15,947 research outputs found
A new orthogonalization procedure with an extremal property
Various methods of constructing an orthonomal set out of a given set of
linearly independent vectors are discussed. Particular attention is paid to the
Gram-Schmidt and the Schweinler-Wigner orthogonalization procedures. A new
orthogonalization procedure which, like the Schweinler- Wigner procedure, is
democratic and is endowed with an extremal property is suggested.Comment: 7 pages, latex, no figures, To appear in J. Phys
Discrimination between evolution operators
Under broad conditions, evolutions due to two different Hamiltonians are
shown to lead at some moment to orthogonal states. For two spin-1/2 systems
subject to precession by different magnetic fields the achievement of
orthogonalization is demonstrated for every scenario but a special one. This
discrimination between evolutions is experimentally much simpler than
procedures proposed earlier based on either sequential or parallel application
of the unknown unitaries. A lower bound for the orthogonalization time is
proposed in terms of the properties of the two Hamiltonians.Comment: 7 pages, 2 figures, REVTe
Incomplete Orthogonal Factorization Methods Using Givens Rotations II: Implementation and Results
We present, implement and test a series of incomplete orthogonal factorization methods based on Givens rotations for large sparse unsymmetric matrices. These methods include: column-Incomplete Givens Orthogonalization (cIGO-method), which drops entries by position only; column-Threshold Incomplete Givens Orthogonalization (cTIGO-method) which drops entries dynamically by both their magnitudes and positions and where the reduction via Givens rotations is done in a column-wise fashion; and, row-Threshold Incomplete Givens Orthogonalization (r-TIGO-method) which again drops entries dynamically, but only magnitude is now taken into account and reduction is performed in a row-wise fashion. We give comprehensive accounts of how one would code these algorithms using a high level language to ensure efficiency of computation and memory use. The methods are then applied to a variety of square systems and their performance as preconditioners is tested against standard incomplete LU factorization techniques. For rectangular matrices corresponding to least-squares problems, the resulting incomplete factorizations are applied as preconditioners for conjugate gradients for the system of normal equations. A comprehensive discussion about the uses, advantages and shortcomings of these preconditioners is given
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