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Time-stepping and preserving orthonormality

Higham, D.J. (1997) Time-stepping and preserving orthonormality. BIT Numerical Mathematics, 37 (1). pp. 24-36. ISSN 0006-3835

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Certain applications produce initial value ODEs whose solutions, regarded as time-dependent 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 time-stepping 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 Gram-Schmidt method). However, the optimal replacement-the one that is closest in the Frobenius norm-is 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 finite-time global error bound fur the ODE solution. Our analysis allows for adaptive time-stepping, where a local error control process is driven by a user-supplied 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.