Showing papers in "Computers & Mathematics With Applications in 1980"
TL;DR: In this paper, two different boundary conditions for the spin are considered: vanishing spin and vanishing surface moment, and the equations of motion are reduced to dimensionless forms which include three dimensionless parameters, and integrated numerically by a Runge-Kutta method.
Abstract: Plane and axially symmetric flows of a micropolar fluid, in contact with an infinite plate, and tending to potential flow at infinity, with a stagnation point on the plate, are considered Two different boundary conditions for the spin are considered: (a), vanishing spin; and (b), vanishing surface moment The equations of motion are reduced to dimensionless forms which include three dimensionless parameters, and integrated numerically by a Runge—Kutta method Results are presented both in tabular and graphical form, and the effects of the values of the parameters on the flow are discussed
173 citations
TL;DR: In this paper, the problem of finding a point on the sphere S 2 = {x = (x, y, z)¦x2 + y2 + z2 = 1} which minimizes the weighted sum of the distances to N given destination points xj on S2 is studied.
Abstract: The problem of finding a point on the sphere S2 = {x = (x, y, z)¦x2 + y2 + z2 = 1} which minimizes the weighted sum of the distances to N given destination points xj on S2 is studied. Three different metrics are considered as distances between points on S2: (A), square of Euclidean distance; (B), Euclidean distance; (C), great circle distance. Non uniqueness of minimizers is demonstrated and some pathological cases are studied. An algorithm, analogous to the Weiszfeld algorithm for the classical unconstrained Weber problem is formulated, and its convergence properties are investigated. A necessary and sufficient condition for a destination point to be a local minimizer is derived. Finally, a modified form of Steffensen's acceleration is given and the results of numerical tests are presented. These results illustrate the predictions of the theory, and confirm the effectiveness of Steffensen's acceleration.
54 citations
TL;DR: A new fourth order method using quintic polynomials for the smooth approximation of the two point boundary value problems involving second order differential equations lacking the first derivative, which outperforms the well-known fourth order Noumerov's finite difference scheme.
Abstract: A new fourth order method using quintic polynomials is designed in this paper for the smooth approximation of the two point boundary value problems involving second order differential equations lacking the first derivative. The present method enables us to approximate the unknown function as well as its derivative at every point of the range of integration and thus it has obvious advantages over other discrete numerical methods. Our present method outperforms the well-known fourth order Noumerov's finite difference scheme. The convergence of the method is briefly outlined using matrix algebra and two numerical illustrations are provided to demonstrate the practical suitability of our approach.
46 citations
TL;DR: In this article, methods of order two and four are developed for the continuous approximation of the solution of a two-point boundary value problem associated with a certain fourth order linear differential equation via quintic and sextic spline functions.
Abstract: Methods of order two and four are developed for the continuous approximation of the solution of a two-point boundary value problem associated with a certain fourth order linear differential equation via quintic and sextic spline functions. Numerical results are summarized for some typical numerical examples and compared with some known finite difference methods of the same order.
39 citations
TL;DR: In this article, the authors compare the corrected inversion method, the log(n)/n-tail method and the record time method for the normal, exponential and gamma densities.
Abstract: Frequently the need arises for the computer generation of variates that are exactly distributed as Z = max(X1, …, Xn) where X1, …, Xn form a sequence of independent identically distributed random variables. For large n the individual generation of the Xi's is unfeasible, and the inversion-of-a-beta-variate is potentially inaccurate. In this paper, we discuss and compare the corrected inversion method, the log(n)/n-tail method and the record time method. The latter two methods have an average complexity 0(log(n)), are very accurate and do not require the inversion of a distribution function. The normal, exponential and gamma densities are treated in detail. The existence of fast and accurate inversion methods for the error function makes the corrected inversion method faster than the other ones for n large enough when the Xi's are normal random variables.
32 citations
TL;DR: In this article, the role of computer architecture on nonlinear filter realization is discussed. Butts et al. present examples of two and three dimensional phase demodulation problems are presented and detailed Monte Carlo performance analysis is presented for the two-dimensional system.
Abstract: Examples of two and three dimensional phase demodulation problems are presented. Computer realizations for the optimal nonlinear phase estimator are discussed in detail, with emphasis on parallel computer architectures. Implementation of the nonlinear filter on various computer architectures, including the CDC6600/7600, CDC STAR-100, Illiac IV, the CRAY-1, and the Floating Point Systems AP120B is reviewed. Detailed Monte Carlo performance analysis is presented for the two-dimensional system, while partial results are included for the three dimensional case. Implications concerning the role of computer architecture upon nonlinear filter realization are discussed. This article is a revision and update of the authors' technical report[1] “New Frontiers in Nonlinear Filtering”. Revisions are primarily involved with the inclusion of up-to-date results and general conclusions.
19 citations
TL;DR: In this article, Laguerre's method has been extended to determine all simple roots of a polynomial together with error bounds at each iteration step, and it has been shown that the convergence rate is of order four.
Abstract: In [1] Laguerre's method has been extended to determine all simple roots of a polynomial together with error bounds at each iteration step. In this paper we extend the previous algorithm to multiple complex roots and show that, also in this case, the convergence rate is of order four. Some numerical examples illustrate the theoretical results.
16 citations
TL;DR: In this paper, an alternating-direction finite element collocation procedure is presented for parabolic boundary value problems posed on rectangular regions, which offers a great savings in time as compared to standard finite element methods during the matrix formation stage of numerical process.
Abstract: An alternating-direction finite element collocation procedure is presented for parabolic boundary value problems posed on rectangular regions. Collocation offers a great savings in time as compared to standard finite element methods during the matrix formation stage of numerical process. With an alternating-direction procedure, multidimensional problems can be solved as a series of one-dimensional problems, which greatly reduces both the work and the storage requirements during the matrix solution phase of the numerical procedure.
15 citations
TL;DR: In this article, an efficient procedure for time-stepping Galerkin methods for approximating smooth solutions of quasilinear second-order hyperbolic equations is considered.
Abstract: Efficient procedures for time-stepping Galerkin methods for approximating smooth solutions of quasilinear second-order hyperbolic equations are considered. The techniques presented can be used to analyze approximation procedures for related second-order-in-time quasilinear partial differential equations which have applications including initial-boundary value problems for vibrations (possibly) with inertia, dynamics of rotating fluids, and nonlinear viscoelasticity. The procedure involves the use of a pre-conditioned iterative method for approximately solving the different linear systems of equations arising at each time step in a discrete-time Galerkin method. Optimal order L 2 spatial errors and almost optimal order work estimates are obtained for the second-order hyperbolic case.
15 citations
TL;DR: In this article, an algorithm which combines the accuracy of linear interpolation and the speed of nearest neighbour interpolation is developed and applied to computed axial tomography by way of illustration.
Abstract: Simple linear interpolation requires numerical multiplication which is time-consuming when a large number of interpolated data points is required. In practice, nearest neighbour interpolation is often employed, even though it is appreciably less accurate. An algorithm which combines the accuracy of linear interpolation and the speed of nearest neighbour interpolation is developed and applied to computed axial tomography by way of illustration.
13 citations
TL;DR: In this article, numerical simulations of the statistics of the amplitude of the response of a lightly damped Van der Pol oscillator under a Gaussian white excitation are presented, and the computation cost of the numerical simulations is considered.
Abstract: Numerical simulations of the statistics of the amplitude of the response of a lightly damped Van der Pol oscillator under a Gaussian white excitation are presented. Computational aspects of several parameters of the numerical model of the Gaussian white process are discussed. The numerical data obtained are used to assess the reliability of an approximate solution for the stationary probability density function of the response amplitude. The computation cost of the numerical simulations is considered.
TL;DR: Finite element methods are introduced for the approximate solution of periodic acoustic problems using a least squares technique and a Galerkin/multigrid technique for second order equations.
Abstract: Finite element methods are introduced for the approximate solution of periodic acoustic problems. A least squares technique is used for those problems which are governed by a first order system of partial differential equations while for second order equations, a Galerkin/multigrid technique is employed. In both cases, the solution process for the algebraic system resulting from discretization is iterative in character.
TL;DR: The partial differential equation ∂u ∂t + ∂ ∂x f(u), where f is a nonnegative diffusion coefficient which may vanish for one or more values of u, has been used to model fluid flow through a porous medium Error estimates for a numerical procedure to approximate the solution of (1) will be derived as mentioned in this paper.
Abstract: The partial differential equation ∂u ∂t + ∂ ∂x f(u)= ∂ ∂x k(u) ∂u ∂x , where k(u) is a non-negative diffusion coefficient which may vanish for one or more values of u, has been used to model fluid flow through a porous medium Error estimates for a numerical procedure to approximate the solution of (1) will be derived
TL;DR: In this paper, it was shown that direct linear multistep methods for Volterra integral equations of the second kind with repetition factor equal to one are always stable and that this result is not true for first kind equations.
Abstract: In this paper we prove that direct linear multistep methods for Volterra integral equations of the second kind with repetition factor equal to one are always stable We show trivially that this result is not true for first kind equations We also demonstrate constructively that direct linear multistep methods for both first and second kind Volterra integral equations can have repetition factors greater than one, and indeed of arbitrary high order, and be numerically stable Finally we explain why the first form of Simpson's rule for second kind equations is stable while the second form is unstable
TL;DR: In this paper, the authors derived the probability generating function for the general Bellman-Harris age dependent branching process with immigration emphasizing the role of immigration parameters and applied it to three particular examples.
Abstract: We derive the probability generating function for the general Bellman—Harris age dependent branching process with immigration emphasizing the role of immigration parameters. The solution requires solving a single scalar-valued integral equation, namely the usual Bellman—Harris equation. Our results are applied to three particular examples. We derive the equations for the immigration of particles governed by a Bellman—Harris process into a second Bellman—Harris process. In the second example we study Poisson immigration of particles into a time continuous Markov branching process. In the third, we derive the equations for a one-time immigration into a Bellman—Harris process.
TL;DR: A class of variable penalty methods for solving the general nonlinear programming problems with mechanisms to control the quality of the approximation for the Hessian matrix, proposed in terms of the constraint functions and their first derivatives.
Abstract: This paper describes a class of variable penalty methods for solving the general nonlinear programming problems. The algorithm poses a sequence of unconstrained optimization problems with mechanisms to control the quality of the approximation for the Hessian matrix. The Hessian matrix is proposed in terms of the constraint functions and their first derivatives. The unconstrained problems are solved using a modified Newton's algorithm. The convergence of the method is accelerated by choosing variable penalty function parameters which in a given constraint environment, during an unconstrained minimization process, best control the error in the approximation of the Hessian matrix. Several possibilities for obtaining such parameters are discussed. The numerical effectiveness of this algorithm is demonstrated on a relatively large set of test problems.
TL;DR: A type-independent finite difference method to solve the transonic flow problem using FORTRAN and type- Independent Finite Difference Analysis.
Abstract: A type-independent finite difference method is given. An application is made to a transonic flow problem. A FORTRAN code listing is included.
TL;DR: In this article, iterative schemes are devised for solving the nonlinear ion-sound wave equation and the improved modified Boussinesq equation separately by using the combined approach of linearization and the finite difference method.
Abstract: Iterative schemes are devised for solving the nonlinear ion-sound wave equation and the improved modified Boussinesq equation separately by using the combined approach of linearization and the finite difference method. After analyzing the schemes for the stability and the accuracy, the dynamics of waves having various initial wave packets is discussed. For both the models, it is found that (i) the solitary waves with amplitudes below a critical value interact elastically with each other, (ii) the solitary waves with amplitudes above the critical value interact inelastically with each other and (iii) the coefficient of inelasticity increases with the increase of the amplitudes of the interacting solitary waves. These results are in conformity with the available results.
TL;DR: In this article, the authors define the class of interpolatory Newton iterations for the computation of a simple zero of a non-linear operator in a Banach space of finite or infinite dimension.
Abstract: The class of interpolatory—Newton iterations is defined and analyzed for the computation of a simple zero of a non-linear operator in a Banach space of finite or infinite dimension. Convergence of the class is established. The concepts of “informationally optimal class of algorithms” and “optimal algorithm” are formalized. For the multivariate case, the optimality of Newton iteration is established in the class of one-point iterations under an “equal cost assumption”.
TL;DR: In this article, the steady flow of a micropolar fluid between two infinite discs, when one is held at rest and the other rotating with constant angular velocity, is considered, and the equations of motion are reduced to a system of ordinary differential equations, which in turn are solved numerically using the Gauss-Seidel iterative procedure and Simpson's rule.
Abstract: The steady flow of a micropolar fluid between two infinite discs, when one is held at rest and the other rotating with constant angular velocity, is considered. The equations of motion are reduced to a system of ordinary differential equations, which in turn are solved numerically using the Gauss—Seidel iterative procedure and Simpson's rule, for four different combinations of the seven parameters involved. Results are given both in tabular and graphical form, and compared with the known theoretical results.
TL;DR: In this paper, the displacement problem of elastostatics in two dimensions is formulated in terms of integral equations via the Airy stress function, and the integral equations are solved numerically using piecewise constant approximations to the unknown functions.
Abstract: The displacement problem of elastostatics in two dimensions is formulated in terms of integral equations via the Airy stress function. The integral equations are solved numerically using piecewise constant approximations to the unknown functions. The validity of the formulation is demonstrated by its application to a simple problem with a known solution.
TL;DR: It is indicated how the results on optimal control can be applied to multiprogrammed computer systems and some numerical examples are given.
Abstract: The paper deals with the derivation of optimal control rules for finite source queueing systems with preemptive resume service discipline. The three performance measures considered are: server utilization, mean queue length and throughput. It is indicated how the results on optimal control can be applied to multiprogrammed computer systems and some numerical examples are given.
TL;DR: In this paper, the authors proposed a method for analyzing linear ordinary differential eigensystems in which a single initial value problem is used, compared with the usual methods that require more than one initial value problems the linear independence of whose solutions must be maintained.
Abstract: A recently proposed computational method for analyzing linear ordinary differential eigensystems in which a single initial value problem is used is compared with the usual methods that require more than one initial value problem the linear independence of whose solutions must be maintained. Since the new method requires only one initial value problem difficulties with parasitic error are avoided and eigenvalues for stiff systems can be obtained using standard integration schemes without resorting to techniques such as orthonormalized integration. Examples consist of a test problem with adjustable stiffness and the Orr-Sommerfeld equation for plane Poiseuille flow.
TL;DR: Program listings are given in the REDUCE 2 Algebraic Programming system that implement the following operations of the exterior calculus: exterior multiplication, exterior differentiation, inner multiplication, Lie differentiation, and Lie multiplication that are used in the computation of isovectors of ideals of exterior forms.
Abstract: Program listings are given in the REDUCE 2 Algebraic Programming system that implement the following operations of the exterior calculus: exterior multiplication, exterior differentiation, inner multiplication, Lie differentiation, and Lie multiplication. These programs realize exterior forms and vector fields by resolving them on unevaluated systems of operators with the necessary algebraic properties. Use of these programs in the computation of isovectors of ideals of exterior forms is illustrated for the case where the ideal is generated by the system of exterior forms of degree two that characterize the equations for shallow water waves.
TL;DR: Implementation of multi-grid algorithm can result in higher accuracy and efficiency than methods with the fixed-grid size and fixed-order of approximation, and specific tests of this are considered in the present study.
Abstract: The multi-grid method which employs a sequence of nested grids in the solution process is a general numerical technique for solving continuous problems. Implementation of multi-grid algorithm can result in higher accuracy and efficiency than methods with the fixed-grid size and fixed-order of approximation, and specific tests of this are considered in the present study. An introduction to the implementation of the multi-grid method in the solution of time dependent partial differential equation is presented. Multi-grid solutions for the advection equation are compared with published results using the usual fixed-grid method.
TL;DR: In this article, an asymptotic solution of initial problem for disturbances of laminar flow obtained in [8] is employed for analysis of autooscillations, and it is shown that there are two autoscillating regimes at values of the wave number α with the range 0.9649 ⌽ α ⌈ 1.0765 in subcritical region.
Abstract: In the present paper an asymptotic solution of initial problem for disturbances of laminar flow obtained in [8] is employed for analysis of autooscillations.
Numerical calculation has shown that there are two autooscillating regimes at values of the wave number α with the range 0.9649 ⩽ α ⩽ 1.0765 in subcritical region. The first regime is a well-known unstable limit cycle, and the second one is a stable autooscillating regime coming from the beyond-critical region. Cycles interflow at some critical value of the supercriticity parameter δ1. The nonlinear critical value of Reynolds number RH(α) corresponds to this value.
All disturbances are quenching at values of α, R lying to the left of nonlinear neutral curve. There exist critical amplitudes A1(α, R) in the region between nonlinear and linear neutral curves. Increasing disturbances are stabilized at amplitude values equal to A2(α, R). Within the linear neutral curve all disturbances increase up to the amplitude A2(α, R).
TL;DR: The complexity index of ϕ* is shown to be close to the lower bound on the minimal complexity index for a system of N nonlinear equations F(x)=0.
Abstract: We study the minimal complexity index of one-point iterations without memory for the solution of a system of N nonlinear equations F(x)=0. We present an iteration ϕ* with maximal order of convergence and with linear combinatory complexity. We show the complexity index of ϕ* is close to the lower bound on the minimal complexity index.
TL;DR: In this article, a generalization of Newton's method can be applied to improve, under certain conditions, such approximation by the recursive algorithm y i +1 = y i − T ′ −1 ( y ǫ) Ty ( i = 0,1,2,…).
Abstract: Let us consider a system of ODE's of the form F ( x , y , y ′, y ″) = 0 where y and F are vector functions. By introducing an operator T such that Tu = F ( x , u , u ′, u ″″) we have Ty = θ . Assuming that y ° is an approximation of the solution y ( x ) a generalization of Newton's method can be applied to improve, under certain conditions, such approximation by the recursive algorithm y i +1 = y i − T ′ −1 ( y ′) Ty ( i = 0,1,2,…). In the present case we use such an approach in a numerical fashion as follows. After obtaining by any method of integration numerical approximations y n on a discrete set of points x n ( n = 1,2,…, N ) we interpolate them by a convenient function R ( x ). By taking this interpolant as the first analytical approximation Newton's process is applied pointwise in order to correct by iterations the discrete approximations y n . This procedure may become rapidly convergent especially in some stiff problems where we have obtained so far promissing results.
TL;DR: In this article, a change of variables and appealing to rational approximation of the cosine matrix was introduced, and two-step approximating schemes of arbitrarily high-order accuracy for the time discretization of homogeneous damped second-order systems of the form M U (t)+2C U(t)+KU(t)=0, where M, C, K are symmetric, M and K are positive definite, and the damping is proportional.
Abstract: By introducing a change of variables and appealing to rational approximation of the cosine matrix, we develop two-step approximating schemes of arbitrarily high-order accuracy for the time discretization of homogeneous damped second-order systems of the form M U (t)+2C U (t)+KU(t)=0 , where M, C, K are symmetric, M and K are positive definite, and the damping is proportional, i.e. C=M ∑ j=0 p-1 a j (M -1 K) j . We perform stability and convergence analysis for the resulting schemes. We exhibit particularly an unconditionally stable fourth-order scheme and discuss its computational implementation. We also present an analogous method for the nonhomogeneous problem.
TL;DR: Attention is focused on the computational experience gained from implementing and using Yohe's interval analysis package with Crary's AUGMENT preprocessor on Control Data 6000, 7000, and Cyber series machines.
Abstract: Computer solutions of scientific and engineering problems involve several sources of floating-point errors which interact with each other and propagate throughout the calculations. One technique for monitoring and bounding the total accumulated error is known as interval analysis and it is the use of this approach via a software package with Fortran preprocessor which is the subject of this paper. Attention is focused on the computational experience gained from implementing and using Yohe's interval analysis package with Crary's AUGMENT preprocessor on Control Data 6000, 7000, and Cyber series machines.