On the Representation of a Basis for the Null Space.
STANFORD UNIV CA SYSTEMS OPTIMIZATION LAB
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Given a rectangular matrix Ax that depends on the independent variables x, many constrained optimization methods involve computations with Z8x, a matrix whose columns form a basis for the null space of Ax. When A is evaluated at a given point, it is well known that a Coleman and Sorensen have recently shown that standard orthogonal factorization methods may produce orthogonal bases that od not vary continuously with x they also suggest several techniques for adapting standard factorization schemes so as to ensure continuity of Z in the neighborhood of a given point. In this note, the authors discuss several aspects of the representation of a basis for the null space. They describe how an explicit matrix Z can be obtained at any point using a method for updating a factorization with either Householder or stabilized elementary transformations. Under a mild non-singularity assumption, the elements of Z are continuous functions of x. They also show that the chosen form of Z is convenient and efficient when implementing certain methods for nonlinearly constrained optimization. Author
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