Computation of normal form coefficients of cycle bifurcations of maps by algorithmic differentiation

Authors:
J. D. Pryce;R. Khoshsiar Ghaziani;V. De Witte;W. Govaerts
Affiliations:
Department of Information Systems, Cranfield University, Shrivenham Campus SN6 8LA, UK;Department of Mathematics, Shahrekord University, P.O. Box 115, Shahrekord, Iran;Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium;Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium
Venue:
Mathematics and Computers in Simulation
Year:
2010

Citing 9
Cited 0

Automatic differentiation in MATLAB

Applied Numerical Mathematics
The Efficient Computation of Sparse Jacobian Matrices Using Automatic Differentiation

SIAM Journal on Scientific Computing
Evaluating derivatives: principles and techniques of algorithmic differentiation

Evaluating derivatives: principles and techniques of algorithmic differentiation
ADMIT-1: automatic differentiation and MATLAB interface toolbox

ACM Transactions on Mathematical Software (TOMS)
ADMIT-1 : Automatic Differentiation and MATLAB Interface Toolbox

ADMIT-1 : Automatic Differentiation and MATLAB Interface Toolbox
Structured automatic differentiation

Structured automatic differentiation
Numerical Normal Forms for Codim 2 Bifurcations of Fixed Points with at Most Two Critical Eigenvalues

SIAM Journal on Scientific Computing
Numerical Methods for Two-Parameter Local Bifurcation Analysis of Maps

SIAM Journal on Scientific Computing
Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation

Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation

Quantified Score

Hi-index	0.00

Visualization

Abstract

As an alternative to symbolic differentiation (SD) and finite differences (FD) for computing partial derivatives, we have implemented algorithmic differentiation (AD) techniques into the Matlab bifurcation software Cl_MatcontM, http://sourceforge.net/projects/matcont, where we need to compute derivatives of an iterated map, with respect to state variables. We use derivatives up to the fifth order, of the iteration of a map to arbitrary order. The multilinear forms are needed to compute the normal form coefficients of codimension-1 and -2 bifurcation points. Methods based on finite differences are inaccurate for such computations. Computation of the normal form coefficients confirms that AD is as accurate as SD. Moreover, elapsed time in computations using AD grows linearly with the iteration number J, but more like J^d for d th derivatives with SD. For small J, SD is still faster than AD.