Order Partial Derivatives Research Articles

Second Order Perturbations of Elliptic Elements with Respect to the Initial Ones

AbstractThe following paper is devoted to the theoretical exposition of the obtention of second order perturbations of elliptic elements and is a follow-up of previous papers (Marco et al., 1996; Marco et al., 1997) where the hypothesis was made that the matrix of the partial derivatives of the orbital elements with respect to the initial ones is the identity matrix at the initial instant only. So, we must compute them through the integration of Lagrange planetary equations and their partial derivatives.Such developments have been applied to the individual corrections of orbits together with the correction of the reference system through the minimization of a quadratic form obtained from the linearized residual. In this state two new targets emerged: 1.To be sure that the most suitable quadratic form was to be considered.2.To provide a wider vision of the behavior of the different orbital parameters in time.Both aims may be accomplished through the consideration of the second order partial derivatives of the elliptic orbital elements with respect to the initial ones.

Computing multiple turning points by using simple extended systems and computational differentiation

A point (x *,λ*) is called a turning point of multiplicity p ≥1 of the nonlinear system if and if the Ljapunov–Schmidt reduced function has the normal form . A minimally extended system F(x, λ)=0 F(x, λ)=0 is proposed for defining turning points of multiplicity p, where is a scalar function which is related to the pth order partial derivatives of g with respect to ξ. When Fdepends on m≤p-1 additional parameters the system F(x, λ α)=0 can be inflated by m + 1 scalar equations f 1(x, λ α)=0,…f m+1(x, λ α)=0 The functions depend on certain partial derivatives of gwith respect to ξ where f m+1 corresponds to f The regular solution (x *, λ *,α *) of the extended system of n+m+1 equations delivers the desired turning point (x *, λ*). For numerically solving these systems, two-stage New tonype methods are proposed, where only one LU decomposition of an (n+1) ×(++1) matrix and some back substitutions have to be preformed per iteration step if Gaussian elimination is used for solving the linear systems. Moreover, the methods require the computation of certain higher order partial derivatives of f with respect to low dimensional subspaces as well as derivatives of implicitly defined related functions. Both tasks are realized via computational differentiation, often also called automatic differentiation. For doing this the special structure of the higher order derivatives is exploited, and the problem is reformulated so that the Taylor coefficients technique of computational differentiation can efficiently be integrated into the algorithm. Some numerical tests show the behavior of the algorithms in case of turning points of multiplicity p= 2,3.

Computational Differentiation M-Files for MATLAB

Computational differentiation is a means of finding the first, second, and higher order partial derivatives of a function. Computational differentiation eliminates the truncation error of numerical differentiation and reduces the roundoff error that often accompanies numerical differentiation. Computational differentiation provides the accuracy of analytical derivatives but does not require the user to find the analytical expressions. MATLAB is a software package oriented around vector and matrix operations. The latest version of MATLAB provides objects and operator overloading. These two features make it particularly attractive for implementing computational differentiation. A basic library of computational derivative routines is presented as MATLAB M-fils. A nonlinear optimization M-file based on Newton's method is given and applied to logit, probit, and ordered probit models as examples.

New boundary element formulas for plane elasticity

A plane elasticity problem leads us to a biharmonic equation with the boundary condition, consisting of first order partial derivatives. We apply the boundary integral equation formulas by Y. Jeon (1996) for the problem. Derivation of the formulas and the concerned analysis and numerical examples are presented. The formulas are especially efficient in evaluating the stress tensor on the boundary as well as inside the domain. It is the stress tensor components that are physically important quantities for the problem.

Relative entropy and mixing properties of infinite dimensional diffusions

Let η be a diffusion process taking values on the infinite dimensional space T Z , where T is the circle, and with components satisfying the equations dη i =σ i (η) dW i +b i (η) dt for some coefficients σ i and b i , i∈Z. Suppose we have an initial distribution μ and a sequence of times t n →∞ such that lim n →∞μS tn =ν exists, where S t is the semi-group of the process. We prove that if σ i and b i are bounded, of finite range, have uniformly bounded second order partial derivatives, and inf i ,ησ i (η)>0, then ν is invariant.

On form-preserving point transformations of partial differential equations

New identities are presented relating arbitrary order partial derivatives of and for the general point transformation , . These identities are used to study the nature of those point transformations which preserve the general form of a wide class of 1 + 1 partial differential equations. These results facilitate the search for point symmetries, both discrete and continuous (Lie), and assist the search for point transformations which reduce equations to canonical, but similar, form. A simple test for the existence of hodograph-type transformations between equations of similar form is given.

Computing Multiple Turning Points

AbstractA point (x*,λ*) is called a turning point of multiplicity p ≥ 1 of the nonlinear system F(x,δ) = 0, F:ℛn × IR1 → ℛn if rank δx F(x*, λ*) = n ‐ 1, rank [δx F(x*, λ*)|λ F (x*, λ*)] = n, and if the Ljapunov‐Schmidt reduced function has the normal form g(ζ,μ) = ζp+1 ± μ, g : ℛ1 × ℛ1 → ℛ1. A minimally extended system F(x,λ) = 0, H(x, λ) = 0 is proposed for defining turning points of multiplicity p, where H : ℛn × ℛ1 → ℛ1 is a scalar function which is related to the p‐th order partial derivatives of g with respect to ζ. When F depends on m ⩽ p ‐‐ 1 additional parameters α ϵ ℛm, the system F(x,λ,α) = 0 can be inflated by m + 1 scalar equations H1 (x,λ, α) = 0,…, Hm+1 (x,λ, α) = 0. The functions H1, : ℛn × ℛ1 × ℛm × ℛ1 depend on certain partial derivatives of g with respect to ζ where Hm+1 corresponds to H in the case m = 0. The regular solution (x*, λ*, α*) of the extended system of n+m+1 equations delivers the desired turning point (x*, λ*) as first part. For numerically solving these systems, two‐stage Newton‐type methods are proposed. They require to compute higher order partial derivatives of F with respect to low dimensional subspaces as well as derivatives of implicitly defined related functions. Both tasks are realized via automatic differentiation. The behavior of the algorithms has been tested in case of turning points of multiplicity p = 2, 3.

On 3D differential operators for detecting point landmarks

Point-based registration of volume image data heavily relies on the selection of suitable landmarks. In this contribution, we study the extraction of 3D point landmarks from images. Point landmarks have a unique position which can be deduced from the intensity variations in a sufficiently large neighborhood around the prominent point. We propose four 3D differential operators which are generalizations of existing 2D operators for detecting points of high intensity variations. In comparison to previous approaches, our operators have the advantage that only low order partial derivatives of the image function are necessary. Therefore, these operators are computationally efficient and do not suffer from instabilities of computing high order partial derivatives. We also describe how the 3D operators can be generalized to be used on images of arbitrary dimension. First experimental results will be presented on medical imagery.

Peak set without peak points

We give an example of a natural Banach function algebra on the unit disc such that a smaller disc is a peak set for the algebra, but it does not contain any peak point. A Banach function algebra on a compact Hausdorff space X is a Banach algebra A consisting of continuous functions on X, such that A separates points of X and contains the constant functions. If the norm of the algebra A coincides with the sup norm on X , it is called a uniform algebra. If any linear and multiplicative functional on A is of the form f 7→ f(x) for some x ∈ X, the algebra is called natural. A subset K of X is a peak set for A if there is an f ∈ A such that f ≡ 1 on K and |f(x)| < 1 for x / ∈ K; if K = {x0} , then x0 is a peak point. If no proper subset of K is a peak set we call it a minimal peak set for A. It is well known [2] that if A is a uniform algebra on a metrizable set X then any peak set contains a peak point. In [1] H. G. Dales constructed a natural Banach function algebra on a compact subset of C having a peak set without any peak point. T. G. Honary [3] provided an example in R, however his algebra is not natural. In this note we give a very simple example of a natural Banach function algebra on the unit disc with peak sets not containing any peak point. Let K be an open nonempty subset of the complex plane C. By C(K) we denote the algebra of all bounded complex valued functions on K with continuous and bounded first order partial derivatives on K. C(K) is a Banach function algebra on K if equipped with the norm ‖f‖ = ‖f‖∞ + ‖fx‖∞ + ‖fy‖∞ , where ‖·‖∞ is the sup norm. By A(K) we denote the uniform algebra of all continuous functions on K which are analytic on K. We put Dr = {z ∈ C : |z| < r} and C− = {z : Re z < 0}. Theorem 1. Let A be a subalgebra of C(D1) consisting of all functions which are analytic on D 1 2 . Then D 1 2 is a minimal peak set for A; in particular D 1 2 does not contain any peak point. Lemma 2. There is no h ∈ A(D1) such that h(1) = 0 and |1 + (z − 1)h(z)| < 1 for z 6= 1. Proof of the lemma. Assume that such a function does exist. If we compose 1 + (z − 1)h(z) with a suitable fractional linear transformation we get an f ∈ A (C−) Received by the editors November 8, 1995. 1991 Mathematics Subject Classification. Primary 46J10. c ©1997 American Mathematical Society

Simultaneous approximations of multivariate functions and their derivatives by neural networks with one hidden layer

Let σ be a non-polynomial activation function of a neural network that has nth order continuous derivatives on R. The objective of this paper is to show that for any compact set K of R s , s ≥ 1, and any multivariate function f defined on an open set containing K, a neural network with one hidden layer can be so constructed that f and all its existing continuous kth order partial derivatives, for k = ( k 1, …, k s ) ϵ Z + s satisfying ∑ i = 1 s k i ≤ n, can be simultaneously and uniformly approximated on K by the network.

EMPIRICAL BAYES ESTIMATION FOR ESTIMABLE FUNCTION OF REGRESSION COEFFICIENT IN A MULTIPLE LINEAR REGRESSION MODEL

In this paper we consider the empirical Bayes (EB) estimation problem for estimable function of regression coefficient in a multiple linear regression model Y = Xβ + e, where e with given β has a multivariate standard normal distribution. We get the EB estimators by using kernel estimation of multivariate density function and its first order partial derivatives. It is shown that the convergence rates of the EB estimators are O(n-2(kλ-1)2k+m) under the condition ∫Θ||β||(m+ξ)λη-λ V(2kλ)dG(β)<∞. where an integer k>1, 12<λ<η<1, ξ>0 is an arbitrary small number and m is the dimension of the vector Y.

Higher-order sensitivity analysis of finite element method by automatic differentiation

To design optimal mechanical structures, design sensitivity analysis technique using higher order derivatives are important. However, usual techniques for computing the derivatives, for example numerical differentiation methods, are hard to apply to real scale structures because of the large amount of computational time and the accumulation of computational errors. To overcome the problem, we have studied a new approach for higher order sensitivity analysis of the finite element method using automatic differentiation techniques. The method automatically transforms FORTRAN code to special purpose code which computes both the value of the given functional dependence and their derivatives. The algorithm used in the method automatically and efficiently computes accurate values of higher order partial derivatives of a given functional dependence on many variables. This paper reports the basic principles of the automatic differentiation method and some experiments on the sensitivity analysis of mechanical structures. The original program of structural analysis by the finite element method is implemented in FORTRAN, which is developed by the first author. Using the proposed method, we get more accurate sensitivity and prediction values compared with usual numerical differentiation. We also discuss the effectiveness of the proposed approach for the sensitivity analysis of the mechanical structures.

Sharp opial-type inequalities in two variables

In this paper we present sharp Opial-type inequalities in two independent variables involving higher order partial derivatives. From the particular cases of our results we then correct as well as improve several recent inequalities.

RECOVERING THE GRADIENTS OF THE SOLUTIONS OF SECOND ORDER HYPERBOLIC EQUATIONS BY INTERPOLATING THE FINITE ELEMENT SOLUTIONS

We present a technique to generate better approximations to the gradients of the solutions of the second order hyperbolic (acoustic wave) equations by postprocessing finite element solutions with higher degree polynomial interpolation. The postprocessing procedure is inexpensive, local, vectorizable, and parallelizable. In addition, the postprocessing procedure is independent of the computation of the finite element solution; therefore it can be easily incorporated with the existing finite element codes to efficiently generate globally better approximations to the gradients or first order partial derivatives of the main quantities of the acoustic field modeled by the acoustic wave equation.

A nodal basis of Cμ-rational spline functions on triangulations

Let ρ be a triangulation of a polygonal domain D⊂R2 with vertices V={vi:l≤i≤Nv} and RSk(D, ρ)={u∈Ck(D): ≠ T∈ρ, u/T is a rational function}. The purpose of this paper is to study the existence and construction of Cμ-rational spline functions on any triangulation ρ for CAGD. The Hermite problem Hμ(V,U)={find u∈U: Dαu(vi)=Dαf(vi),|α|≤μ} is solved by the generalized wedge function method in rational spline function family, i.e. U=RSμ. this solution needs only the knowledge of partial derivatives of order≤μ at vi. The explicit repesentations of all Cμ-GWF(generalized wedge functions)and the interpolating operator with degree of precision at least 2μ+1 for any triangulation are given.

On analyticity of functions involving eigenvalues

Let A( z) be an n × n complex matrix whose elements depend analytically on z ∈ C m . It is well known that any individual eigenvalue of A( z) may be nondifferentiable when it coalesces with others. In this paper, we investigate the analycity property of functions on the eigenvalues λ( z) = ( λ 1( z),…, λ n ( z)) of A( z). We first introduce the notion of functions that are symmetric with respect to partitions. It is then shown that if a function ƒ : C n → C is analytic at λ( a), where a ϵ C m , and is symmetric with respect to a certain partition induced by λ( a), then the composite function g(z) = ƒ(λ 1(z),…,λ n(z)) is analytic at a. When z is real, A( z) is symmetric or Hermitian, and the aforementioned assumptions hold, so that g( z) is analytic at a, we also derive formulae for its first and second order partial derivatives. We apply the results to several problems involving eigenvalues.

Convergence rates for single hidden layer feedforward networks

By allowing the training set to become arbitrarily large, appropriately trained and configured single hidden layer feedforward networks converge in probability to the smooth function that they were trained to estimate. A bound on the probabilistic rate of convergence of these network estimates is given. The convergence rate is calculated as a function of the sample size n. If the function being estimated has square integrable m th order partial derivatives then the L 2-norm estimation error approaches O p ( n − 1 2 ) for large m. Two steps are required for determining these bounds. A bound on the rate of convergence of approximations to an unknown smooth function by members of a special class of single hidden layer feedforward networks is determined. The class of networks considered can embed Fourier series. Using this fact and results on approximation properties of Fourier series yields a bound on L 2-norm approximation error. This bound is less than O( q − 1 2 ) for approximating a smooth function by networks with q hidden units. A modification of existing results for bounding estimation error provides a general theorem for calculating estimation error convergence rates. Combining this result with the bound on approximation rates yields the final convergence rates.

A parallel tensor algorithm for nonlinear optimization

A new iterative algorithm for solving unconstrained optimization problems is introduced. It is based on the construction, at each iteration, of a curvilinear path to be searched for a local solution. Since the curvilinear path is denned by using a tensor of third order partial derivatives of the objective function, efficient and reliable implementations can benefit of powerful computational tools like parallel computing and automatic differentiation. Computational experiments were carried out with the aim to compare the proposed algorithm with well known Newton type algorithms. It turns out that the proposed algorithm is very efficient especially in the case of badly scaled and ill-conditioned problems.

C 1 surface interpolation with constraints

Givenn pairwise distinct and arbitrarily spaced pointsP i in a domainD of thex−y plane andn real numbersf i, consider the problem of computing a bivariate functionf(x, y) of classC 1 inD whose values inP i are exactlyf i,i=1,…,n, and whose first or second order partial derivatives satisfy appropriate equality and inequality constraints on a given set ofp pointsQ l inD. In this paper we present a method for solving the above problem, which is designed for extremely large data sets. A step of this method requires the solution of a large scale quadratic programming (QP) problem. The main purpose of this work is to analyse an iterative method for determining the solution of this QP problem: such a method is very efficient and well suited for parallel implementation on a multiprocessor system.

Refinements in Asymptotic Expansions for Sums of Weakly Dependent Random Vectors

Let $S_n$ denote the $n$th normalized partial sum of a sequence of mean zero, weakly dependent random vectors. This paper gives asymptotic expansions for $Ef(S_n)$ under weaker moment conditions than those of Gotze and Hipp (1983). It is also shown that an expansion for $Ef(S_n)$ with an error term $o(n^{-(s - 2)/2})$ is valid without any Cramer-type condition, if $f$ has partial derivatives of order $(s - 1)$ only. This settles a conjecture of Gotze and Hipp in their 1983 paper.