{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

dhb-iccopt

# dhb-iccopt - Experimental Mathematics and Optimization...

This preview shows pages 1–7. Sign up to view the full content.

1 Experimental Mathematics and Optimization David H Bailey Lawrence Berkeley National Laboratory http://crd.lbl.gov/~dhbailey

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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:
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern