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Showing papers in "ACM Transactions on Mathematical Software in 1987"


Journal ArticleDOI
TL;DR: A new global optimization algorithm for functions of continuous variables is presented, derived from the “Simulated Annealing” algorithm recently introduced in combinatorial optimization, which is quite costly in terms of function evaluations, but its cost can be predicted in advance, depending only slightly on the starting point.
Abstract: A new global optimization algorithm for functions of continuous variables is presented, derived from the “Simulated Annealing” algorithm recently introduced in combinatorial optimization.The algorithm is essentially an iterative random search procedure with adaptive moves along the coordinate directions. It permits uphill moves under the control of a probabilistic criterion, thus tending to avoid the first local minima encountered.The algorithm has been tested against the Nelder and Mead simplex method and against a version of Adaptive Random Search. The test functions were Rosenbrock valleys and multiminima functions in 2,4, and 10 dimensions.The new method proved to be more reliable than the others, being always able to find the optimum, or at least a point very close to it. It is quite costly in term of function evaluations, but its cost can be predicted in advance, depending only slightly on the starting point.

1,598 citations


Journal ArticleDOI
TL;DR: HOMPACK provides three qualitatively different algorithms for tracking the homotopy zero curve: ordinary differential equation-based, normal flow, and augmented Jacobian matrix.
Abstract: There are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points, i.e., with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map. HOMPACK provides three qualitatively different algorithms for tracking the homotopy zero curve: ordinary differential equation-based, normal flow, and augmented Jacobian matrix. Separate routines are also provided for dense and sparse Jacobian matrices. A high-level driver is included for the special case of polynomial systems.

393 citations


Journal ArticleDOI
TL;DR: The principal objective is the development of testing tools that can be used to assess the efficiency and reliability of a standard numerical method without requiring significant modifications to the method and without the tools themselves affecting the performance of the method.
Abstract: We present a discussion and description of a collection of FORTRAN routines designed to aid in the assessment of initial value methods for ordinary differential equations. Although the overall design characteristics are similar to those of earlier testing packages [2,6] that were used for the comparison of methods [5,7], the details and objectives of the current collection are quite different. Our principal objective is the development of testing tools that can be used to assess the efficiency and reliability of a standard numerical method without requiring significant modifications to the method and without the tools themselves affecting the performance of the method.

170 citations


Journal ArticleDOI
TL;DR: This paper systematically compares various forms of generalized bisection algorithms to themselves, to continuation methods, and to hybrid steepest descent/quasi-Newton methods.
Abstract: This paper addresses the task of reliably finding approximations to all solutions to a system of nonlinear equations within a region defined by bounds on each of the individual coordinates. Various forms of generalized bisection were proposed some time ago for this task. This paper systematically compares such generalized bisection algorithms to themselves, to continuation methods, and to hybrid steepest descent/quasi-Newton methods. A specific algorithm containing novel “expansion” and “exclusion” steps is fully described, and the effectiveness of these steps is evaluated. A test problem consisting of a small, high-degree polynomial system that is appropriate for generalized bisection, but very difficult for continuation methods, is presented. This problem forms part of a set of 17 test problems from published literature on the methods being compared; this test set is fully described here.

95 citations


Journal ArticleDOI
TL;DR: The mathematical theory is presented together with short algorithms for parametric interpolation to “tighten” the weighted v-spline on intervals and/or at the interpolation points.
Abstract: Various methods have been developed to control the shape of an interpolating curve for computer-aided design applications. Some methods are better suited for controlling the tension of the curve on an interval, while others are better suited for controlling the tension at the individual interpolation points. The weighted v-spline is a C1 piecewise cubic polynomial interpolant that generalizes C2 cubic splines, weighted splines, and v-splines. Shape controls are available to “tighten” the weighted v-spline on intervals and/or at the interpolation points. The mathematical theory is presented together with short algorithms for parametric interpolation.

87 citations


Journal ArticleDOI
TL;DR: This work examines several methods for drawing a sequential random sample of n records from a file containing N records and recommends method D, which is on-line, has a small constant memory requirement, and is easy to program.
Abstract: We examine several methods for drawing a sequential random sample of n records from a file containing N records. Method D is recommended for general use. The algorithm is on-line (so that CPU time can be overlapped with I/O), has a small constant memory requirement, and is easy to program. An improved implementation is detailed in the Appendix.

77 citations


Journal ArticleDOI
TL;DR: A specialization of the primal truncated Newton algorithm for solving nonlinear optimization problems on networks with gains, able to capitalize on the special structure of the constraints.
Abstract: We describe a specialization of the primal truncated Newton algorithm for solving nonlinear optimization problems on networks with gains. The algorithm and its implementation are able to capitalize on the special structure of the constraints. Extensive computational tests show that the algorithm is capable of solving very large problems. Testing of numerous tactical issues are described, including maximal basis, projected line search, and pivot strategies. Comparisons with NLPNET, a nonlinear network code, and MINOS, a general-purpose nonlinear programming code, are also included.

54 citations


Journal ArticleDOI
TL;DR: It is shown that the passed boxes tend to cluster in geometrical configura-tions whose number is stable under subdivision, which implies for many problems that the work required to do simple bisection may be prohibitive.
Abstract: Box-bisection is a method for solving nonlinear systems. Space is subdivided into boxes of smaller and smaller diameter, and each subbox is tested for the existence of solutions by a test that either eliminates it from further consideration or marks it for subdivision. Simple bisection uses a test for the exclusion of subboxes, but no test that guarantees the existence of a unique solution in a subbox. Using this simple bisection, we show that the passed boxes tend to cluster in geometrical configura- tions whose number is stable under subdivision. This implies for many problems that the work required to do simple bisection may be prohibitive. However, improvements may be possible by grouping clusters and dynamically redefining the box proportions. The restriction to second-degree systems is sufficient to display this behavior.

42 citations


Journal ArticleDOI
TL;DR: Fourth-order-accurate compact discretizations of the Helmholtz equation on rectangular domains in two and three dimensions with any combination of Dirichlet, Neumann, or periodic boundary conditions are presented.
Abstract: We present fourth-order-accurate compact discretizations of the Helmholtz equation on rectangular domains in two and three dimensions with any combination of Dirichlet, Neumann, or periodic boundary conditions. The resulting systems of linear algebraic equations have the same block-tridiagonal structure as traditional central differences and hence may be solved efficiently using the Fourier method. The performance of the method for a variety of test cases, including problems with nonsmooth solutions, is presented. The method is seen to be roughly twice as fast as deferred corrections and, in many cases, results in a smaller discretization error.

36 citations


Journal ArticleDOI
TL;DR: ForTRAN subroutines are presented that implement the method described in [3] for the stable evaluation of the weights of interpolatory quadratures with prescribed simple or multiple knots.
Abstract: We present FORTRAN subroutines that implement the method described in [3] for the stable evaluation of the weights of interpolatory quadratures with prescribed simple or multiple knots. Given a set of knots and their multiplicities, the package generates the weights by using the zeroth moment m0 of w, the weight function in the integrand, and the (symmetric tridiagonal) Jacobi matrix J associated with the polynomials orthogonal on (a, b) with respect to w. There are utility routines that generate m0 and J for classical weight functions, but quadratures can be generated for any m0 and J supplied by the user. Utility routines are also provided that (1) evaluate a computed quadrature, applied to a user-supplied integrand, (2) check the polynomial order of precision of a quadrature formula, and (3) compute the knots and weights of simple Gaussian quadrature formula.

34 citations


Journal ArticleDOI
TL;DR: A method is presented for the generation of test problems for global optimization algorithms that constructs nonconvex quadratic functions whose global minimum is attained at the selected vertex.
Abstract: A method is presented for the generation of test problems for global optimization algorithms. Given a bounded polyhedron in R and a vertex, the method constructs nonconvex quadratic functions (concave or indefinite) whose global minimum is attained at the selected vertex. The construction requires only the use of linear programming and linear systems of equations.

Journal ArticleDOI
TL;DR: This paper discusses several problems from the viewpoint of software development and user interface, for example, how to deliver the PDEs and BCs to FIDISOL and how to allow a flexible use by a suitable parameter list.
Abstract: FIDISOL is a program package for the solution of nonlinear systems of two-dimensional and three-dimensional elliptic and parabolic partial differential equations (PDEs) with nonlinear boundary conditions (BCs) on the boundaries of a rectangular domain. A finite difference method (FDM) with an arbitrary grid and arbitrary consistency order is used, these are either prescribed by the user or are self-adapted for a given relative tolerance. FIDISOL has been designed to be fully vectorizable on vector computers. In this paper we discuss several problems from the viewpoint of software development and user interface, for example, how to deliver the PDEs and BCs to FIDISOL and how to allow a flexible use by a suitable parameter list.

Journal ArticleDOI
TL;DR: Interactive ELLPACK is a versatile, very high-level language for solving elliptic partial differential equations that provides true interactive elliptic problem solving by allowing the user to interactively build grids, choose solution methods, and analyze computed results.
Abstract: ELLPACK is a versatile, very high-level language for solving elliptic partial differential equations. Solving elliptic problems with ELLPACK typically involves a process in which one repeatedly computes a solution, analyzes the results, and modifies the solution technique. Although this process is best suited for an interactive environment, ELLPACK itself is batch oriented. With this in mind, we have developed Interactive ELLPACK, an extension of ELLPACK that provides true interactive elliptic problem solving by allowing the user to interactively build grids, choose solution methods, and analyze computed results. Interactive ELLPACK features a sophisticated interface with window- ing, color graphics output, and graphics input.

Journal ArticleDOI
TL;DR: An algorithm is presented for generating random variables from the chi family of distributions withdegrees of freedom parameter LY 2 1 based on the ratio of uniforms method and can be usedeffectively for the gamma family.
Abstract: An algorithm is presented for generating random variables from the chi family of distributions with degrees of freedom parameter LY 2 1. It is based on the ratio of uniforms method and can be used effectively for the gamma family.

Journal ArticleDOI
TL;DR: This research presents a meta-analyses of the determinants of high-resolution 3D image recognition algorithms and their applications in the image recognition system, and some of the methods used to identify and characterize these systems are simple and straightforward.
Abstract: Authors’ address: U.S. Naval Surface Weapons Center, Dahlgren, VA 22448. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission.

Journal ArticleDOI
TL;DR: A threshold version of the partial pivoting strategy for sparse symmetric matrix decomposition is explored in the multifrontal method of Duff and Reid for sparse indefinite systems and is at least as effective as the existing pivot strategy used in the current multifrontal implementation.
Abstract: It is well known that the partial pivoting strategy by Bunch and Kaufman is very effective for factoring dense symmetric indefinite matrices using the diagonal pivoting method. In this paper, we study a threshold version of the strategy for sparse symmetric matrix decomposition. The use of this scheme is explored in the multifrontal method of Duff and Reid for sparse indefinite systems. Experimental results show that it is at least as effective as the existing pivoting strategy used in the current multifrontal implementation.

Journal ArticleDOI
TL;DR: A new pseudorandom number generator, the Multiple Prime Random Number Generator, has been developed; it is efficient, conceptually simple, flexible, and easy to program.
Abstract: A new pseudorandom number generator, the Multiple Prime Random Number Generator, has been developed; it is efficient, conceptually simple, flexible, and easy to program. The generator utilizes cycles around prime numbers to guarantee the length of the period, which can easily be programmed to surpass the maximum period of any other presently available random number generator. There are minimum limits placed on the seed values of the variables because the period of the generator is not a function of the initial values of the variables. The generator passes thirteen standard random number generator tests. It requires only about fifteen lines of FORTRAN code to program and utilizes programming language constructs found in most major languages. Finally, it compares very favorably to the fastest of the other available generators.

Journal ArticleDOI
TL;DR: Two square root algorithms (for integer and floating point data types) are presented, which are simpler and more efficient than standard procedures, and could be effectively used as the basis ofware-based square root generators as well as for software implementations.
Abstract: Two square root algorithms (for integer and floating point data types) are presented, which are simpler and more efficient than standard procedures. These could be effectively used as the basis of hardware-based square root generators as well as for software implementations. One possible appli- cation for an efficient square root routine would be in calculating trigonometric and exponential functions. (This application may be primarily of academic interest, however, since standard tran- scendental function generators would generally be more efficient.) Three accompanying MC68000 implementations of the algorithm for 32-bit integer and IEEE single- and double-precision data are available on the CALGO listing. These programs return rounding status in the condition code register, and they exhibit the following approximate runtime performance at 8 MHz: 105-134 ps (integer); 180-222 ps (single precision); 558-652 ps (double precision).

Journal ArticleDOI
TL;DR: The user interface to HFFT is described and an example of its usage and several details of its implementation are presented.
Abstract: HFFT is a software package for solving the Helmholtz equation on bounded two- and three-dimensional rectangular domains with Dirichlet, Neumann, or periodic boundary conditions. The software is the result of combining new fourth-order accurate compact finite difference (HODIE) discretizations and a fast-direct solution technique (the Fourier method). In this paper we briefly describe the user interface to HFFT and present an example of its usage and several details of its implementation.

Journal ArticleDOI
TL;DR: An improved version of this practical parallel algorithm for solving band symmetric positive definite systems of linear equations in O(m* log) time usingnm/log/log processors is given.
Abstract: We give a practical parallel algorithm for solving band symmetric positive definite systems of linear equations in O(m* log n) time using nm/log n processors. Here n denotes the system size and m its bandwidth. Hence, the algorithm is efficient. For tridiagonal systems, the algorithm runs in O(log n) time using n/log n processors. Furthermore, an improved version runs in O(log m log n) time using nm2/(log m log n) processors.

Journal ArticleDOI
TL;DR: A simple modification to the numerical pivot selection criteria in the multifrontal scheme of Duff and Reid for sparse symmetric matrix factorization is presented, which allows a broader choice of block 2 X 2 pivots owing to a less restrictive pivoting condition.
Abstract: A simple modification to the numerical pivot selection criteria in the multifrontal scheme of Duff and Reid for sparse symmetric matrix factorization is presented. For a given threshold value, the modification allows a broader choice of block 2 X 2 pivots owing to a less restrictive pivoting condition. It also extends the range of permissible threshold values from [0, 1/2) to [0, 0.6404). Moreover, the bound on element growth for stability consideration in the modified scheme is nearly the same as that of the original strategy.

Journal ArticleDOI
TL;DR: The independence of the mathematical operations from the particulars of the compiler was achieved by the judicious use of Assembly language macros, and it is now a relatively easy job to write these macros for a FORTRAN compiler that is not on the authors' list.
Abstract: The Basic Linear Algebra Subprograms (BLAS) are described in [l]. The particular implementation documented here is intended for any of the FORTRAN compilers, [2-41, that run on MS-DOS and PC-DOS operating systems. Source code is provided for an Assembly language implementation of these subprograms, which are designed so that the computation is independent of the interface with the calling program unit. In fact, each of the compilers have different methods of passing pointers to input argument lists and returning results for functions. The independence of the mathematical operations from the particulars of the compiler was achieved by the judicious use of Assembly language macros. We believe that it is now a relatively easy job to write these macros for a FORTRAN compiler that is not on our list. The Assembly language versions of the PC-BLAS are generally more efficient when used in applications than are the FORTRAN versions. (See Appendix B for a brief rationale based on efficiency.) Usage of this code requires that the machine have an 8087 or 80287 Numeric Data Processor. The Assembly code for this translation can be assembled using the product of [5]. That product must be acquired by the reader separately; it is not included here. FORTRAN

Journal ArticleDOI
TL;DR: An effective procedure for solving single commodity problems is described based on a variable dimension, complementary pivoting algorithm of Jones, Saigal, and Schneider based on an implementation of this approach called the expanding equilibrium algorithm using network data structures and sparse graphs.
Abstract: Spatial-equilibrium models are the primary framework for applied equilibrium modeling and policy analysis. An effective procedure for solving single commodity problems is described based on a variable dimension, complementary pivoting algorithm of Jones, Saigal, and Schneider. An implementation of this approach called the expanding equilibrium algorithm is described using network data structures and sparse graphs. The data structure that is used presorts the graph to avoid using linked lists while maintaining the flexibility to add arcs to the graph as the algorithm is executing. A computational experiment is developed showing the algorithm's ability to exploit the problem's underlying network and economic structure. Computational results are presented for a computer code written in C and tested on a large-scale, randomly generated problem.

Journal ArticleDOI
TL;DR: An improved adaptive hybrid algorithm for multiplying dense multivariate polynomials that is both time and space efficient and shows most of the theoretical superiority is maintained in actual implementation.
Abstract: This paper presents an improved adaptive hybrid algorithm for multiplying dense multivariate polynomials that is both time and space efficient. The hybrid algorithm makes use of two families of univariate algorithms, one Karatsuba based and the other DFT based, which are applied recursively to solve the multivariate problem. The hybrid algorithm is adaptive in that particular univariate algorithms are selected at run time to minimize the time complexity; an order-of-magnitude speedup with respect to classical multiplication is achieved over the entire practical range except for very small problems. Empirical investigation shows that most of the theoretical superiority is maintained in actual implementation. The largest contribution to the space requirements of the total algorithm is determined by the univariate algorithm used for the outermost variable; except for quite small problems, selecting univariate algorithms to minimize run time almost always leads to situations where the space requirements of the total algorithm are extremely close to the space required merely to store the result.

Journal ArticleDOI
TL;DR: The use of STDTST is described, which is designed for assessing the performance of solvers suitable for nonstiff systems and which is similar to NSDTST in that the requirements and calling sequence is almost identical.
Abstract: Two packages are provided for assessing the performance of initial value solvers. The first package, whose main routine is NSDTST, is designed for assessing the performance of solvers suitable for nonstiff systems, whereas the second package, whose main routine is STDTST, is designed for assessing the performance of solvers suitable for stiff systems. Each package consists of a number of routines, but the user need only be aware of the appropriate main routine. In this document we describe the use of STDTST. The requirements and calling sequence of NSDTST is almost identical. In Enright and Pryce [3] the design of the testing package is discussed and guidance is given to aid in the interpretation of the reported results. A set of test problems, described in detail in Enright and Hull [2] and Enright et al. [4], is incorporated into the stiff package. The code being tested is run on

Journal ArticleDOI
TL;DR: A package that allows the computation of the trigonometric Fourier coefficients of a smooth function and can be provided as a subprogram or as a data list of function values at equally spaced points is presented.
Abstract: We present a package that allows the computation of the trigonometric Fourier coefficients of a smooth function. The function can be provided as a subprogram or as a data list of function values at equally spaced points.The computational cost of the algorithm does not depend on the required number of Fourier coefficients. Numerical results of comparative tests with a standard integrator for oscillatory functions are also reported.