Faulkner, F.D.

Seibel, E.W.

English

Calhoun, Institutional Archive of the Naval Postgraduate School

Calhoun: The NPS Institutional ArchiveDSpace RepositoryFaculty and Researchers Faculty and Researchers' Publications1963A Comparison of Methods for DeterminingOptimum Paths in the Problem of BolzaFaulkner, F.D.; Seibel, E.W.FAULKNER, FD, and E. W. Seibel. "A comparison of methods for determiningoptimum paths in the problem of Bolza." In COMMUNICATIONS OF THE ACM, vol. 6,no. 7, pp. 357-357. 1515 BROADWAY, NEW YORK, NY 10036: ASSOC COMPUTINGMACHINERY, 1963.http://hdl.handle.net/10945/60004This publication is a work of the U.S. Government as defined in Title 17, UnitedStates Code, Section 101. Copyright protection is not available for this work in theUnited States.Downloaded from NPS Archive: Calhoun1 8 {v3 8 } v f(x) dx ~ g h(i) 6 ~ [f(x2;-1) - f(x2;)] + f(x2s+1) is exact for polynomials of degree 2 or less, where Vis the s-dimen-sional hyper-rectangle r<i) = !h<i> ~ x<il ~ r<i> + !h(i) and (") (") h(i) [v3 J x2}-1 = r ' + 2 6 (1 - 13;;) - Oii (") n h (i) [v3 J X2} = r ' + 2 (\ (1 - /Ji;) + Oii xi!~1 = r<i> + ~3 h<i>. When this formula is used for subdivisions of a larger region, the contributions from the summation cancel on all interior bounda-ries, leaving only one point to be evaluated for each interior subregion. Errors for the formula and numerical comparisons with other methods will be given. 2B.4: Computational Procedure for the Calculation of the Subdivisions of the Components of the Analysis of Variance. M. C. Mn.LER, III, University of Oklahoma Medi-cal School, Oklahoma City, Okla. A computational procedure for the calculation of the subdi-visions of the components of the analysis of variance is given. The method is essentially one of matrix operations and is shown to be quite general in its applicability. Special emphasis is placed on the use of the method as a means of obtaining the linear, quadratic, etc., components of quantitative main effects and the linear by linear, linear by quadratic, etc., components of the interactions of quantitative by quantitative factors. Also dis-cussed, is the breakdown of the sum of squares of the interaction of qualitative by quantitative factors. 2B.5: A Fast Direct Solution of Poisson's Equation Using Fourier Analysis. R. HOCKNEY AND 0. BuNEMAN, Stanford University, Stanford, Calif. The motion of interacting charged particles in two dimensions (e.g., in plasmas, electron tubes and ion guns) may be simulated on a computer by a time-stepping process. To do this it is neces-sary to solve Poisson's equation on a mesh of several thousand nodes in a few seconds. A direct method of solution has been developed involving Fourier analysis in one dimension and Gaussian elimination in the other that solves Poisson's equation on a 48 X 48 mesh in 2.75 seconds (IBM 7090). The maximum observed error in the potential for a random charge distribution is rv5 X 10-7 of the maximum potential difference. 3B TUTORIAL 3B.l: On the Construction of a Compiler for ALGOL 60. AR'rHUR EVANS, JR., Carnegie Institute of Technology, Pitts-burgh, Pa. It is possible to use a computer language, such as ALGOL, for a variety of programs with no concept of the functions of the com-piler used with the machine for which the programs are being written. The compiler becomes in the mind of such a programmer, an integral part of the machine. More flexibility in programming and wiser use of computing machinery can be achieved if one has some knowledge of compilers and, in some cases, machine language. ThiR tutorial is presented with the intent of providing a stronger background to the programmer, showing the intricacies of the compiler and mechanism whereby the object program is derived in machine language. 4A PANEL ON INFORMATION RETRIEVAL Moderator: Louis C. RAY, System Development Corp., Santa Monica, Calif. Panelists: JACK BELZER, Western Reserve University, Cleve-land, Ohio; JoHN C. CowrELLO, JR., Ratte/le Memorial Volume 6 / Number 7 / July, 1%:l Institute, Columbus, Ohio; and RoNALD E. WYLLYS, System Development Corp., Santa Monica, Calif. The panel will survey current approaches to developing a theoretical base for Information Retrieval work. Special atten-tion will be given to underdeveloped areas of the subject and to pragmatic approaches to the design of IR systems. Audience discussion will be invited. 4B NUMERICAL ANALYSIS II 4B.l: A Comparison of Methods for Determining Optimum Paths in the Problem of Bolza. F. D. FAULKNER AND E. W. SEIBEL, United States Naval Postgraduate School, Monte-rey, Calif. A comparison is made of two methods of determining optimum trajectories. The methods are (1) the differential method due to Bliss, using only extremals and (2) the method of steepest descent, based on the fundamental lemma of the calculus of variations. The study so far has been confined to low order systems. It indi-cates the differential method converges more rapidly if it con-verges. Convergence is a problem and it seems that there are problems where the latter method will converge but the former may not. Optimum correction of perturbed paths is a minor problem if the former method is used. The latter method seems to require less luck or skill in getting started. 4B.2: lnterpolatory Iteration Functions. J. F. TRAUB, Bell Telephone Laboratories, Murray Hill, N. J. Let (xi) be a sequence of approximants to a zero of a real function f. The x; are generated by an iteration function (I.F.) which samples f(x) and its derivatives at one or more values of x. An important class of I.F., which includes the most commonly used I.F. and their generalizations, arc classified and uniformly studied by considering them as generated from hyperosculatory interpolation off or its inverse at a number of points. In particu-lar, a rigorous derivation of the order of any interpolatory I.F. will be discussed. The derivation requires no a priori as-sumptions about the existence of an order. 4B.3: Matrix Symmetrization Method for the Algehraic Eigenproblem. J. L. HOWLAND AND F. J. FARRELL, Uni-versity of Ottawa, Ottawa, Ont., Canada An arbitrary matrix M is said to be symmetrized by the sym-metric matrix Bin case the matrix product A = BM is symmetric. When a symmetrizing matrix B is known, the eigenproblem (M - H) x = 0 may be reformulated as (A - A.B) x = 0, and several methods due to S. H. Crandall, and related to Wielandt's "broken iteration" may be applied. In this paper it is shown that Lanczos' method of minimized iterations may be adapted to the calculation of symmetrizing matrices. Moreover, this method produces the symmetrizing matrix B and the symmetric product A in a coordinate system in which R is diagonal and A is tri-diagonal. 4B.4: Rapid Methods of Structural Change Analysis. BER'l'RAM KLEIN, Hughes Aircraft Co., Culver City, Calif. Methods are investigated for determining rapidly the effects of changes in matrix problems of the form AX = L without again solving the entire problem. Special techniques are discussed arnl examples given for treating the approximate methods. The tech-niques facilitate the parametric analysis of many complex prob-lems, particularly in the field of structures. 4B.5: On the Numerial Solution of Certain Classes of Boundary Value Problems. L. E. HULBER'l' AND F. W. NIEDENFUHR, Ohio State University, Columbus, Ohio The boundary collocation technique is a powerful but little used method for solving boundary value problems. A slight modification of this technique has been used in the computer solution of a variety of boundary value problems involving regions of complicated shapes. A particular advantage of the approach is that the computer program requires only minor modifications to Communications of the ACM 35i 

A Comparison of Methods for Determining Optimum Paths in the Problem of Bolza

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A Comparison of Methods for Determining Optimum Paths in the Problem of Bolza

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