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dhb-iccopt - Experimental Mathematics and Optimization...

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1 Experimental Mathematics and Optimization David H Bailey Lawrence Berkeley National Laboratory http://crd.lbl.gov/~dhbailey
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2 Outline 1. Introduction Application: the uncertainty principle. 1. Integer relation detection The PSLQ algorithm. Applications: bifurcation constants, sculpture. 1. Sequence limit extrapolation Application: the Quinn-Rand-Strogatz constant. 1. Infinite series summation The Euler-Maclaurin formula. Euler’s transformation. Applications: Euler sums, Apery-like series, Ramanujan-like series. 1. Numerical integration Gaussian quadrature. Tanh-sinh quadrature. Applications: tetrahedral integral, box integrals, Ising integrals, spin integrals. 1. Summary
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3 Methods Used in Experimental and Computer-Assisted Mathematics Computer-based symbolic computation (e.g., Mathematica, Maple, Magma ). High-precision (hundreds or thousands of digits) numerical computation. Integer relation detection algorithms (notably the PSLQ algorithm). Computer-based sequence and constant recognition facilities. Minimization and maximization schemes. Prime verification algorithms and other computational number theory methods. The Wilf-Zeilberger algorithm for proving certain infinite series identities. Formal verification methods – for instance, the effort to verify Thomas Hales’ recent proof of the Kepler conjecture using symbolic logic.
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4 The Uncertainty Principle where Uncertainty principle from quantum mechanics: The uncertainty in position times the uncertainty in momentum must be at least some minimum value. Uncertainty principle from signal processing: A signal cannot be both time- limited and band-limited -- the dispersion of the signal times the dispersion of the Fourier transform must be at least some minimum value. The mathematical formulations of these two principles are identical:
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5 A Numerical Minimization Approach to the Uncertainty Principle A formal proof of the uncertainty principle (UP) is not very difficult, but it is hardly intuitive at first reading. Here is a more computationally and intuitively appealing approach: 1. Start with a simple “tent” function. 2. Evaluate the UP product for the all perturbations of the current function on a grid with the current resolution. 3. Select the function from step 2 that has the smallest UP product. 4. Refine the grid and go to 2. 5. End when grid is sufficiently fine. Minimizing function: Gaussian function Minimum UP product: 0.25 Jonathan M. Borwein and David H. Bailey, Mathematics by Experiment , AK Peters, 2004, pg. 183-188.
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6 The Integer Relation Problem Let (x n ) be a given vector of real numbers. An integer relation algorithm finds integers (a n ) such that (or at least within “epsilon” of zero). This can be viewed as an integer programming problem: find a set of integers (a i ) that minimizes for some M.
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