Electrical computers: arithmetic processing and calculating – Electrical digital calculating computer – Particular function performed
Reexamination Certificate
2005-07-19
2005-07-19
Ngo, Chuong Dinh (Department: 2124)
Electrical computers: arithmetic processing and calculating
Electrical digital calculating computer
Particular function performed
Reexamination Certificate
active
06920472
ABSTRACT:
One embodiment of the present invention provides a system for finding the roots of a system of nonlinear equations within an interval vector X=(X1, . . . , Xn), wherein the system of non-linear equations is specified by a vector function ƒ=(ƒ1, . . . , ƒn). The system operates by receiving a representation of the interval vector X (which is also called a box), wherein for each dimension, i, the representation of Xiincludes a first floating-point number, αi, representing the left endpoint of Xi, and a second floating-point number, bi, representing the right endpoint of Xi. Next, the system performs an interval Newton step on X to produce a resulting interval vector, X′, wherein the point of expansion of the interval Newton step is a point, x, within the interval X, and wherein performing the interval Newton step involves evaluating ƒ(x) to produce an interval result ƒ1(x). The system then evaluates a first termination condition, wherein the first termination condition is TRUE if: zero is contained within ƒ1(x), J(x,X) is regular (wherein J(x,X) is the Jacobian of the function ƒevaluated with respect to x over the box X); and X is contained within X′. If the first termination condition is TRUE, the system terminates the interval Newton method and records X′ as a final bound.
REFERENCES:
patent: 5220524 (1993-06-01), Hesson
patent: 5631858 (1997-05-01), Narita et al.
patent: 2004/0015830 (2004-01-01), Reps
E.R. Hansen, “Global Optimization Using Interval Analysis,” Marcel Dekker, Inc., New York, NY, 1992.
R.B. Kearfott, “A Fortran 90 Environment for Research and Prototyping of Enclosure Algorithms for Nonlinear Equations and Global Optimization,” ACM Transactions on Mathematical Software, vol. 21, No. 1, Mar. 1995, pp. 63-78 http://interval.louisiana.edu/preprints.html.
R. B. Kearfott, Algorithm 763: Interval Arithmetic: A Fortran 90 Module for an Interval Data Type, ACM Trans. Math. Software, 22, vol. 4, 1996, pp. 385-392. http://interval.louisiana.edu/preprints.html.
R. B. Kearfott and M. Novoa III, “Algorithm 681: INTBIS, A portable interval Newton/bisection package”, ACM Trans. Math Software, vol. 16, No. 2, pp. 152-147. http://www.netlib.org/toms/681.
R. B. Kearfott, M. Dawande, K.S. Du, and C. Hu, “Algorithm 737: INTLIB: A Portable Fortran 737 Interval Standard Function Library,” ACM Trans. Math. Software, 20, vol. 4, Dec. 1994, pp. 447-458.
R. B. Kearfott and G.W. Walster, “On Stopping Criteria in Verified Nonlinear Systems or Optimization Algorithms,” ACM Trans. Math. Software, 26, vol. 3, Sep. 2000, pp. 323-351. The publication itself says Received: Jul. 1999; revised: Mar. 2000; accepted: Mar. 2000. http://interval.louisiana.edu/preprints.html.
R.E. Moore and S.T. Jones “Safe Starting Regions for Iterative Methods”, SIAM Journal on Numerical Analysis, vol. 14, No. 6 (Dec. 1977), pp. 1051-1065.
A. Neumaier, “The Enclosure of Solutions of Parameter-Dependent Systems of Euqations,” Cambridge University Press, Cambridge, 1990, ISBN: 0-12-505630-3, Reliability in Computing pp. 269-286.
S.M. Rump, “Verification Methods for Dense and Sparse Systems of Equations,” in Topics in Validated Computations: Proceedings of the IMACS-GAMM International Workshop on Validated Computations, University of Oldenburg, J. Herzberger, ed., Elsevier Studies in Computational Mathematics, Elsevier, 1994, pp. 63-136.
Publication: “Design, implementation and evaluation of the constraint language cc (FD)” by Pascal Van Hentenryck et al., The Journal of Logic Programming 37, 1998, pp. 139-164.
No affiliations
Park Vaughan & Fleming LLP
Sun Microsystems Inc
LandOfFree
Termination criteria for the interval version of Newton's... does not yet have a rating. At this time, there are no reviews or comments for this patent.
If you have personal experience with Termination criteria for the interval version of Newton's..., we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Termination criteria for the interval version of Newton's... will most certainly appreciate the feedback.
Profile ID: LFUS-PAI-O-3395122