Date of Award


Degree Type


Degree Name

Master of Science


Applied Mathematics


Robert Corless


The idea of backward error analysis is to assess the quality of a numerical solution by regarding it as the exact solution of a nearby problem. This method of error analysis was developed by Wilkinson in the context of numerical linear algebra, and has been extended widely to other areas of numerical analysis. The subject of this work is a reconsideration of the use of backward error analysis for the numerical solution of ordinary differential equations, focusing mainly on initial value problems. The three main types, viz. defect control, shadowing, and the method of modified equations, are surveyed and algorithms for implementing these methods are considered. The asymptotic relationship between the local error, which is normally used to control the step-size of variable step-size numerical methods for initial value problems, and the defect, the difference between the specified problem and the problem exactly solved by the numerical method, is considered. Finally, the advantages of using backward error analysis when using ordinary differential equations to model real world phenomena, including chaotic systems, are considered. In the light of the omnipresence of physical and modeling error, the conditions under which a numerical solution can be regarded as the exact solution to just as valid a problem as the one originally posed are discussed.



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