Automatic computation of partial derivatives and rounding error estimates with applications to large-scale systems of nonlinear equations
type: Item
instance of: scholarly article
title: Automatic computation of partial derivatives and rounding error estimates with applications to large-scale systems of nonlinear equations
zbMATH Open document ID: 0677.65015
DOI: 10.1016/0377-0427(88)90298-1
author: Masao Iri
author: Takashi Tsuchiya
author: Mamoru Hoshi
published in: Journal of Computational and Applied Mathematics
publication date: 1988-01-01T00:00:00Z
review text: The problem of computing the Jacobian of functions is studied in some detail. The main idea is to evaluate formulas and its (first partial) derivatives by sequences of very simple steps of the form \(v_ i=\phi_ i(u_{i1},u_{i2},...,u_{im})\) where the chain rule is used repeatedly for the derivatives and where the \(\phi_ i\) are predefined basic computations. The order of the computation is described by a graph. The authors distinguish between bottom-up and top-down traversing this graph, where the classical method is represented by the bottom-up traversing. The authors develop absolute and probabilistic error estimates. They perform various tests with respect to running time and rounding errors using involved models from chemistry. They report that their algorithm for computing the accurate Jacobian is six times faster than that by numerical differentiation.
Mathematics Subject Classification ID: 65D25
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 65Yxx
zbMATH DE Number: 4109931
zbMATH Keywords: automatic computation
zbMATH Keywords: partial derivatives
zbMATH Keywords: large-scale systems
zbMATH Keywords: Jacobian
zbMATH Keywords: error estimates
zbMATH Keywords: rounding errors
zbMATH Keywords: algorithm
zbMATH Keywords: numerical differentiation
reviewed by: Gerhard Opfer
MaRDI profile type: Publication
full work available at URL: 0377-0427(88)90298-1
OpenAlex ID: W2064643369
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description: scientific article; zbMATH DE number 4109931