Fouriera: Automated Spectral Methods for Multiphysics Problems via Symbolic Computing
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Investigations into the shape-preserving interpolants using symbolic computation
Shape representation is a central issue in computer graphics and computer-aided geometric design. Many physical phenomena involve curves and surfaces that are monotone (in some directions) or are convex. The corresponding representation problem is given some monotone or convex data, and a monotone or convex interpolant is found. Standard interpolants need not be monotone or convex even though they may match monotone or convex data. Most of the methods of investigation of this problem involve the utilization of quadratic splines or Hermite polynomials. In this investigation, a similar approach is adopted. These methods require derivative information at the given data points. The key to the problem is the selection of the derivative values to be assigned to the given data points. Schemes for choosing derivatives were examined. Along the way, fitting given data points by a conic section has also been investigated as part of the effort to study shape-preserving quadratic splines.
Lam, Maria↗
The symbolic computation and automatic analysis of trajectories
Research was generally done on computation of trajectories of dynamical systems, especially control systems. Algorithms were further developed for rewriting expressions involving differential operators. The differential operators involved arise in the local analysis of nonlinear control systems. An initial design was completed of the system architecture for software to analyze nonlinear control systems using data base computing.
Grossman, Robert↗
Applications of symbolic computation in fracture mechanics
A FORTRAN program for calculating the stresses of n cracks embedded in an isotropic plate is presented. Formulas are given for calculating the stresses of one crack, two cracks, and n cracks in an isotropic plate. Then the program code that accomplishes this is provided.
Tan, Hui-Qian↗
Symbolic Computation of Strongly Connected Components Using Saturation
Finding strongly connected components (SCCs) in the state-space of discrete-state models is a critical task in formal verification of LTL and fair CTL properties, but the potentially huge number of reachable states and SCCs constitutes a formidable challenge. This paper is concerned with computing the sets of states in SCCs or terminal SCCs of asynchronous systems. Because of its advantages in many applications, we employ saturation on two previously proposed approaches: the Xie-Beerel algorithm and transitive closure. First, saturation speeds up state-space exploration when computing each SCC in the Xie-Beerel algorithm. Then, our main contribution is a novel algorithm to compute the transitive closure using saturation. Experimental results indicate that our improved algorithms achieve a clear speedup over previous algorithms in some cases. With the help of the new transitive closure computation algorithm, up to 10(exp 150) SCCs can be explored within a few seconds.
Zhao, Yang↗
Proceedings of the 1968 Summer Institute on Symbolic Mathematical Computation
Investigating symbolic mathematical computation using PL/1 FORMAC batch system and Scope FORMAC interactive system
Tobey, R. G.↗
Mathematical modelling by symbolic mathematical computation - A cosmological application
Cosmology application of mathematical modeling by symbolic computation
Howard, J. C.↗
The analysis of control trajectories using symbolic and database computing
The research broadly concerned the symbolic computation, mixed numeric-symbolic computation, and data base computation of trajectories of dynamical systems, especially control systems. It was determined that trees can be used to compute symbolically series which approximate solutions to differential equations.
Grossman, Robert↗
Computers for symbolic processing
A detailed survey on the motivations, design, applications, current status, and limitations of computers designed for symbolic processing is provided. Symbolic processing computations are performed at the word, relation, or meaning levels, and the knowledge used in symbolic applications may be fuzzy, uncertain, indeterminate, and ill represented. Various techniques for knowledge representation and processing are discussed from both the designers' and users' points of view. The design and choice of a suitable language for symbolic processing and the mapping of applications into a software architecture are then considered. The process of refining the application requirements into hardware and software architectures is treated, and state-of-the-art sequential and parallel computers designed for symbolic processing are discussed.
Wah, Benjamin W.↗
The transformation of aerodynamic stability derivatives by symbolic mathematical computation
The formulation of mathematical models of aeronautical systems for simulation or other purposes, involves the transformation of aerodynamic stability derivatives. It is shown that these derivatives transform like the components of a second order tensor having one index of covariance and one index of contravariance. Moreover, due to the equivalence of covariant and contravariant transformations in orthogonal Cartesian systems of coordinates, the transformations can be treated as doubly covariant or doubly contravariant, if this simplifies the formulation. It is shown that the tensor properties of these derivatives can be used to facilitate their transformation by symbolic mathematical computation, and the use of digital computers equipped with formula manipulation compilers. When the tensor transformations are mechanised in the manner described, man-hours are saved and the errors to which human operators are prone can be avoided.
Howard, J. C.↗
HyperCLIPS
Integration of CLIPS computer program into HyperCard computer program combines intuitive, interactive user interface of MacIntosh computer with powerful symbolic computation of expert-system interpreter. HyperCard provides excellent programming environment for quickly developing front end of application program with buttons, dialogs, and pictures, while CLIPS interpreter provides powerful inference engine for analysis and solution of complicated problems. Advantages of both packages available for wide range of uses: rapid prototyping of knowledge-based expert-systems software, interactive simulations of physical systems, and intelligent control of hypertext processes. Written in C language and Pascal.
Hill, Randall W., Jr.↗
The explicit computation of integration algorithms and first integrals for ordinary differential equations with polynomials coefficients using trees
This note is concerned with the explicit symbolic computation of expressions involving differential operators and their actions on functions. The derivation of specialized numerical algorithms, the explicit symbolic computation of integrals of motion, and the explicit computation of normal forms for nonlinear systems all require such computations. More precisely, if R = k(x(sub 1),...,x(sub N)), where k = R or C, F denotes a differential operator with coefficients from R, and g member of R, we describe data structures and algorithms for efficiently computing g. The basic idea is to impose a multiplicative structure on the vector space with basis the set of finite rooted trees and whose nodes are labeled with the coefficients of the differential operators. Cancellations of two trees with r + 1 nodes translates into cancellation of O(N(exp r)) expressions involving the coefficient functions and their derivatives.
Crouch, P. E.↗
Time-domain finite elements in optimal control with application to launch-vehicle guidance
A time-domain finite element method is developed for optimal control problems. The theory derived is general enough to handle a large class of problems including optimal control problems that are continuous in the states and controls, problems with discontinuities in the states and/or system equations, problems with control inequality constraints, problems with state inequality constraints, or problems involving any combination of the above. The theory is developed in such a way that no numerical quadrature is necessary regardless of the degree of nonlinearity in the equations. Also, the same shape functions may be employed for every problem because all strong boundary conditions are transformed into natural or weak boundary conditions. In addition, the resulting nonlinear algebraic equations are very sparse. Use of sparse matrix solvers allows for the rapid and accurate solution of very difficult optimization problems. The formulation is applied to launch-vehicle trajectory optimization problems, and results show that real-time optimal guidance is realizable with this method. Finally, a general problem solving environment is created for solving a large class of optimal control problems. The algorithm uses both FORTRAN and a symbolic computation program to solve problems with a minimum of user interaction. The use of symbolic computation eliminates the need for user-written subroutines which greatly reduces the setup time for solving problems.
Bless, Robert R.↗
In-flight research applications of an analog computer and symbol generator to determine display and control requirements for VTOL instrument landings
This paper describes the design and application of an airborne analog computer and display generator. The system was developed as an aid to investigate the interaction of display system complexity and control system complexity on pilot rating and performance for VTOL aircraft instrument landings.
Beilman, J. L.↗
The AutoBayes Program Synthesis System: System Description
AUTOBAYES is a fully automatic program synthesis system for the statistical data analysis domain. Its input is a concise description of a data analysis problem in the form of a statistical model; its output is optimized and fully documented C/C++ code which can be linked dynamically into the Matlab and Octave environments. AUTOBAYES synthesizes code by a schema-guided deductive process. Schemas (i.e., code templates with associated semantic constraints) are applied to the original problem and recursively to emerging subproblems. AUTOBAYES complements this approach by symbolic computation to derive closed-form solutions whenever possible. In this paper, we concentrate on the interaction between the symbolic computations and the deductive synthesis process. A statistical model specifies for each problem variable (i.e., data or parameter) its properties and dependencies in the form of a probability distribution, A typical data analysis task is to estimate the best possible parameter values from the given observations or measurements. The following example models normal-distributed data but takes prior information (e.g., from previous experiments) on the data's mean value and variance into account.
Fischer, Bernd↗
The analysis of control trajectories using symbolic and database computing
This final report comprises the formal semi-annual status reports for this grant for the periods June 30-December 31, 1993, January 1-June 30, 1994, and June 1-December 31, 1994. The research supported by this grant is broadly concerned with the symbolic computation, mixed numeric-symbolic computation, and database computation of trajectories of dynamical systems, especially control systems. A review of work during the report period covers: trajectories and approximating series, the Cayley algebra of trees, actions of differential operators, geometrically stable integration algorithms, hybrid systems, trajectory stores, PTool, and other activities. A list of publications written during the report period is attached.
Grossman, Robert↗
Computer algebra and operators
The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.
Fateman, Richard↗
Design consideration in constructing high performance embedded Knowledge-Based Systems (KBS)
As the hardware trends for artificial intelligence (AI) involve more and more complexity, the process of optimizing the computer system design for a particular problem will also increase in complexity. Space applications of knowledge based systems (KBS) will often require an ability to perform both numerically intensive vector computations and real time symbolic computations. Although parallel machines can theoretically achieve the speeds necessary for most of these problems, if the application itself is not highly parallel, the machine's power cannot be utilized. A scheme is presented which will provide the computer systems engineer with a tool for analyzing machines with various configurations of array, symbolic, scaler, and multiprocessors. High speed networks and interconnections make customized, distributed, intelligent systems feasible for the application of AI in space. The method presented can be used to optimize such AI system configurations and to make comparisons between existing computer systems. It is an open question whether or not, for a given mission requirement, a suitable computer system design can be constructed for any amount of money.
Dalton, Shelly D.↗