Yousuo Joseph Zou, Professor of University of Guam,USA, CEO of Blue Pacific Technology Com, delivers a speech at Track E: World Forum on High-end Cooperation of the 2nd WORLD EMERGING INDUSTRIES SUMMIT (WEIS 2013)

outline of the speech:

ewly developed SINC methods 

SINC methods are mainly developed by the author’s PhD adviser, Professor Frank Stenger and his graduate students at University of Utah, USA.  SINC methods are based on linear algebra, Complex analytical functions, Fourier transformation, Newton iteration and etc. SINC methods are highly efficient, more accurate high-performance computational algorithms compared to currently existing computational methods.  SINC methods can re-write the whole sets of existing computational algorithms and make a revolution in future computational engineering and computational science (Please see details in the following).

Frank Stenger Explains:  

         What are Sinc Methods?

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Sinc methods are a new family of formulas for computation, which were developed primarily by Frank Stenger and his students.  These formulas enable simple, yet accurate approximations for every type of operation of calculus, such as function approximation, the approximation of derivatives, the approximation of definite and indefinite integrals, and the approximation of definite and indefinite convolutions.  Efficient Sinc methods even exist for approximating or inverting every type of transform, such as the Laplace, the Mellin, the Hankel, the Hilbert, and the Fourier transforms.

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Algorithmically, all of the Sinc approximations involve the use of known linear algebra routines, such as computation of inner products, matrix multiplication, the solution of linear algebraic equations, or at worst, the computation of eigenvalues and eigenvectors of relatively small matrices.  Moreover, Sinc algorithms have close to optimal complexity.

No other method of computation accomplishes such breadth of applicability.  Indeed, due to their efficiency, these methods make possible the accurate solution of frequently occurring computational problems which were hitherto extremely difficult, or impossible to carry out.  Examples of such problems are:

§  The accurate approximation of integrals with singular integrand, or over infinite intervals;

§  The accurate approximation of boundary layer problems;

§  The accurate solution of ordinary, or partial differential equations that have coefficients with singularities, or the solution of such problems over infinite domains, and

§  The accurate approximate solution of one, or multidimensional, Fredholm or Volterra, linear or nonlinear integral equations, especially those with singular kernels, or over infinite regions, or both.  For example, the solution of Weiner-Hopf integral equations can be very simply carried out via Sinc methods.

Parallel computation, for e.g., the solution of multidimensional problems falls out very simply and efficiently for algorithms based on Sinc methods.

If low accuracy is required, then Sinc methods do not have an appreciable advantage over classical methods, such as finite difference or finite element methods. On the other hand, while high accuracy is either computationally very intensive, or impossible to achieve via algorithms based on classical methods of computation, the same Sinc-based program, written in terms of an easily specified parameter can be used to achieve arbitrary accuracy.  Here, too, Sinc methods fill a gap, by enabling the solution of computational problems which were hitherto impossible to carry out.  Furthermore, the fact that Sinc methods can easily achieve arbitrary accuracy will make these methods very suitable for use in the setting of arbitrary precision arithmetic, which is expected to replace finite precision arithmetic during the next 10 years.

An important Computer Science – Sinc research area thus is the development of Sinc based computer algorithms for the solution of problems which were hitherto difficult or impossible.  Some of these projects are in the nature of general algorithms, such as the following, some of which have already been produced, while the last 2 are in the process of being completed:

§  “ALGORITHM 614. A FORTRAN Subroutine for Numerical Integration in H^p”,

by K. Sikorski, J. Schwing, and F. Stenger, ACM TOMS 10 (1984) 152-160.  (For integration over finite, semi-infinite, or infinite intervals, with singularities allowed.)

§  “ODE-IVP-PACK, via Sinc  Indefinite Integration and Newton Iteration”, by F. Stenger,

B. Keys, M. O’Reilly, and K. Parker, to appear in ‘Numerical Algorithms’.   This is like Fear’s package, except that we can also efficiently compute solutions with singularities, boundary layers, or over infinite intervals, and moreover, stability and stiffness are no problem for this package.  An explicit description of this routine, as well as the computer package itself can be downloaded.  Click Computer Packages for more information.

§  “PTOLEMY, A Sinc Collocation Sub-System”, Ph.D. Thesis of Kenneth Parker, to be completed at end of fall quarter, 1995.  This package is a sub-package of MAPLE, for solving PDE.  The user starts MAPLE, which he runs ‘with (Ptolemy)’.  He types in the PDE in MAPLE, then he types in the boundary of the region, and the boundary conditions.  Parker’s program then replaces the PDE with a system of algebraic equations, based on Sinc collocation.  Domain subdivision is carried out in such a way to preserve the exponentially small error of Sinc approximation.  The description of PTOLEMY as well as the programs for PTOLEMY can be downloaded.  Click Computer Packages for more information.

§  “Sinc-PACK, A Package of Sinc Algorithms”, by M. O’Reilly, F. Stenger, and T. Zhang, to be completed in the Spring quarter of 1996.  This toolbox is written to run independently, or with MATLAB.  It consists of methods for approximating every operation of calculus.  As mentioned above, the package is undergoing revision, although the present form of the package can be downloaded.   Click Computer Packages for more information.

The textbooks cited at the outset of these web pages are of course the best sources for learning about Sinc methods.  On the other hand, the paper “Matrices of Sinc Methods” gives an excellent summary of these methods.