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"Certain Integration Formulae Useful in Numerical Computation" is an article from The American Mathematical Monthly, Volume 19 . View more articles from The American Mathematical Monthly . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2972443

The reduction of circuit equations to normal form for numerical integration is considered for the general circuit where excess reactive elements, all types of linearly dependent sources, and nonlinear dissipative and reactive element are present. For the linear circuit, necessary and sufficient conditions for the existence of numerical solutions are considered and stated. For the nonlinear circuit, reduction to normal form is not always possible. Numerical solution is shown to be simplified if certain a priori conditions are satisfied in formulating the original circuit equations. A new systematic reduction procedure is presented for obtaining the normal form equations. This procedure is also extended to a new procedure for obtaining transfer and immittence functions of the linear circuit from a proper tree formulation.

viii, pages 157-372

viii, pages 157-372

This work unit is in the third year of a three year coupled rapid transition plan 6.2-6.4 development effort to transition aerosol prediction and EO tactical decision aid capability to the Fleet. This work is carried out jointly by NRL-Monterey and SPAWAR-SSC. The program will improve the components of the Target Acquisition Weather Software (TAWS), and link operational EO tactical decision aids (TDA's) to numerical weather prediction models. The NRL Aerosol Analysis and Predictions System (NAAPS) will be transitioned to the Fleet Numerical and Oceanography Center (FNMOC) at Monterey, CA for operational implementation. Specific elements of this 6.2-6.4 program (and the responsible personnel) include: (a) Develop and validate operational, diagnostic, and prognostic versions of the Navy Aerosol Analysis and Prediction System (NAAPS) for analysis of airborne dust loads (Westphal/NRL); (b) Modify existing radiative transfer codes to ingest NAAPS forecasts and produce practical products that depict the state of the atmosphere in terms of visibility and range (Reid/NRL; Tsay/GSFC; Westphal/NRL); (c) Improve the current marine target radiance models by developing new surface wake models and adding improved ship target models that include wake production (Doss-Hammel/SSC); and (d) Develop and improve operation of TAWS in maritime scenarios by utilizing numerical weather analyses and forecasts, and surface and satellite observations (Goroch/NRL).

A new scheme for numerical integration of motion for classical systems composed of rigid polyatomic molecules is proposed. The scheme is based on a matrix representation of the rotational degrees of freedom. The equations of motion are integrated within the Verlet framework in velocity form. It is shown that, contrary to previous methods, in the approach introduced the rigidity of molecules can be conserved automatically without any additional transformations. A comparison of various techniques with respect to numerical stability is made.

An indexed-based object recognition system using geometric invariance techniques has been designed, and used to recognize buildings in an image of a military site and for recognizing curved planar objects including gasless. New invariants and indexing techniques for polyhedral and curved objects with repetition or bilateral symmetry and objects with the imaged outline of a surface of revolution have been developed. A method to distinguish projectively equivalent but Euclidean distinct objects in an uncalibrated view has been investigated. A group-theoretic framework for relating quasi-invariants to invariants has been formulated. Computing invariants can be formulated as an algebraic manipulation problem involving variable elimination and solving nonlinear polynomial equations. Based on Dixon's formulation of resultants, new methods for eliminating variables have been developed and implemented. These methods are much faster and superior than other elimination techniques. A branch and prune approach for numerically solving polynomial equations has been developed. A simple algorithm for separating invariant relations among object and image features to compute invariants of object features has been designed. These algorithms can serve as a basis for building an invariant work-bench that would enable researchers to experiment with geometric configurations and investigate their geometric invariants.

In the calculation of cross sections for infrared-safe observables in high energy collisions at next-to-leading order, one approach is to perform all of the integrations, including the virtual loop integration numerically. One would use a subtraction scheme that removes infrared and collinear divergences from the integrand in a style similar to that used for real emission graphs. Then one would perform the loop integration by Monte Carlo integration along with the integrations over final state momenta. In this paper, we have explored how one can perform the numerical integration. We have studied the N-photon scattering amplitude with a massless electron loop in order to have a case with a singular integrand that is not, however, so singular as to require the subtractions. We report results for N = 4, N = 5 with left-handed couplings, and N=6.

Power series expansions (with coefficients obtained by recurrence formulas) are more efficient than other integration procedures for computing concurrently an orbit and the resolvent matrix of its variational equations, in the Restricted Problem of Three Bodies. For the same requirements on accuracy, the series expansions use only about 30 per cent of the computing time of the multi-step procedures, and only 12 to 15 per cent of the computing time of the Runge-Kutta-Nystrom method.

A set of nonlinear and coupled equations governing the thermomechanical deformations of a viscoplastic body undergoing simple shearing deformations is integrated in time by using the Forward-Difference-Galerkin-Finite-Element (FDGFE) method and the Crank-Nicolson-Galerkin-Finite-Element (CNGFE) method. In the latter scheme the number of unknown functions is increased so that the governing equations involve only first order spatial derivatives. It is shown that the solutions obtained by the two methods agree qualitatively but the CNGFE method seems to introduce considerable damping into the system. However, the time increment needed to obtain a stable solution by the CNGFE method is 200 times that required by the FDGFE method. Keywords: Shear bands, Finite element method, Crank nicolson method.

Stability of multistep methods in numerical integration

Two general purpose numerical integration schemes were built into the NASA-JSC computer system. The state-of-the-art of numerical integration, the particular integrators built into the JSC computer system, and the use of the new integration packages are described. Background information about numerical integration and the variable-order, variable-stepsize Adams numerical integration technique is discussed. Results concerning the PEACE parameter optimization program are given along with recommendations and conclusions.

A numerical algorithm suitable for rapid numerical solution of advanced turbulence-model equations on elliptic regions has been devised. The algorithm is particularly useful for obtaining accurate solutions when (a) one of the dependent variables is singular approaching a solid boundary, and (b) the number of mesh points available to resolve the flowfield is small. The new algorithm is used in conjunction with the MacCormack semi-implicit scheme for solving the time-averaged Navier-Stokes equations to obtain solutions for a laminar Mach 2 boundary layer and for a turbulent Mach 3 boundary layer. In both cases, numerical accuracy is shown to be excellent for meshes having less than 20 points normal to the surfaces. As a corollary result of this study we have shown that, using the new algorithm and the MacCormack scheme, time-averaged Navier-Stokes solutions are now feasible using a microcomputer. (Author)

We consider the geometric numerical integration of Hamiltonian systems subject to both equality and "hard" inequality constraints. As in the standard geometric integration setting, we target long-term structure preservation. We additionally, however, also consider invariant preservation over persistent, simultaneous and/or frequent boundary interactions. Appropriately formulating geometric methods to include such conditions has long-remained challenging due to the inherent nonsmoothness they impose. To resolve these issues we thus focus both on symplectic-momentum preserving behavior and the preservation of additional structures, unique to the inequality constrained setting. Leveraging discrete variational techniques, we construct a family of geometric numerical integration methods that not only obtain the usual desirable properties of momentum preservation, approximate energy conservation and equality constraint preservation, but also enforce multiple simultaneous inequality constraints, obtain smooth unilateral motion along constraint boundaries and allow for both nonsmooth and smooth boundary approach and exit trajectories. Numerical experiments are presented to illustrate the behavior of these methods on difficult test examples where both smooth and nonsmooth active constraint modes persist with high frequency.

We consider the geometric numerical integration of Hamiltonian systems subject to both equality and "hard" inequality constraints. As in the standard geometric integration setting, we target long-term structure preservation. We additionally, however, also consider invariant preservation over persistent, simultaneous and/or frequent boundary interactions. Appropriately formulating geometric methods to include such conditions has long-remained challenging due to the inherent nonsmoothness they impose. To resolve these issues we thus focus both on symplectic-momentum preserving behavior and the preservation of additional structures, unique to the inequality constrained setting. Leveraging discrete variational techniques, we construct a family of geometric numerical integration methods that not only obtain the usual desirable properties of momentum preservation, approximate energy conservation and equality constraint preservation, but also enforce multiple simultaneous inequality constraints, obtain smooth unilateral motion along constraint boundaries and allow for both nonsmooth and smooth boundary approach and exit trajectories. Numerical experiments are presented to illustrate the behavior of these methods on difficult test examples where both smooth and nonsmooth active constraint modes persist with high frequency.

Some results obtained with a digital computer program written at Goddard Space Flight Center to obtain electromagnetic fields scattered by perfectly reflecting surfaces are presented. For purposes of illustration a paraboloidal reflector was illuminated at radio frequencies in the simulation for both receiving and transmitting modes of operation. Fields were computed in the Fresnel and Fraunhofer regions. A dual-reflector system (Cassegrain) was also simulated for the transmitting case, and fields were computed in the Fraunhofer region. Appended results include derivations which show that the vector Kirchhoff-Kottler formulation has an equivalent form requiring only incident magnetic fields as a driving function. Satisfaction of the radiation conditions at infinity by the equivalent form is demonstrated by a conversion from Cartesian to spherical vector operators. A subsequent development presents the formulation by which Fresnel or Fraunhofer patterns are obtainable for dual-reflector systems. A discussion of the time-average Poynting vector is also appended.

Report describes techniques for the numerical integration of differential equations of various orders. Modified multistep predictor-corrector methods for general initial-value problems are discussed and new methods are introduced.

Predictor-corrector algorithm applicable to numerical integration of satellite orbits in multirevolution steps

The field method is developed for arbitrary open branch domains subjected to general linear boundary conditions. Although closed branches are within the scope of the method, they are not treated here. The numerical feasibility of the method has been demonstrated by implementing it in a computer program for the linear static analysis of open branch shells of revolution under asymmetric loads. For such problems the field method eliminates the well-known numerical problem of long subintervals associated with the rapid growth of extraneous solutions. Also, the method appears to execute significantly faster than other numerical integration methods.

Report discusses several methods of performing numerical integration with computer. When data can be expressed as state vector that is dependent variable in a differential equation, self-starting integrators can be used to predict future data.

For abstract, see N76-33552.

Cowell numerical integration and satellite orbit calculation

Algorithm for use in estimating accumulated numerical integration errors

The term 'trajectory problem' is taken to include problems that can arise, for instance, in connection with contour plotting, or in the application of continuation methods, or during phase-plane analysis. Geometrical techniques are used to construct difference methods for these problems to produce in turn explicit and implicit circularly exact formulae. Based on these formulae, a predictor-corrector method is derived which, when compared with a closely related standard method, shows improved performance. It is found that this latter method produces spurious limit cycles, and this behavior is partly analyzed. Finally, a simple variable-step algorithm is constructed and tested.

Due to the stringent design requirement for aerospace or nuclear structural components, considerable research interests have been generated on the development of constitutive models for representing the inelastic behavior of metals at elevated temperatures. In particular, a class of unified theories (or viscoplastic constitutive models) have been proposed to simulate material responses such as cyclic plasticity, rate sensitivity, creep deformations, strain hardening or softening, etc. This approach differs from the conventional creep and plasticity theory in that both the creep and plastic deformations are treated as unified time-dependent quantities. Although most of viscoplastic models give better material behavior representation, the associated constitutive differential equations have stiff regimes which present numerical difficulties in time-dependent analysis. In this connection, appropriate solution algorithm must be developed for viscoplastic analysis via finite element method.

Using a spherical finite-difference grid system covering the hemisphere, numerical solutions of the non-divergent barotropic vorticity equation have be n obtained for periods up to 10 days by rela ation methods. In the case of analytic initial conditions, the solutions are sensitive to abrupt changes of the longitudinal mesh size in the regions of appreciable amplitude of the streamfunction tendency. A doubling of the longitudinal mesh size from 5 to 10 deg at 45 deg N produces a progressive tearing or shearing in the stream function which is quite noticeable at 10 days. When the grid variation is removed to 70 degrees N, however, the solutions at 10 day are free of this defect, although exhibiting the expected phase lag relative to the analytic solution. When applied to hemispherical barotropic prediction with actual 500 mb data, this spherical grid scheme appears to perform as satisfactorily as do conventional (rectangular) grid formulations.

A workshop on the integration of symbolic and numerical computing methods was held on July 9 to 11, 1990 in Saratoga Springs, New York. The workshop was supported by a grant from the Air Force Office of Scientific Research and the National Science Foundation along with partial funding from G. E. Corporate Research and Development, and the State University of New York at Albany. The workshop was hosted by the Institute of Programming and Logics, the State University of New York at Albany. Twenty five research papers on symbolic methods, numerical methods, interface between symbolic and numerical methods, applications of symbolics methods in machine vision, robotics, computer aided design, computational geometry, and related topics were presented. Over forty researchers and students participated in the workshop.

Multirevolution predictor-corrector algorithm applicability to numerical integration of orbits

The efficiency and accuracy of several algorithms recently developed for the efficient numerical integration of stiff ordinary differential equations are compared. The methods examined include two general-purpose codes, EPISODE and LSODE, and three codes (CHEMEQ, CREK1D, and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes are applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code currently available for the integration of combustion kinetic rate equations. An important finding is that an interactive solution of the algebraic energy conservation equation to compute the temperature does not result in significant errors. In addition, this method is more efficient than evaluating the temperature by integrating its time derivative. Significant reductions in computational work are realized by updating the rate constants (k = at(supra N) N exp(-E/RT) only when the temperature change exceeds an amount delta T that is problem dependent. An approximate expression for the automatic evaluation of delta T is derived and is shown to result in increased efficiency.

Operators of numerical integration of functions are constructed. Also considered is the question of estimating the total error. The operators are applied to the solution of a problem with initial conditions for ordinary linear differential equations with constant coefficients. It was determined that the process of finding the solution of a differential equation of the type being examined by means of numerical integration operators reduces to arithmetic operations with numerical matrices and, consequently, is highly suitable for computer realization. An example is given, illustrating the advantages of the proposed method.

Many engineering studies require the evaluation of a Hilbert Transform of a tabular function. The function is also considered as an even function in order to conform to physical reality. A numerical procedure, incorporated into a Fortran subroutine HTRAN, has been developed to accomplish this task. The procedure has been designed to avoid the theoretical and numerical singularities and employs cubic polynomial interpolation to enhance numerical accuracy.

Many engineering studies require the evaluation of a Hilbert Transform of a tabular function. The function is also considered as an even function in order to conform to physical reality. A numerical procedure, incorporated into a Fortran subroutine HTRAN, has been developed to accomplish this task. The procedure has been designed to avoid the theoretical and numerical singularities and employs cubic polynomial interpolation to enhance numerical accuracy.

New Lie series method for numerical integration of ordinary differential equations, and restricted three-body problem application

Numerical integration orbits and Brouwer and modified Brouwer orbits

Numerical integration orbits and Brouwer and modified Brouwer orbits

The Air Force Institute of Techology (AFIT) is developing a simulation framework to model a wide variety of existing and proposed Quantum Key Distribution (QKD) systems. This research investigates using graphical processing unit (GPU) technology to more efficiently integrate optical pulses modeled within this framework. The goal is to reduce the simulation execution time. A GPU algorithm is presented for performing numerical integration of optical pulses described by Gaussian curves to improve pulse energy and power calculations. In order to measure the performance of the algorithm a optimal timing method is needed. A timer using Comute Unified Device Architecture (CUDA) events is selected over a Windows system application programming interface (API) timer. The problem sizes studied produce speedups greater than 60x on the NVIDIA Tesla C2075 compared to the Intel i7-3610QM CPU.

We present a comparison between finite differences schemes and a pseudospectral method applied to the numerical integration of stochastic partial differential equations that model surface growth. We have studied, in 1+1 dimensions, the Kardar, Parisi and Zhang model (KPZ) and the Lai, Das Sarma and Villain model (LDV). The pseudospectral method appears to be the most stable for a given time step for both models. This means that the time up to which we can follow the temporal evolution of a given system is larger for the pseudospectral method. Moreover, for the KPZ model, a pseudospectral scheme gives results closer to the predictions of the continuum model than those obtained through finite difference methods. On the other hand, some numerical instabilities appearing with finite difference methods for the LDV model are absent when a pseudospectral integration is performed. These numerical instabilities give rise to an approximate multiscaling observed in the numerical simulations. With the pseudospectral approach no multiscaling is seen in agreement with the continuum model.

In this report we have analyzed the errors introduced by table-look-up when an electromagnetic diffraction integral is performed numerically. We have shown that, in order to keep the error of both the real and imaginary components to be less than epsilon, the required table should have at least K entries uniformily distributed within a 2 pi range K or = 3.85/epsilon square root of N and N is the number of terms used for numerical integration. A numerical example is presented that confirms this prediction. (Author)

This report discusses the computer simulation of the ATACMS fin actuators. These subroutines are one of many possible methods of representing actuator behavior in a 6DOF (six degree of freedom) or 5DOF missile simulation. Also included is a section to discuss the accuracy of a version of the Runge- Kutta integration method. This particular simulation was run on a Zenith personal computer with Ryan-McFarland FORTRAN.

The venerable 2D point-vortex model plays an important role as a simplified version of many disparate physical systems, including superfluids, Bose-Einstein condensates, certain plasma configurations, and inviscid turbulence. This system is also a veritable mathematical playground, touching upon many different disciplines from topology to dynamic systems theory. Point-vortex dynamics are described by a relatively simple system of nonlinear ODEs which can easily be integrated numerically using an appropriate adaptive time stepping method. As the separation between a pair of vortices relative to all other inter-vortex length scales decreases, however, the computational time required diverges. Accuracy is usually the most discouraging casualty when trying to account for such vortex motion, though the varying energy of this ostensibly Hamiltonian system is a potentially more serious problem. We solve these problems by a series of coordinate transformations: We first transform to action-angle coordinates, which, to lowest order, treat the close pair as a single vortex amongst all others with an internal degree of freedom. We next, and most importantly, apply Lie transform perturbation theory to remove the higher-order correction terms in succession. The overall transformation drastically increases the numerical efficiency and ensures that the total energy remains constant to high accuracy.

We prove the curse of dimensionality in the worst case setting for numerical integration for a number of classes of smooth $d$-variate functions. Roughly speaking, we consider different bounds for the derivatives of $f \in C^k(D_d)$ and ask whether the curse of dimensionality holds for the respective classes of functions. We always assume that $D_d \subset \R^d$ has volume one and consider various values of $k$ including the case $k=\infty$ which corresponds to infinitely many differentiable functions. We obtain necessary and sufficient conditions, and in some cases a full characterization for the curse of dimensionality. For infinitely many differentiable functions we prove the curse if the bounds on the successive derivatives are appropriately large. The proof technique is based on a volume estimate of a neighborhood of the convex hull of $n$ points which decays exponentially fast if $n$ is small relative to $d$. For $k=\infty$, we also also study conditions for quasi-polynomial, weak and uniform weak tractability.

A quasi-one dimensional, nonequlibrium, streamtube model and finite difference computer code has been developed to study nonisentropic supercritical, real gas flows. A system of unsteady Euler equations is integrated to a steady state solution utilizing MacCormack's explicit/implicit numerical scheme. Implcit characteristic boundary conditions are employed at the subsonic inlet to enable aut omatic solution of inlet conditons consistent with the critical mass flow rate. A numerical damping technique for the explicity evaluated reaction rate has been developed such that nonequilibrium flows with large reaction rate coefficients have been solved numerically with significant reduction in computation time.

AD743748

We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE can be decomposed into several pieces and each of them is integrable. This class of integrators are explicit, simple to implement and preserve structural properties of the system. In consequence, they are specially useful in geometric numerical integration. In addition, the numerical solution obtained by splitting schemes can be seen as the exact solution to a perturbed system of ODEs possessing the same geometric properties as the original system. This backward error interpretation has direct implications for the qualitative behavior of the numerical solution as well as for the error propagation along time. Closely connected with splitting integrators are composition methods. We analyze the order conditions required by a method to achieve a given order and summarize the different families of schemes one can find in the literature. Finally, we illustrate the main features of splitting and composition methods on several numerical examples arising from applications.

Three schemes, whose expressions are not too complex, are selected for the numerical integration of a system of stochastic differential equations in the Stratonovich interpretation: the integration methods of Heun, Milstein, and derivative-free Milstein. The strong (path-wise) convergence is studied for each method by comparing the final points after integrating with $2^n$ and $2^{n-1}$ time steps. We also compare the time that the computer takes to carry out the integration with each scheme. Putting both things together, we conclude that, at least for our system, the Heun method is by far the best performing one.

We study numerically classical 1-dimensional Hamiltonian lattices involving inter-particle long range interactions that decay with distance like 1/r^alpha, for alpha>=0. We demonstrate that although such systems are generally characterized by strong chaos, they exhibit an unexpectedly organized behavior when the exponent alpha

The rational Landen transformations are used to produce a highly efficient numerical method for the integration of rational functions.

A mathematical model developed for digital gunfire control using the NSWC/DL effective drag functions was modified. the NSWC/DL functions were replaced with fourth-order Rungs-Kutta numerical integration of particle trajectory equations for calculation of gun elevation angle, projectile deflection, and projectile time of flight. Projectile drag coefficient is obtained during the integration process by table look-up of prestored values. The sequence of computations is given for the present position, prediction, ballistics and gun order sections of the model. The current version of the modified model was designed not for use in a gunfire control system but as a tool for evaluation of gunfire control models.

A mathematical model developed for digital gunfire control using the NSWC/DL effective drag functions was modified. the NSWC/DL functions were replaced with fourth-order Rungs-Kutta numerical integration of particle trajectory equations for calculation of gun elevation angle, projectile deflection, and projectile time of flight. Projectile drag coefficient is obtained during the integration process by table look-up of prestored values. The sequence of computations is given for the present position, prediction, ballistics and gun order sections of the model. The current version of the modified model was designed not for use in a gunfire control system but as a tool for evaluation of gunfire control models.

A mathematical model developed for digital gunfire control using the NSWC/DL effective drag functions was modified. the NSWC/DL functions were replaced with fourth-order Rungs-Kutta numerical integration of particle trajectory equations for calculation of gun elevation angle, projectile deflection, and projectile time of flight. Projectile drag coefficient is obtained during the integration process by table look-up of prestored values. The sequence of computations is given for the present position, prediction, ballistics and gun order sections of the model. The current version of the modified model was designed not for use in a gunfire control system but as a tool for evaluation of gunfire control models.