Timestepping and preserving orthonormality
Higham, D.J. (1997) Timestepping and preserving orthonormality. BIT Numerical Mathematics, 37 (1). pp. 2436. ISSN 00063835
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Certain applications produce initial value ODEs whose solutions, regarded as timedependent matrices, preserve orthonormality. Such systems arise in the computation of Lyapunov exponents and the construction of smooth singular value decompositions of parametrized matrices. For some special problem classes, there exist timestepping methods that automatically inherit the orthonormality preservation. However, a more widely applicable approach is to apply a standard integrator and regularly replace the approximate solution by an orthonormal matrix. Typically, the approximate solution is replaced by the factor Q from its QR decomposition (computed, for example, by the modified GramSchmidt method). However, the optimal replacementthe one that is closest in the Frobenius normis given by the orthonormal polar factor. Quadratically convergent iteration schemes can be used to compute this factor. In particular, there is a matrix multiplication based iteration that is ideally suited to modern computer architectures. Hence, we argue that perturbing towards the orthonormal polar factor is an attractive choice, and ne consider performing a fixed number of iterations. Using the optimality property we show that the perturbations improve the departure from orthonormality without significantly degrading the finitetime global error bound fur the ODE solution. Our analysis allows for adaptive timestepping, where a local error control process is driven by a usersupplied tolerance. Finally, using a recent result of Sun, we show how the global error bound tarries through to the case where the orthonormal QR factor is used instead of the orthonormal polar factor.


Item type: Article ID code: 187 Dates: DateEventMarch 1997PublishedSubjects: Science > Mathematics > Electronic computers. Computer science
Science > MathematicsDepartment: Faculty of Science > Mathematics and Statistics Depositing user: Ms Sarah Scott Date deposited: 02 Mar 2006 Last modified: 08 Apr 2024 15:16 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/187