Evaluation Of First Derivatives Research Articles

Optimal fourth order iterative method for solving nonlinear equation

The principal objective of this work is to propose a fourth order scheme for solving a nonlinear equation. In terms of computational cost, per iteration, the fourth order method uses two evaluation of the function and one evaluation of first derivative. So the method has satisfied the Kung-Traub optimality conjecture. In addition, the theoretical convergence properties of our schemes are fully explored with the help of the main theorem that demonstrates the convergence order. The performance and effectiveness of our optimal iteration functions have compared with the existing competitors on some standard academic problems.

An improved parameterization procedure for NDDO-descendant semi-empirical methods

ConceptMNDO-based semi-empirical methods in quantum chemistry have found widespread application in the modelling of large and complex systems. A method for the analytic evaluation of first and second derivatives of molecular properties against semi-empirical parameters in MNDO-based NDDO-descendant models is presented, and the resultant parameter Hessian is compared against the approximant currently used in parameterization for the PMx models.MethodsAs a proof of concept, the exact parameter Hessian is employed in a limited reparameterization of MNDO for the elements C, H, N, O and F using 1206 molecules for reference data (heats of formation, ionization energies, dipole moments and reference geometries). The correctness of our MNDO implementation was verified by comparing the calculated molecular properties with the MOPAC program.

Analytic evaluation of energy first derivatives for spin-orbit coupled-cluster singles and doubles augmented with noniterative triples method: General formulation and an implementation for first-order properties.

A formulation of analytic energy first derivatives for the coupled-cluster singles and doubles augmented with noniterative triples [CCSD(T)] method with spin-orbit coupling included at the orbital level and an implementation for evaluation of first-order properties are reported. The standard density-matrix formulation for analytic CC gradient theory adapted to complex algebra has been used. The orbital-relaxation contributions from frozen core, occupied, virtual, and frozen virtual orbitals to analytic spin-orbit CCSD(T) gradients are fully taken into account and treated efficiently, which is of importance to calculations of heavy elements. Benchmark calculations of first-order properties including dipole moments and electric-field gradients using the corresponding exact two-component property integrals are presented for heavy-element containing molecules to demonstrate the applicability and usefulness of the present analytic scheme.

$LDL^T$ Direction Interior Point Method for Semidefinite Programming

We present an interior point method for semidefinite programming where the semidefinite constraints on a matrix $X$ are formulated as nonnegative constraints on $d_{[1]}(X),\ldots,d_{[n]}(X)$ obtained from the $LDL^T$ factorization $X = L{\rm Diag}(d_{[1]}(X),\ldots,d_{[n]}(X))L^T$. The approach was first proposed by Fletcher [SIAM J. Control Optim., 23 (1985), pp. 493--513], who also provided analytic expressions for the derivatives of the factors in terms of $X$, and the approach was subsequently utilized in an interior point algorithm by Benson and Vanderbei [Math. Program. Ser. B, 95 (2003), pp. 279--302]. However, the evaluation of first and second derivatives of $d_{[i]}(X)$ has been a bottleneck in such an algorithm. In this paper, we (i) derive formulae for the first and second derivatives of $d_{[i]}(X)$ that are efficient and numerically stable to compute, (ii) show that the $LDL^T$ based search direction can be viewed in the standard framework of interior point methods for semidefinite programs (SDPs) with comparable computational cost per iteration, (iii) characterize the central path, and (iv) analyze the numerical conditioning of the linear system arising in the algorithm. We provide detailed numerical results on 79 SDP instances from the SDPLIB test set.

Derivation of three-derivative Runge-Kutta methods

We introduce an algorithm for a numerical integration of ordinary differential equations in the form of yź = f(y). We extend the two-derivative Runge-Kutta methods (Chan and Tsai, Numer. Algor. 53, 171---194, 2010) to three-derivative Runge-Kutta methods by including the third derivative yźźź=ăź(y)=fźź(y)(f(y),f(y))+fź(y)fź(y)f(y)$y^{\prime \prime \prime }=\hat {g}(y)=f^{\prime \prime }(y)(f(y), f(y))+f^{\prime }(y)f^{\prime }(y)f(y)$. We present an approach based on the algebraic theory of Butcher (Math. Comp. 26, 79---106, 1972) and the źź$\mathcal {B}-$ series theory of Hairer and Wanner (Computing 13, 1---15 (1974)) combined with the methodology of Chan and Chan (Computing 77(3), 237---252, 2006). In this study, special explicit three-derivative Runge-Kutta methods that possess one evaluation of first derivative, one evaluation of second derivative, and many evaluations of third derivative per step are introduced. Methods with stages up to six and of order up to ten are presented. The numerical calculations have been performed on some standard problems and comparisons made with the accessible methods in the literature.

Very fast averaging of thermal properties of crystals by molecular simulation.

Knowledge of approximate harmonic behavior of crystals is introduced into a new "mapped averaging" framework to yield alternative expressions for the thermodynamic properties of crystalline systems. The expressions separate the known harmonic behavior from residual averages, which thus encapsulate anharmonic contributions to the properties. With harmonic contributions removed, direct measurement of these anharmonic contributions by molecular simulation can be accomplished without contamination by noise produced by the already-known harmonic behavior. We show with application to the Lennard-Jones model that first-derivative properties (pressure, energy) can be obtained to a given precision via this harmonically mapped averaging at least 10 times faster than by using conventional averaging, and second-derivative properties (e.g., heat capacity) are obtained at least 100 times faster; in more favorable cases, the speedup exceeds a millionfold. Free-energy calculations are accelerated by 50 to 1000 times. Data obtained using these formulations are rigorous and not subject to any added approximation, and in fact are less sensitive to inaccuracies relating to finite-size effects, potential truncation, equilibration, and similar considerations. Moreover, the approach does not require any alteration in how sampling is performed during the simulation, so it may be used with standard Monte Carlo or molecular dynamics methods. However, the mapped averages do require evaluation of first and second derivatives of the intermolecular potential, for evaluation of first and second thermodynamic-derivative properties, respectively. Apart from its usefulness to simulation, the formalism developed here may constitute a basis for new theoretical treatments of crystals.

A new velocity–vorticity formulation for direct numerical simulation of 3D transitional and turbulent flows

Accuracy of velocity–vorticity (V→,ω→)-formulations over other formulations in solving Navier–Stokes equation has been established in recent times. However, the issue of non-satisfaction of solenoidality conditions on vorticity is not addressed in the literature which can possibly lead to non-physical solution. In this respect, here, we have developed and reported conservative rotational form of the (V→,ω→)-formulation which preserves the solenoidality condition on vorticity in a much simpler way compared to other formulations. Superiority of rotational form over the conventional Laplacian form of (V→,ω→)-formulation is also shown [by comparing the results for flows inside cubical lid driven cavity (LDC)]. For solving the 3D Navier–Stokes equation using a staggered grid, we use optimized compact schemes for (a) interpolation and (b) evaluation of first and second derivatives. As illustrations, we have solved problems of (i) flow inside a 3D lid driven cavity (LDC), whose solutions are compared with experimental results reported by Koseff and Street [19] and (ii) 3D transitional flow of an equilibrium zero pressure gradient (ZPG) boundary layer over a flat plate as demonstration of the effectiveness of the rotational form of (V→,ω→)-formulation.

A priori mesh quality metrics for three-dimensional hybrid grids

Use of general hybrid grids to attain complex-geometry field simulations poses a challenge on estimation of their quality. Apart from the typical problems of non-uniformity and non-orthogonality, the change in element topology is an extra issue to address. The present work derives and evaluates an a priori mesh quality indicator for structured, unstructured, as well as hybrid grids consisting of hexahedra, prisms, tetrahedra, and pyramids. Emphasis is placed on deriving a direct relation between the quality measure and mesh distortion. The work is based on use of the Finite Volume discretization for evaluation of first order spatial derivatives. The analytic form of the truncation error is derived and applied to elementary types of mesh distortion including typical hybrid grid interfaces. The corresponding analytic expressions provide relations between the truncation error and the degree of stretching, skewness, shearing, torsion, expansion, as well as the type of grid interface.

Analytic first derivatives for a spin-adapted open-shell coupled cluster theory: evaluation of first-order electrical properties.

An analytic scheme is presented for the evaluation of first derivatives of the energy for a unitary group based spin-adapted coupled cluster (CC) theory, namely, the combinatoric open-shell CC (COSCC) approach within the singles and doubles approximation. The widely used Lagrange multiplier approach is employed for the derivation of an analytical expression for the first derivative of the energy, which in combination with the well-established density-matrix formulation, is used for the computation of first-order electrical properties. Derivations of the spin-adapted lambda equations for determining the Lagrange multipliers and the expressions for the spin-free effective density matrices for the COSCC approach are presented. Orbital-relaxation effects due to the electric-field perturbation are treated via the Z-vector technique. We present calculations of the dipole moments for a number of doublet radicals in their ground states using restricted open-shell Hartree-Fock (ROHF) and quasi-restricted HF (QRHF) orbitals in order to demonstrate the applicability of our analytic scheme for computing energy derivatives. We also report calculations of the chlorine electric-field gradients and nuclear quadrupole-coupling constants for the CCl, CH2Cl, ClO2, and SiCl radicals.

Third-order geomorphometric variables (derivatives): definition, computation and utilization of changes of curvatures

Third-order geomorphometric variables (based on third derivatives of the altitudinal field) have been neglected in geomorphometry, but their application to the delimitation of surface objects will lead to their increasing significance in future. New techniques of computation, presented and evaluated here, facilitate their use. This paper summarizes recent knowledge concerning definition, computation and geomorphologic interpretation of these variables. Formulae defining various third-order variables are unified based on the physical definition of slope gradient. Methods for their computation are compared from the point of view of method error and error generated by digital elevation model (DEM) inaccuracy. For exact mathematical test surfaces, the most natural and simple variant of the method of central differences (CD2) shows a method error 2–3 times smaller than the other methods used recently in geomorphometry. However, success in coping with DEM inaccuracy depends (for a given grid mesh) on the number and weighting of points from which the derivative is computed. This was tested for surfaces with varying degrees of random error. Here least squares-based methods are the most effective for mixed derivatives (especially for finer grids and less accurate DEMs), while a variant of the CD method, that repeats numerical evaluation of first derivatives (CD1), is the most successful for derivatives in cardinal directions. The CD2 method is generally the most successful for coarser grids where the method error is dominant. Utilization of third-order variables is documented from examples of terrain feature (ridge, valley and edge) extraction and from a first statistical test of the hypothesis that real segments of the land surface have a tendency to a constant value of some morphometric variable. For detection of (sharp) ridges and valleys, it is shown that the rate of change of tangential curvature is inadequate: rate of change of normal curvature is also required. A basic confirmation of the constant-value tendency is provided.

Critical Issues in the Numerical Treatment of the Parameter Estimation Problems in Immunology

A robust and reliable parameter estimation is a critical issue for modeling in immunology. We developed a computational methodology for analysis of the best-fit parameter estimates and the information-theoretic assessment of the mathematical models formulated with ODEs. The core element of the methodology is a robust evaluation of the first and second derivatives of the model solution with respect to the model parameter values. The critical issue of the reliable estimation of the derivatives was addressed in the context of inverse problems arising in mathematical immunology. To evaluate the first and second derivatives of the ODE solution with respect to parameters, we implemented the variational equations-, automatic differentiation and complex-step derivative approximation methods. A comprehensive analysis of these approaches to the derivative approximations is presented to understand their advantages and limitations. Mathematics subject classification: 34K29, 92-08, 65K10.

A new family of modified Ostrowski’s methods with accelerated eighth order convergence

Based on Ostrowski’s fourth order method, we derive a family of eighth order methods for the solution of nonlinear equations. In terms of computational cost the family requires three evaluations of the function and one evaluation of first derivative. Therefore, the efficiency index of the present methods is 1.682 which is better than the efficiency index 1.587 of Ostrowski’s method. Kung and Traub conjectured that multipoint iteration methods without memory based on n evaluations have optimal order 2n − 1. Thus, the family agrees with Kung–Traub conjecture for the case n = 4. The efficacy of the present methods is tested on a number of numerical examples. It is observed that our methods are competitive with other similar robust methods and very effective in high precision computations.

A priori mesh quality estimation via direct relation between truncation error and mesh distortion

The purpose of the present work is the derivation and evaluation of a priori mesh quality indicators for structured, unstructured, as well as hybrid grids. Emphasis is placed on deriving direct relations between the indicators and mesh distortion. The work is based on use of the finite volume discretization for evaluation of first order spatial derivatives. The analytic form of the truncation error is derived and applied to elementary types of mesh distortion including typical hybrid grid interfaces. The corresponding analytic expressions provide direct relations between computational accuracy and the degree of stretching, skewness, shearing and non-alignment of the mesh.

Genetic algorithm for the determination of binodal curves in ternary systems polymer–liquid(1)–liquid(2) and polymer(1)–polymer(2)–solvent

A simple genetic algorithm for the numerical evaluation of binodal curves in ternary systems polymer-liquid (1)-liquid (2) and polymer (1)-polymer (2)-solvent is presented. The technique exploits a specifically developed restarting technique based on a combined elitist and zooming strategy on the last population at each iteration. The objective function (fitness) is represented by the weighted sum of the squared differences of chemical potentials of the two phases of each component, obtained evaluating first derivatives of Gibbs free energy of the mixture with respect to the number of moles of the components. The method proposed (a) is numerically stable since it does not require the evaluation of first derivatives of the objective function and (b) can be applied in a wide range of cases changing the equation of state. Several comparisons with simplified iterative procedures presented in the past in the technical literature both for mixtures of two polymers with identical characteristics in a solvent and for mixtures of solvent-nonsolvent-polymer with solvent-polymer interaction parameter equal to zero are reported. Finally, a comparison between present results and the "alternating tangent approach" is reported for two technically meaningful binary systems, when a simplified PC-SAFT equation of state is adopted. The comparisons show that reliable results can be obtained by means of the algorithm proposed and suggest that the procedure presented can be used for practical purposes.

An efficient overloaded implementation of forward mode automatic differentiation in MATLAB

The Mad package described here facilitates the evaluation of first derivatives of multidimensional functions that are defined by computer codes written in MATLAB. The underlying algorithm is the well-known forward mode of automatic differentiation implemented via operator overloading on variables of the class fmad. The main distinguishing feature of this MATLAB implementation is the separation of the linear combination of derivative vectors into a separate derivative vector class derivvec. This allows for the straightforward performance optimization of the overall package. Additionally, by internally using a matrix (two-dimensional) representation of arbitrary dimension directional derivatives, we may utilize MATLAB's sparse matrix class to propagate sparse directional derivatives for MATLAB code which uses arbitrary dimension arrays. On several examples, the package is shown to be more efficient than Verma's ADMAT package [Verma 1998a].

Exchange functionals with improved long-range behavior and adiabatic connection methods without adjustable parameters: The mPW and mPW1PW models

Starting from an analysis of the low-density and large gradient regions which dominate van der Waals interactions, we propose a modification of the exchange functional introduced by Perdew and Wang, which significantly enlarges its field of applications. This is obtained without increasing the number of adjustable parameters and retaining all the asymptotic and scaling properties of the original model. Coupling the new exchange functional to the correlation functional also proposed by Perdew and Wang leads to the mPWPW model, which represents the most accurate generalized gradient approximation available until now. We next introduce an adiabatic connection method in which the ratio between exact and density functional exchange is determined a priori from purely theoretical considerations and no further parameters are present. The resulting mPW1PW model allows to obtain remarkable results both for covalent and noncovalent interactions in a quite satisfactory theoretical framework encompassing the free electron gas limit and most of the known scaling conditions. The new functionals have been coded with their derivatives in the Gaussian series of programs, thus allowing fully self-consistent computations of energy and properties together with analytical evaluation of first and second geometry derivatives.

Kinematic and Kinetic Derivatives in Multibody System Analysis*

Analytical formulas for kinematic and kinetic derivatives needed in multibody system analyses are derived. A broad spectrum of problems, including implicit numerical integration, dynamic sensitivity analysis, and kinematic workspace analysis, require evaluation of first derivatives of generalized inertia and force expressions and at least three derivatives of algebraic constraint functions. In the setting of a formulation based on Cartesian generalized coordinates with Euler parameters for orientation, basic identities are developed that enable practical and efficient computation of all derivatives required for a large number of multibody mechanical system analyses. The formulation is verified through application to a spatial slider crank mechanism and a 14 body vehicle model. Efficiency of computation using the expressions derived is compared with results obtained employing finite differences, showing significant computational advantage using the analytically derived expressions.

Creation of intensity theory in vibrational spectroscopy: Key role of ab initio quantum mechanical calculations

The development of effective theoretical approaches for analytic evaluation of first, second, and third derivatives of molecular properties, in particular energy, dipole moment, and polarizability, has contributed to increased accuracy of ab initio methods in predicting vibrational spectral parameters. It has become possible to devise and test theoretical approaches for analysis of vibrational intensities using consistent and reliable results from high-level ab initio calculations. There is a second important side of the application of ab initio molecular orbital (MO) calculations to the study of vibrational intensities. As for many other physical and chemical phenomena, the quantum mechanical studies provide essential information about the fine mechanisms and factors determining the observed physical quantities, in our case the intensities in infrared and Raman spectra of molecules. On this basis, new theoretical formulations for infrared and Raman intensities were recently developed. Infrared intensities are transformed into quantities termed effective bond charges that are derived from dipole moment derivatives with respect to atomic Cartesian coordinates. Raman intensities are reduced to effective induced bond charges following an analogous approach. In the present work we review the relation between advances in analytic derivative ab initio quantum mechanical calculations and the development of methods for analysis and interpretation of experimental or ab initio dipole and polarizability derivatives. Brief presentation of the effective bond charge and effective induced bond charge formulations is given. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 331–339, 1998

RI-MP2: first derivatives and global consistency

The evaluation of RI-MP2 first derivatives with respect to nuclear coordinates or with respect to an external electric field is described. The prefix RI indicates the use of an approximate resolution of identity in the Hilbert space of interacting charge distributions (Coulomb metric), i.e., the use of an auxiliary basis set to approximate charge distributions. The RI technique is applied to first derivatives of the MP2 correlation energy expression while the (restricted) Hartree-Fock reference is treated in the usual way. Computational savings by a factor of 10 over conventional approaches are demonstrated in an application to porphyrin. It is shown that the RI approximation to MP2 derivatives does not entail any significant loss in accuracy. Finally, the relative energetic stabilities of a representative sample of closed-shell molecules built from first and second row elements have been investigated by the RI-MP2 approach, and thus it is tested whether such properties that refer to potential energy hypersurfaces in a more global way can be described with similar consistency to the more locally defined derivatives.

Algorithm 755: ADOL-C

The C++ package ADOL-C described here facilitates the evaluation of first and higher derivatives of vector functions that are defined by computer programs written in C or C++. The resulting derivative evaluation routines may be called from C/C++, Fortran, or any other language that can be linked with C. The numerical values of derivative vectors are obtained free of truncation errors at a small multiple of the run-time and randomly accessed memory of the given function evaluation program. Derivative matrices are obtained by columns or rows. For solution curves defined by ordinary differential equations, special routines are provided that evaluate the Taylor coefficient vectors and their Jacobians with respect to the current state vector. The derivative calculations involve a possibly substantial (but always predictable) amount of data that are accessed strictly sequentially and are therefore automatically paged out to external files.