Complex-step Method Research Articles

Review and Unification of Methods for Computing Derivatives of Multidisciplinary Computational Models

This paper presents a review of all existing discrete methods for computing the derivatives of computational models within a unified mathematical framework. This framework hinges on a new equation, the unifying chain rule, from which all the methods can be derived. The computation of derivatives is described as a two-step process: the evaluation of the partial derivatives and the computation of the total derivatives, which are dependent on the partial derivatives. Finite differences, the complex-step method, and symbolic differentiation are discussed as options for computing the partial derivatives. It is shown that these are building blocks with which the total derivatives can be assembled using algorithmic differentiation, the direct and adjoint methods, and coupled analytic methods for multidisciplinary systems. Each of these methods is presented and applied to a common numerical example to demonstrate and compare the various approaches. The paper ends with a discussion of current challenges and possible future research directions in this field.

The Fractional Complex Step Method

It is well known that the complex step method is a tool that calculates derivatives by imposing a complex step in a strict sense. We extended the method by employing the fractional calculus differential operator in this paper. The fractional calculus can be taken in the sense of the Caputo operator, Riemann-Liouville operator, and so forth. Furthermore, we derived several approximations for computing the fractional order derivatives. Stability of the generalized fractional complex step approximations is demonstrated for an analytic test function.

205 複素数階微分法による高延性接着材の弾粘塑性変形挙動有限要素解析

205 複素数階微分法による高延性接着材の弾粘塑性変形挙動有限要素解析

On the generalization of the Complex Step Method

We generalize the well known Complex Step Method for computing derivatives by introducing a complex step in a strict sense. Exploring different combinations of terms, we derive 52 approximations for computing the first order derivatives and 43 for the second order derivatives. For an appropriate combination of terms and appropriate choice of the step size in the real and imaginary directions, fourth order accuracy can be achieved in a very simple and efficient scheme on a compact stencil. New different ways of computing second order derivatives in one single step are shown. Many of the first order derivative approximations avoid the problem of subtractive cancellation inherent to the classic finite difference approximations for real valued steps, and the superior accuracy and stability of the generalized complex step approximations are demonstrated for an analytic test function.

Using Multicomplex Variables for Automatic Computation of High-Order Derivatives

The computations of the high-order partial derivatives in a given problem are often cumbersome or not accurate. To combat such shortcomings, a new method for calculating exact high-order sensitivities using multicomplex numbers is presented. Inspired by the recent complex step method that is only valid for firstorder sensitivities, the new multicomplex approach is valid to arbitrary order. The mathematical theory behind this approach is revealed, and an efficient procedure for the automatic implementation of the method is described. Several applications are presented to validate and demonstrate the accuracy and efficiency of the algorithm. The results are compared to conventional approaches such as finite differencing, the complex step method, and two separate automatic differentiation tools. The multicomplex method performs favorably in the preliminary comparisons and is therefore expected to be useful for a variety of algorithms that exploit higher order derivatives.

Statistical Applications of the Complex-Step Method of Numerical Differentiation

The complex-step method is a clever way of obtaining a numerical approximation to the first derivative of a function, avoiding the round-off error that plagues standard finite difference approximations. An extension of the method allows second derivatives to be calculated with reduced round-off error. This article provides an overview of the method, discusses its practical implementation, with particular reference to R, and studies its effectiveness in several statistical examples.

Nonlinear robust performance analysis using complex-step gradient approximation

In this paper, the complex-step method is applied in the setting of numerical optimisation problems involving dynamical systems modelled as nonlinear differential equations. The main advantage of the complex-step method for gradient approximation is that it entails no subtractive cancellation error, and therefore the truncation error can be made arbitrarily (to machine precision) small. The method is applied to two robust performance analysis problems. The accuracy and convergence rate of the solutions computed using the proposed approach are seen to be significantly better than those achieved using standard gradient approximation methods.

COMPLEX-STEP GRADIENT APPROXIMATION FOR ROBUSTNESS ANALYSIS OF NONLINEAR SYSTEMS

In this paper, the complex-step perturbation method is extended to the setting of optimisation problems involving dynamical systems modelled as nonlinear differential equations. The main advantage of the complex-step method for gradient approximation is that it entails no subtraction cancellation error, and therefore the truncation error can be made arbitrarily small. The method is applied to two robust performance analysis problems and is shown to provide more accurate solutions and improved convergence times when compared with standard finite difference-based approaches.

The complex-step derivative approximation

The complex-step derivative approximation and its application to numerical algorithms are presented. Improvements to the basic method are suggested that further increase its accuracy and robustness and unveil the connection to algorithmic differentiation theory. A general procedure for the implementation of the complex-step method is described in detail and a script is developed that automates its implementation. Automatic implementations of the complex-step method for Fortran and C/C++ are presented and compared to existing algorithmic differentiation tools. The complex-step method is tested in two large multidisciplinary solvers and the resulting sensitivities are compared to results given by finite differences. The resulting sensitivities are shown to be as accurate as the analyses. Accuracy, robustness, ease of implementation and maintainability make these complex-step derivative approximation tools very attractive options for sensitivity analysis.