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IBM Journal of Research and Development

Issue 4 • Date July 1972

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Displaying Results 1 - 20 of 20
  • Automatic Computation of Exponentials, Logarithms, Ratios and Square Roots

    Page(s): 380 - 388
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (609 KB)  

    It is shown how a relatively simple device can evaluate exponentials, logarithms, ratios and square roots for fraction arguments, employing only shifts, adds, high-speed table lookups, and bit counting. The scheme is based on the cotransformation of a number pair (x,y) such that the F(x,y) = f(x0) is invariant; when x is driven towards a known value xω , y is driven towards the result. For an N-bit fraction about N/4 iterations are required, each involving two or three adds; then a termination algorithm, based on an add and an abbreviated multiply, completes the process, for a total cost of about one conventional multiply time. Convergence, errors and simulation using APL are discussed. View full abstract»

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  • Preface: Mathematics of Numerical Computation

    Page(s): 334
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (168 KB)  

    The extensive use of computers for solving scientific and engineering problems has had a profound effect on the evolution of the mathematical techniques of numerical computation. It has given new impetus to research in numerical integration of differential equations: it has stimulated fresh approaches in the areas of approximation and optimization; and, in general, it has increased interest in numerical methods for the solution of a variety of problems. This special issue of the Journal reports some recent developments and innovations in the computational methods of numerical analysis. View full abstract»

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  • A-stable, Accurate Averaging of Multistep Methods for Stiff Differential Equations

    Page(s): 335 - 348
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (781 KB)  

    Several low-order numerical solutions of stiff systems of ordinary differential equations are computed by repeated integration, using a multistep formula with parameters. By forming suitable linear combinations of such solutions, higher-order solutions are obtained. If the parameters are properly chosen the underlying solutions, and thus the higher-order one, can be made A-stable and strongly damping with respect to the stiff components of the system. A detailed description is given of an algorithmic implementation of the method, which is computationally efficient. Numerical experiments are carried out on some test problems, confirming the validity of the method. View full abstract»

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  • Hopscotch Difference Methods for Nonlinear Hyperbolic Systems

    Page(s): 349 - 353
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (450 KB)  

    In a recent series of papers, one of the authors has developed and demonstrated properties of a computational algorithm for solving partial differential equations. This process, known as the hopscotch algorithm, has been studied particularly with reference to the efficient integration of parabolic and elliptic problems. In the present paper attention is directed to the application of the technique to the numerical integration of first-order nonlinear hyperbolic systems. While maintaining the properties of the hopscotch process as applied to parabolic problems, it is shown that one of the novel schemes generated by this approach has an added bonus, namely, maximum stability for a variable choice of damping or pseudoviscous term. This property should be of particular value in the solution of problems with shocks. A class of hopscotch Lax-Wendroff schemes is also studied. View full abstract»

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  • Parallel Shooting Method for Boundary-value Problems: Application to the Neutron Transport Equation

    Page(s): 354 - 364
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (736 KB)  

    A direct method is given for the solution of the spherical harmonics approximation to the Boltzmann equation for neutron transport in slab geometry. The roundoff instability of the problem is eliminated by performing linear transformations of the matrices involved, which ensure that the matrix columns are linearly independent. The novelty of the method lies in that only a minimum number of matrix transformations is performed, the precise number being determined dynamically, efficiently, and in a new way by the program itself in the course of the computation. View full abstract»

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  • Finite Difference Formulas for Neumann Conditions on Irregularly Shaped Boundaries

    Page(s): 365 - 371
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (387 KB)  

    A method, based on finite element techniques, is described for obtaining finite differences for Neumann conditions on irregularly shaped boundaries. The resultant difference equations may be used in both time-dependent and -independent problems. As an example, the propagation of an elastic surface wave (Rayleigh wave) around a 90° corner is studied. The results of this simulation compare favorably with laboratory experiments. View full abstract»

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  • Turán Formulae and Highest Precision Quadrature Rules for Chebyshev Coefficients

    Page(s): 372 - 379
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (405 KB)  

    Expansions of functions in series of Chebyshev polynomials are frequently used in numerical analysis. The coefficients occurring in the expansion are definite integrals; the purpose of this paper is to investigate numerical integration formulae for the coefficients of highest degree of precision. View full abstract»

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  • Enveloping an Iteration Scheme

    Page(s): 389 - 392
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (393 KB)  

    The problem of determining the optimal values of relaxation parameters for a linear iteration is solved. The optimal iteration scheme is achieved by a second-order linear iteration method. View full abstract»

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  • Numerical Properties of a Multivariate Ritz-Trefftz Method

    Page(s): 393 - 400
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (570 KB)  

    In this paper the numerical properties of the Ritz-Trefftz algorithm are discussed in the context of the numerical approximation to the linear parabolic regulator problem using multivariate splines. The algorithm is first derived in the problem context and the resulting linear algebraic system is discussed. Such properties as definiteness and band structure are treated. The algorithm is applied to a number of sample control problems, and it is shown that the method yields efficient and highly accurate continuous approximations to the solutions of the selected sample problems. Computer implementation of the general algorithm is also discussed. View full abstract»

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  • Recursive Evaluation of Padé Approximants for Matrix Sequences

    Page(s): 401 - 406
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (434 KB)  

    An algorithm is described for calculating the existing Padé approximants to any sequence A0, A1, ⋅ ⋅ ⋅ of s × t-matrices. As an application the algorithm gives a new way for finding the minimal polynomial of any square matrix A and the inverse of the characteristic matrix xI − A. View full abstract»

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  • On the Convergence of Gradient Methods under Constraint

    Page(s): 407 - 411
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (370 KB)  

    The mathematical programming problem discussed is the convergence of a certain popular type of gradient procedure for maximizing a function under inequality constraints. An example shows that convergence to a solution need not always occur, and a theorem shows that under certain circumstances the gradient method does converge. View full abstract»

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  • Finding All Shortest Distances in a Directed Network

    Page(s): 412 - 414
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (353 KB)  

    A new method is given for finding all shortest distances in a directed network. The amount of work (in performing additions, subtractions, and comparisons) is slightly more than half of that required in the best of previous methods. View full abstract»

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  • Maintaining a Sparse Inverse in the Simplex Method

    Page(s): 415 - 423
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (578 KB)  

    Improved methods are discussed for handling sparse matrices in practical linear programming. An analytical comparison is made of four methods for updating the inverse in the iterations following a reinversion. Of these, one technique using the elimination form of inverse is selected for some computational experiments and its advantages in terms of speed and sparseness demonstrated. View full abstract»

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  • Mixed-integer Algorithms for the (0,1) Knapsack Problem

    Page(s): 424 - 430
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (443 KB)  

    An enumerative scheme is presented for the (0,1) knapsack problem as a specialization of the state enumeration method. Techniques are explored for rendering search procedures more efficient by systematic use of information generated during execution of the algorithm. The inequalities of Benders and Gomory-Johnson are exploited to yield implicit enumeration tests in the special case of the knapsack problem. In a comparative study of eight algorithms and of the utility of certain approximations and inequalities, computational results are given for twelve knapsack problems, each having ten (0,1) variables. The effectiveness of these enumerative algorithms are thus tested in a relatively simple framework. View full abstract»

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  • Linear Convergence of the Conjugate Gradient Method

    Page(s): 431 - 433
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (295 KB)  

    There are two procedures for applying the method of conjugate gradients to the problem of minimizing a convex, nonlinear function: the “continued” method, and the “restarted” method in which all the data except the best previous point are discarded, and the procedure is begun anew from that point. It is demonstrated by example that in the absence of the standard initial starting condition on a quadratic function, the continued conjugate gradient method will converge to the solution no better than linearly. Furthermore, it is shown that for a general nonlinear function, the nonrestarted conjugate gradient method converges no worse than linearly. View full abstract»

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  • On the Davidenko-Branin Method for Solving Simultaneous Nonlinear Equations

    Page(s): 434 - 436
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (483 KB)  

    It has been conjectured that the Davidenko–Branin method for solving simultaneous nonlinear equations is globally convergent, provided that the surfaces on which each equation vanishes are homeomorphic to hyperplanes. We give an example to show that this conjecture is false. A more complicated example shows that the method may fail to converge to a zero of the gradient of a scalar function, so the associated method for function minimization is not globally convergent. View full abstract»

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  • Recent Papers by IBM Authors

    Page(s): 437 - 441
    Save to Project icon | Click to expandQuick Abstract | PDF file iconPDF (354 KB)  

    Reprints of the papers listed here can usually be obtained by writing directly to the uuthors. The authors' IBM divisions are identified as follows: ASDD is the Advanced Systems Development Division; CD, Components Division; DPD, Data Processing Division; DPG, Data Processing Group; FED, Field Engineering Division; FSD, Federal Systems Division; GSD. General Systems Division; OPD, Office Products Division; RES, Research Dlvision; SDD, Systems Development Division; SMD, Systems Manufacturing Division; and WTC, World Trade Corporation. Papers are listed alphabetically by name of journal View full abstract»

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  • Recent IBM Patents

    Page(s): 442 - 445
    Save to Project icon | PDF file iconPDF (333 KB)  
    Freely Available from IEEE
  • Authors

    Page(s): 446 - 448
    Save to Project icon | PDF file iconPDF (324 KB)  
    Freely Available from IEEE
  • Contents of previous is

    Page(s): 449
    Save to Project icon | PDF file iconPDF (190 KB)  
    Freely Available from IEEE

Aims & Scope

The IBM Journal of Research and Development is a peer-reviewed technical journal, published bimonthly, which features the work of authors in the science, technology and engineering of information systems.

Full Aims & Scope

Meet Our Editors

Editor-in-Chief
Clifford A. Pickover
IBM T. J. Watson Research Center