301,228 research outputs found
Exact Real Arithmetic with Perturbation Analysis and Proof of Correctness
In this article, we consider a simple representation for real numbers and
propose top-down procedures to approximate various algebraic and transcendental
operations with arbitrary precision. Detailed algorithms and proofs are
provided to guarantee the correctness of the approximations. Moreover, we
develop and apply a perturbation analysis method to show that our approximation
procedures only recompute expressions when unavoidable.
In the last decade, various theories have been developed and implemented to
realize real computations with arbitrary precision. Proof of correctness for
existing approaches typically consider basic algebraic operations, whereas
detailed arguments about transcendental operations are not available. Another
important observation is that in each approach some expressions might require
iterative computations to guarantee the desired precision. However, no formal
reasoning is provided to prove that such iterative calculations are essential
in the approximation procedures. In our approximations of real functions, we
explicitly relate the precision of the inputs to the guaranteed precision of
the output, provide full proofs and a precise analysis of the necessity of
iterations
Rational approximations in Analytic QCD
We consider the ``modified Minimal Analytic'' (mMA) coupling that involves an
infrared cut to the standard MA coupling. The mMA coupling is a Stieltjes
function and, as a consequence, the paradiagonal Pade approximants converge to
the coupling in the entire -plane except on the time-like semiaxis below
the cut. The equivalence between the narrow width approximation of the
discontinuity function of the coupling, on the one hand, and this Pade
(rational) approximation of the coupling, on the other hand, is shown. We
approximate the analytic analogs of the higher powers of mMA coupling by
rational functions in such a way that the singularity region is respected by
the approximants.Several comparisons, for real and complex arguments ,
between the exact and approximate expressions are made and the speed of
convergence is discussed. Motivated by the success of these approximants, an
improvement of the mMA coupling is suggested, and possible uses in the
reproduction of experimental data are discussed.Comment: 12 pages,9 figures (6 double figures); figs.6-8 corrected due to a
programming error; analysis extended to two IR cutoffs; Introduction
rewritten; to appear in J.Phys.
A Mathematical Basis for an Interval Arithmetic Standard
Basic concepts for an interval arithmetic standard are discussed
in the paper. Interval arithmetic deals with closed and connected sets of real
numbers. Unlike floating-point arithmetic it is free of exceptions. A complete
set of formulas to approximate real interval arithmetic on the computer
is displayed in section 3 of the paper. The essential comparison relations and
lattice operations are discussed in section 6. Evaluation of functions for interval
arguments is studied in section 7. The desirability of variable length
interval arithmetic is also discussed in the paper. The requirement to adapt
the digital computer to the needs of interval arithmetic is as old as interval
arithmetic. An obvious, simple possible solution is shown in section 8
On the complete instability of empirically implemented dynamic Leontief models
On theoretical grounds, real world implementations of forward-looking dynamic Leontief systems were expected to be stable. Empirical work, however, showed the opposite to be true: all investigated systems proved to be unstable. In fact, an extreme form of instability ('complete instability') appeared to be the rule. In contrast to this, backward-looking models and dynamic inverse versions appeared to be exceptionally stable. For this stability-instability switch a number of arguments have been put forward, none of which was convincing. Dual (in)stability theorems only seemed to complicate matters even more. In this paper we offer an explanation. We show that in the balanced growth case--under certain conditions--the spectrum of eigenvalues of matrix D equivalent to (I - A)-1B, where A stands for the matrix of intermediate input coefficients and B for the capital matrix, will closely approximate the spectrum of a positive matrix of rank one. From this property the observed instability properties are easily derived. We argue that the employed approximations are not unrealistic in view of the data available up to now
Algorithmic Polynomials
The approximate degree of a Boolean function is
the minimum degree of a real polynomial that approximates pointwise within
. Upper bounds on approximate degree have a variety of applications in
learning theory, differential privacy, and algorithm design in general. Nearly
all known upper bounds on approximate degree arise in an existential manner
from bounds on quantum query complexity. We develop a first-principles,
classical approach to the polynomial approximation of Boolean functions. We use
it to give the first constructive upper bounds on the approximate degree of
several fundamental problems:
- for the -element
distinctness problem;
- for the -subset sum problem;
- for any -DNF or -CNF formula;
- for the surjectivity problem.
In all cases, we obtain explicit, closed-form approximating polynomials that
are unrelated to the quantum arguments from previous work. Our first three
results match the bounds from quantum query complexity. Our fourth result
improves polynomially on the quantum query complexity of the
problem and refutes the conjecture by several experts that surjectivity has
approximate degree . In particular, we exhibit the first natural
problem with a polynomial gap between approximate degree and quantum query
complexity
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