Showing papers on "Computation published in 1973"

A Parallel Algorithm for the Efficient Solution of a General Class of Recurrence Equations

Abstract: An mth-order recurrence problem is defined as the computation of the series x 1 , x 2 , ..., X N , where x i = f i (x i-1 , ..., x i-m ) for some function f i . This paper uses a technique called recursive doubling in an algorithm for solving a large class of recurrence problems on parallel computers such as the Iliac IV. Recursive doubling involves the splitting of the computation of a function into two equally complex subfunctions whose evaluation can be performed simultaneously in two separate processors. Successive splitting of each of these subfunctions spreads the computation over more processors. This algorithm can be applied to any recurrence equation of the form x i = f(b i , g(a i , x i-1 )) where f and g are functions that satisfy certain distributive and associative-like properties. Although this recurrence is first order, all linear mth-order recurrence equations can be cast into this form. Suitable applications include linear recurrence equations, polynomial evaluation, several nonlinear problems, the determination of the maximum or minimum of N numbers, and the solution of tridiagonal linear equations. The resulting algorithm computes the entire series x 1 , ..., x N in time proportional to [log 2 N] on a computer with N-fold parallelism. On a serial computer, computation time is proportional to N.

An Efficient Parallel Algorithm for the Solution of a Tridiagonal Linear System of Equations

Abstract: Tridiagonal linear systems of equations can be solved on conventional serial machines in a time proportional to N, where N is the number of equations The conventional algorithms do not lend themselves directly to parallel computation on computers of the ILLIAC IV class, in the sense that they appear to be inherently serial An efficient parallel algorithm is presented in which computation time grows as log2N The algorithm is based on recursive doubling solutions of linear recurrence relations, and can be used to solve recurrence relations of all orders

Computation and Deduction.

Abstract: Draft notes for a course given at Carnegie Mellon University during the fall semester of 1994. Please send comments to fp@cs.cmu.edu. Do not cite, copy, or distribute without the express written consent of Frank Pfenning and the National Football League.

A Comparison of Some Theoretical Models of Parallel Computation

Abstract: In this paper we briefly describe and compare a number of theoretical models for parallel computation; namely, Petri nets, computation graphs, and parallel program schemata. We discuss various problems and properties of parallel computation that can be studied within these formulations and indicate the ties between these properties and the more practical aspects of parallel computation. We show how marked graphs, a particular type of Petri net, are a restricted type of computation graph and indicate how some results of marked graphs can be obtained from known results of computation graphs. Also, for schemata we discuss the decidability versus undecidability of various properties and several techniques of schemata composition.

Pitfalls in digital computation of the impulse response of vascular beds from indicator-dilution curves.

Abstract: Many methods are available for digital computation of the impulse response from indicator-dilution measurements representing input and output signals. In all instances, the only criterion for validity of the computation is comparison of the reconvolution of the computed impulse response and the input with the actual output. In this paper, a model mathematical system was constructed with a known impulse response; noise and time variation could be introduced independently or simultaneously in the input and the output data. Six methods for digital computation of the impulse response were applied to data from this system and to actual dye-dilution data. Precision of reconvolution did not assure that the computed response would resemble the actual response of the system. Some numerical considerations also significantly affected the digital computation of a valid response.

Berechnung und Programm. II

Abstract: Let A be an algebraic structure and assign to each operation of A a nonnegative real number as the performance time of the operation on a given computer. The notion of a computation (or straight line program) in A yields two functions from finite subsets of A to nonnegative real numbers, namely the computational length (or complexity), and the computational depth. We characterize these functions in a quasiaxiomatic way and prove a number of general results, which will be applied to concrete problems elsewhere (see [12]---[15]).

Dynamic programming and parallel computers

Abstract: The computational theory of dynamic programming is examined from the viewpoint of parallel computation. A discussion of various forms of parallelism, the corresponding parallel algorithms, the applicability of the algorithms to different types of optimization problems, and their advantages over serial computation is presented. In addition, parallel aspects of various dimensionality reduction techniques such as state increment dynamic programming, successive approximations, and shift vectors are also given.

Computation of bounds for digital filter quantization errors

Abstract: Spectral theory of operators is used to determine a general procedure for determining bounds on the difference between the states and outputs of a finite precision fixed-point digital filter and its infinite precision ideal counterpart. The results bound quantization errors for transients as well as limit cycles and apply when input signals are present. The procedure is extended to digital filters associated with difference equations, including the important special case of the basic second-order section.

Computer Implementation of the Finite-Element Procedure

Abstract: Publisher Summary This chapter aims at presenting fundamental concepts for the implementation of the finite-element method, with primary emphasis on static analysis based upon the displacement method. The correct interpretation of information feedback from the user's side in the course of the development of a large-scale system should have a significant effect upon its applicability. Thus the ideal developers of computer software for the application of finite element methods would be design engineers with several years of experience in the development of large computer systems and with some basic understanding of numerical analysis. The internal data are handled in the internal machine representation, thus excluding transferability between computers with different word length. All internal data handling can be concentrated in a separate data retrieval package with sufficient parametric generality to enable its adaptation to a large spectrum of computer configurations. The solution of the linear load-deflection equations is usually the most time-consuming computation step in the displacement method. As the finite-element method is ideally suited for analyzing highly complex structures, problems with several thousand unknowns occur frequently.

On a combined sturm sequence and inverse iteration technique for eigenproblem solution of spinning structures

Abstract: This paper describes an efficient numerical algorithm for the accurate computation of specific eigenvalues and related vectors of spinning structures. The Sturm sequence technique is first applied to isolate the required roots, which are then individually located by an inverse iteration technique adopting an efficient matrix formulation specially developed for this purpose. The associated digital computer procedure is numerically stable, which also fully exploits the banded form of relevant matrices. Furthermore, the algorithm proves to be much faster than other related existing procedures. A computer program, based on the current algorithm, has been developed in FORTRAN V for the JPL UNIVAC 1108 computer. Numerical results for representative structures, computed by the program, are also presented in detail. Extensive applications of the program are envisaged in the attitude control of spacecraft and in the natural frequency analysis of spinning structures, discretized by the finite element method.

Computer technique for solving 3-dimensional electron-optics and capacitance problems

Abstract: Electron-optics problems involving planar and axial symmetry are frequently analysed by using iterative procedures to solve Laplace's equation within a specified boundary. The computation time and storage requirement for these procedures may be prohibitive when it is necessary to extend the analysis to three dimensions for problems involving asymmetric fields. An alternative approach is described, known as the method of moments, which does not use an iterative method, but calculates the field directly from the charges induced on the electrodes by the excitation potentials. This technique provides a further facility, which permits the capacitances between arbitrary 3-dimensional electrodes to be obtained. The basic theory and operation of a computer program which employs this method are described. As a practical application, the program has been used to investigate the electron-optical properties of a mesh with rectangular apertures.

A note on base-2 logarithm computations

Abstract: The well-known methods of iteration used suitably can yield a more accurate and rapidly convergent computation for the values of many functions. One typical example is the base-2 logarithm-antilogarithm computation. Some recently proposed techniques are reviewed and compared with an iterative process for computation using a fixed number of iterations. It is shown that the latter yields improved accuracy and can be implemented either as a subroutine or as a hardwired peripheral device for, say, a minicomputer.

An Example of Information and Computation Resource Trade-Off

Abstract: This paper is concerned with establishing the existence of a recursive sequence x which exhibits a "continuous" trade-off between the length of programs (information) which compute the finite initial segments of x and the computation resources (e.g. computation time) required by these programs for such computations.

Accuracy and speed of real and complex interpolation

Abstract: Because complex arithmetic takes only a little more time than real arithmetic on modern computers it is advisable to take a closer look at some of the known mathematical tools and compare the advantages in complex computations.

Note on Design of an Optimized Computation Scheme For Kinematic Vertical Motion Fields1,,2

Abstract: The design of an optimized computation of bias error accumulation in the kinematic w profiles seems scheme for kinematic vertical motion fields with an upper to have been practically overcome. The discussions air network is discussed with a scheme that incorporates include the general construction of the scheme in terms 14 degrees of optimization in the evaluation of divergence. of the sequential flow of the computation, the quality The provided range of optimization enables the computa- of the vertical motion field over the synoptic network tion scheme to suit wide varieties of data dispostion in through the depth of the atmosphere, and the utilization the observation network, and the well-known problem and application of the scheme under various circumstances. As an integral part of our observational study of at- mospheric energetics, the development of a scheme for the kinematic estimate of the vertical motion field is under way. Since it is based on no assumptions about the nature of the atmospheric circulation except the hydro- static relationship, the kinematic estimate of the vertical p velocity, w, has paramount importance in the objective energy diagnosis. In a preceding paper (Kung 1972), a new approach to the kinematic estimate of the vertical motion field with an upper air network was presented. The scheme has been designed under the presumption that the vertical profile of w should converge to a near-zero value at the top of the atmosphere without being so required when a sensitive balance between the analysis scheme and the data dis- position is achieved. Thus, a range of optimization is provided in a computation scheme to suit wide varieties of data disposition in the observation network. The pre- liminary computation with eight degrees of optimization in a scheme, as reported in the previous paper, has sup- ported this presumption, and our extensive computational studies subsequent to the paper also confirm the approach. Thus, it may be concluded that the well-known problem of bias error accumulation in the kinematic w profiles can be practically overcome, and meaningful fields of kinematic vertical p velocity can be obtained through the depth of the atmosphere with observed mind data over a

Efficient computation of large change multiparameter sensitivity

Abstract: In many applications, a network has to be solved repeatedly while certain element values are changed. The computation is reduced significantly if the network functions are generated in terms of these variable parameters explicitly. This paper presents a method of doing this which requires the solution of the network with at most m+1 excitations when there are m variable parameters.

On the computation of semiconductor device current characteristics by finite difference methods

Abstract: An analysis is presented of the error in numerical approximations to a system of elliptic equations describing the steady-state distribution of mobile carriers in a semiconductor device. Although this system has been extensively studied by finite difference methods, the accuracy of the numerical methods employed has not been previously established. Computation schemes are presented for which suitable error estimates are obtained, without assuming an unreasonably small mesh size. In addition, for the one-dimensional problem, the effect of the inexact solution of the discrete equations is estimated.

Improved computation of cubic natural splines with equi-spaced knots

Abstract: An improved algorithm is given for the computation of the coefficients of the interpolating polynomials for cubic natural splines with equi-spaced knots. By solving the continuity equation recursively, a gain in computation efficiency is obtained and the requirement of previous techniques for exact computation is eliminated. 1. Introduction. Considerable interest has developed over the last several years in the use of spline functions for interpolation, largely as a result of the discovery of their extremal properties (1). Cubic splines have become popular because they combine a fair degree of approximation (continuity of the function and its first two derivatives at the knots) with relative ease of determination of the spline parameters (only a second-order difference equation need be solved). Two recent papers have shown that the parameters may be even more simply determined for the case of natural splines defined at equi-spaced knots ((2), (3)). Unfortunately, the techniques given in both these papers suffer from the defect that they require exact computation. To retain any accuracy requires multiple precision computation once the number of data points exceeds the limiting value nL = (n8/log 4) - 2, where n8 is the number of significant decimal digits carried by the computer employed. In the present paper, an improved algorithm is presented which does not require exact computation and which also displays an improved efficiency in determining the spline parameters.