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Solving quadratically constrained least squares problems using a differential-geometric approach
Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Beräkningsvetenskap.ORCID-id: 0000-0003-2281-856X
2002 (Engelska)Ingår i: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 42, nr 2, s. 323-335Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

A quadratically constrained linea least squares problem is usually solved using a Lagrange multiplier for the constraint and then solving iteratively a nonlinear secular equation for the optimal Lagrange multiplier. It is well-known that, due to the closeness to a pole for the secular equation, standard methods for solving the secular equation can be slow, and sometimes it is not easy to select a good starting value for the iteration. The problem can be reformulated as that of minimizing the residual of the least squares problem on the unit sphere. Using a differential-geometric approach we formulate Newton's method on the sphere, and thereby avoid the difficulties associated with the Lagrange multiplier formulation. This Newton method on the sphere can be implemented efficiently, and since it is easy to find a good starting value for the iteration, and the convergence is often quite fast, it has a clear advantage over the Lagrange multiplier method. A numerical example is given.

Ort, förlag, år, upplaga, sidor
2002. Vol. 42, nr 2, s. 323-335
Nyckelord [en]
ill-conditioned, Lagrange multiplier, least squares, Newton's method, quadratic constraint, Stiefel manifold
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Identifikatorer
URN: urn:nbn:se:liu:diva-48903OAI: oai:DiVA.org:liu-48903DiVA, id: diva2:269799
Tillgänglig från: 2009-10-11 Skapad: 2009-10-11 Senast uppdaterad: 2017-12-12

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