiallyquadratic binary space and time complexity via Storjohann’s algorithms.
The “Seven Dwarfs ” of Symbolic Computation*
, 2010
"... We present the Seven Dwarfs of Symbolic Computation, which are sequential and parallel algorithmic methods that today carry a great majority of all exact and hybrid symbolic compute cycles. SymDwf 1. Exact linear algebra, integer lattices SymDwf 2. Exact polynomial and differential algebra, Gröbner ..."
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We present the Seven Dwarfs of Symbolic Computation, which are sequential and parallel algorithmic methods that today carry a great majority of all exact and hybrid symbolic compute cycles. SymDwf 1. Exact linear algebra, integer lattices SymDwf 2. Exact polynomial and differential algebra, Gröbner bases SymDwf 3. Inverse symbolic problems, e.g., interpolation and parameterization SymDwf 4. Tarski’s algebraic theory of real geometry SymDwf 5. Hybrid symbolicnumeric computation SymDwf 6. Computation of closed form solutions SymDwf 7. Rewrite rule systems and computational group theory We will elaborate on each dwarf and compare with Colella’s seven and the Berkeley team’s thirteen dwarfs of scientific computing.
unknown title
, 2011
"... 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 proofread them, or checked ..."
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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 proofread them, or checked
On the Generation of Positivstellensatz Witnesses in Degenerate Cases ⋆
"... Abstract. One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some linear equality (Positivstellensatz). This produce ..."
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Abstract. One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some linear equality (Positivstellensatz). This produces a witness for the desired property, from which it is reasonably easy to obtain a formal proof of the property suitable for a proof assistant such as Coq. The problem of finding a witness reduces to a feasibility problem in semidefinite programming, for which there exist numerical solvers. Unfortunately, this problem is in general not strictly feasible, meaning the solution can be a convex set with empty interior, in which case the numerical optimization method fails. Previously published methods thus assumed strict feasibility; we propose a workaround for this difficulty. We implemented our method and illustrate its use with examples, including extractions of proofs to Coq. 1