Representation of a basis for the null space

Abstract : Given a rectangular matrix A(x) 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 A(x). 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)