A one-sweep numerical method for vector-matrix difference equations with two-point boundary conditions

A new method is proposed for reducing two-point boundaryvalue problems for vector-matrix systems of linear difference equations to initial-value problems. The method has the advantage that only one sweep is required, and memory requirements are minimal. Applications to potential theory are discussed.This research was supported by the National Institutes of Health under Grants Nos. GM-16197-01 and GM-16437-01 and by the Atomic Energy Commission under Contract No. AT(11-1)-113, Project No. 19.     

Exact solution of Pontryagin's equations of optimal control—Part 1

In the treatment of constrained optimal control processes, it is customary to employ the Pontryagin maximum principle, which requires the solution of a two-point boundary-value problem. Various economic, mechanical, and biological control processes are of this type, including optimization of hemodialysis. Generally speaking, two-point boundary-value problems are more difficult to treat computationally than initial-value or Cauchy problems. In this paper, a Cauchy system is derived for a class of optimal control processes, and it is then shown that the solution of the Cauchy problem satisfies the Pontryagin equations.This research was supported by the National Science Foundation, Grant No. GF-294, and the National Institutes of Health, Grants Nos. GM-16197-01 and GM-16437-01.     

Imbedding methods for the integral equations for the optimal control of time-lag systems,I: Parameter dependence

The problem of the optimal control of systems with time lags can be reduced to a Fredholm integral equation. In this paper, it is shown that the Fredholm resolvent satisfies an initial-value problem in which a parameter serves as the independent variable. This reduction may be of computational utility.This work was partially supported by Sigma Delta Epsilon—Graduate Women in Science and NIH Grant No. GM-23732.     

Identification of parameters in nonlinear boundary-value problems

This paper presents a method for identification of parameters in nonlinear boundary-value problems. The successive approximations technique proposed uses the theory of Lagrange multipliers and the Newton-Raphson method. This method does not require storage of functions and is quadratically convergent. Numerical results are presented.This research was sponsored by the National Institutes of Health, Grant No. GM-16197-01. Computing assistance was obtained from the Health Sciences Computing Facility, University of California at Los Angeles, NIH Grant No. FR-3.     

A stability theorem for symmetrically rational counterplanning

In the absence of contrary information, it would seem prudent for a competitor to attribute to his opponents the same level of rationality that he himself employs. In the context of a general, linear-quadratic, nonzero-sum, two-person game, it is shown that a counterplanning procedure consistent with this principle of symmetrical rationality always converges to the unique Nash equilibrium for the game.The work of R. Kalaba and L. Tesfatsion was partially supported by the National Science Foundation under Grant No. ENG-77-28432 and by the National Institutes of Health under Grant No. GM-23732-03.     

Transcendental geometric programs

This paper treats a class of posynomial-like functions whose variables may appear also as exponents or in logarithms. It is shown that the resulting programs, called transcendental geometric programs, retain many useful properties of ordinary geometric programs, although the new class of problems need not have unique minima and cannot, in general, be transformed into convex programs. A duality theory, analogous to geometric programming duality, is formulated under somewhat more restrictive conditions. The dual constraints are not all linear, but the notion ofdegrees of difficulty is maintained in its geometric programming sense. One formulation of the dual program is shown to be a generalization of the chemical equilibrium problem where correction factors are added to account for nonideality. Some of the computational difficulties in solving transcendental programs are discussed briefly.This research was partially supported by the National Institute of Health Grant No. GM-14789; Office of Naval Research under Contract No. N00014-75-C-0276; National Science Foundation Grant No. MPS-71-03341 A03; and the US Atomic Energy Commission Contract No. AT(04-3)-326 PA #18.     

Variational equations for the eigenvalues and eigenvectors of nonsymmetric matrices

The tracking of eigenvalues and eigenvectors for parameterized matrices is of major importance in optimization and stability problems. In the present paper, we consider a one-parameter family of matrices with distinct eigenvalues. A complete system of differential equations is developed for both the eigenvalues and the right and left eigenvectors. The computational feasibility of the differential system is demonstrated by means of a numerical example.The work of R. Kalaba and L. Tesfatsion was partially supported by the National Science Foundation under Grant No. ENG-77-28432 and by the National Institutes of Health under Grant No. GM-23732-03.     

A proof of the Pontryagin maximum principle for initial-value problems

Many optimization problems in economic analysis, when cast as optimal control problems, are initial-value problems, not two-point boundary-value problems. While the proof of Pontryagin (Ref. 1) is valid also for initial-value problems, it is desirable to present the potential practitioner with a simple proof specially constructed for initial-value problems. This paper proves the Pontryagin maximum principle for an initial-value problem with bounded controls, using a construction in which all comparison controls remain feasible. The continuity of the Hamiltonian is an immediate corollary. The same construction is also shown to produce the maximum principle for the problem of Bolza.     

Identification of linear systems using long periods of observation

A new method is presented for identifying parameters in a linear differential system arising, e.g., from compartment models in drug kinetics. The linearity of the system is used to produce a series of recurrence relations that help reduce the computational load. The method is especially useful when a long period of observation is used to identify the parameters. Numerical experiments are described.This research was sponsored by the National Institutes of Health, Grant No. GM-16197-01. Computing assistance was obtained from the Health Sciences Computing Facility, sponsored by NIH Grant No. FR-3.     

Convergence of the shifted algorithm for unitary Hessenberg matrices

This paper shows that for unitary Hessenberg matrices the algorithm, with (an exceptional initial-value modification of) the Wilkinson shift, gives global convergence; moreover, the asymptotic rate of convergence is at least cubic, higher than that which can be shown to be quadratic only for Hermitian tridiagonal matrices, under no further assumption. A general mixed shift strategy with global convergence and cubic rates is also presented.

     

New initial-value method for singularly perturbed boundary-value problems

An initial-value method is given for second-order singularly perturbed boundary-value problems with a boundary layer at one endpoint. The idea is to replace the original two-point boundary value problem by two suitable initial-value problems. The method is very easy to use and to implement. Nontrivial text problems are used to show the feasibility of the given method, its versatility, and its performance in solving linear and nonlinear singularly perturbed problems.This work was supported in part by the Consiglio Nazionale delle Ricerche, Contract No. 86.02108.01, and in part by the Ministero della Pubblica Istruzione.     

A sequential method for nonlinear filtering: Numerical implementation and comparisons

Exact equations are presented for sequentially updating the optimal solution for a discrete-time analog of the basic Sridhar nonlinear filtering problem as the process length increases and new observations are obtained. A tabular method is described for implementing numerically the sequential filtering equations. The accuracy and efficiency of the tabular method are illustrated by means of several numerical examples.Dedicated to R. SridharThe work of R. Kalaba and L. Tesfatsion was partially supported by the National Science Foundation under Grant No. ENG-77-28432 and by the National Institutes of Health under Grant No. GM-23732-03.     

An initial-value method for the determination of Green's functions

The convergence of an initial-value method for computing the Green's function of a class of second-order differential operators is established. The proof relies on an interpolation procedure which is shown to generalize the Nyström method for Fredholm integral equations. The approximate Green's function is related to the solution of a discrete summation equation. The results of Anselone and Moore on collectively compact operators are then applied.This research was partially supported by UNLV Grant No. 001-060-4573.     

Asymptotic behaviour of solutions of linear integro-differential equations

Given the linear integro-differential equation (P_o) on a reflexive Banach space, we prove the existence of unbounded solutions with an exponential growth rate for a class of initial-value problems. Since the appearing kernel functions are of convolution type on a semi-axis, abstract Wiener-Hopf techniques, recently developed by Feldman [3,4,5], are used for the construction of the resolving operator associated with the problems under consideration. Applicability of the results is shown to initial boundary-value problems arising in the theory of generalized heat conduction in materials with memory and of viscoelasticity.     

Riccati scheme for integrating nonlinear systems of differential equations

A modification of the Riccati system that makes it possible to reduce to linear problems the initial-value problem for systems of ordinary differential equations with bilinear nonlinearity is discussed. It is shown that from the algebraic point of view it is natural, in the framework of the scheme, to consider functions that take values in an algebra with two multiplications related by a condition of the type of associativity.State University, St. Petersburg. Translated from Teoretcheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 352–363, March, 1995.     

Invariant imbedding and the numerical determination of Green's functions

A Cauchy system for a Green's function is derived. This is investigated numerically, and applications to the determination of eigenvalues and stochastic differential operators are sketched.This research was supported by the National Institutes of Health under Grants Nos. GM-16437-02 and GM-01724-04.     

Solving Volterra Integral Equations with ODE Codes

Some nonlinear Volterra integral equations are equivalent toan initial-value problem for a system of ordinary differentialequations (ODEs). Because effective ODE codes are widely available,some authors have sought to exploit this connection for thenumerical solution of the integral equations. There are twomajor difficulties: One would like to solve large systems ofODEs. The initial-value problem for the ODEs may be stiff. Theinitial-value problems are shown to have a remarkable structurewhich can be exploited to overcome these two difficulties.     

Some relationships between Green's functions and invariant imbedding

The method of invariant imbedding has been used to resolve the solution of linear two-point boundary-value problems into contributions associated with the homogeneous equation with homogeneous boundary conditions, with inhomogeneous boundary conditions, and with an inhomogeneous source term in the equation. The relationship between the Green's function and the invariant imbedding equations is described, and it is shown that the Green's function can be determined from an initial-value problem. Several numerical examples are given which illustrate the efficacy of the initial-value algorithm.This work was supported by the US Atomic Energy Commission.     

Initial-value technique for self-adjoint singular perturbation boundary value problems

We have developed an initial-value technique for self-adjoint singularly perturbed two-point boundary value problems. The original problem is reduced to its normal form, and the reduced problem is converted into first-order initial-value problems. These initial-value problems are solved by the cubic spline method. Numerical illustrations are given at the end to demonstrate the efficiency of our method. Graphs are also depicted in support of the results. An erratum to this article can be found at     

On the uniqueness of Nash strategies for a class of analytic differential games

The uniqueness of Nash equilibria is shown for the case where the data of the problem are analytic functions and the admissible strategy spaces are restricted to analytic functions of the current state and time.This work was supported in part by the Joint Services Electronics Program (US Army, US Navy, and US Air Force) under Contract No. DAAB-07-72-C-0259, in part by the National Science Foundation under Grant No. ENG-74-20091, and in part by the Department of Energy, Electric Energy Systems Division under Contract No. US ERDA EX-76-C-01-2088.