# An analytical approach to computing step sizes for finite-difference derivatives

 dc.contributor.advisor Ocampo, Cesar dc.creator Mathur, Ravishankar dc.date.accessioned 2012-06-29T19:42:25Z dc.date.available 2012-06-29T19:42:25Z dc.date.created 2012-05 dc.date.issued 2012-06-29 dc.date.submitted May 2012 dc.identifier.uri http://hdl.handle.net/2152/ETD-UT-2012-05-5275 dc.description.abstract Finite-difference methods for computing the derivative of a function with respect to an independent variable require knowledge of the perturbation step size for that variable. Although rules of thumb exist for determining the magnitude of the step size, their effectiveness diminishes for complicated functions or when numerically solving difficult optimization problems. This dissertation investigates the problem of determining the step size that minimizes the total error associated with finite-difference derivative approximations. The total error is defined as the sum of errors from numerical sources (roundoff error) and mathematical approximations (truncation error). Several finite-difference approximations are considered, and expressions are derived for the errors associated with each approximation. Analysis of these errors leads to an algorithm that determines the optimal perturbation step size that minimizes the total error. A benefit of this algorithm is that the computed optimal step size, when used with neighboring values of the independent variable, results in approximately the same magnitude of error in the derivative. This allows the same step size to be used for several successive iterations of the independent variable in an optimization loop. A range of independent variable values for which the optimal step size can safely remain constant is also computed. In addition to roundoff and truncation errors within the finite-difference method, numerical errors within the actual function implementation are also considered. It is shown that the optimal step size can be used to compute an upper bound for these condition errors, without any prior knowledge of the function implementation. Knowledge of a function's condition error is of great assistance during the debugging stages of simulation design. Although the fundamental analysis assumes a scalar function of a scalar independent variable, it is later extended to the general case of a vector function of a vector independent variable. Several numerical examples are shown, ranging from simple polynomial and trigonometric functions to complex trajectory optimization problems. In each example, the step size is computed using the algorithm developed herein, a rule-of-thumb method, and an alternative statistical algorithm, and the resulting finite-difference derivatives are compared to the true derivative where available. dc.format.mimetype application/pdf dc.language.iso eng dc.subject Finite-difference dc.subject Finite dc.subject Difference dc.subject Derivative dc.subject Optimal dc.subject Step-size dc.subject Step dc.subject Size dc.subject Condition dc.subject Error dc.title An analytical approach to computing step sizes for finite-difference derivatives dc.date.updated 2012-06-29T19:42:37Z dc.identifier.slug 2152/ETD-UT-2012-05-5275 dc.contributor.committeeMember Hull, David G. dc.contributor.committeeMember Fowler, Wallace T. dc.contributor.committeeMember Marchand, Belinda dc.contributor.committeeMember Senent, Juan dc.description.department Aerospace Engineering dc.type.genre thesis dc.type.material text thesis.degree.department Aerospace Engineering thesis.degree.discipline Aerospace Engineering thesis.degree.grantor University of Texas at Austin thesis.degree.level Doctoral thesis.degree.name Doctor of Philosophy

## Files in this work

Size: 2.102Mb
Format: application/pdf