By Emilio Spedicato (auth.), Christodoulos A. Floudas, Panos M. Pardalos (eds.)

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**Example text**

The class is obtained by deﬁning V = AU, where U is an arbitrary nonsingular matrix in Rn . Indeed at the point xn+1 the satisﬁed Petrov–Galerkin condition is just equivalent to the normal equations of Gauss. If U = P then the orthogonally scaled class is obtained, implying, as already stated in section 2, that the methods of this class can be applied to solve linear least squares (but a suitable modiﬁcation See also ABS Algorithms for Optimization Cholesky Factorization Gauss–Newton Method: Least Squares, Relation to Newton’s Method Generalized Total Least Squares Interval Linear Systems Large Scale Trust Region Problems Large Scale Unconstrained Optimization Least Squares Orthogonal Polynomials Least Squares Problems Linear Programming Nonlinear Least Squares: Newton-type Methods Nonlinear Least Squares Problems Nonlinear Least Squares: Trust Region Methods Orthogonal Triangularization Overdetermined Systems of Linear Equations QR Factorization Solving Large Scale and Sparse Semideﬁnite Programs Symmetric Systems of Linear Equations References 1.

Applications to Quasi-Newton Methods ABS methods have been used to provide the general solution of the quasi-Newton equation, also with the additional conditions of symmetry, sparsity and positive deﬁniteness. While the general solution of only the quasi-Newton equation was already known from [2], the explicit formulas obtained for the sparse symmetric case are new, and so is the way of constructing sparse SPD updates. Let us consider the quasi-Newton equation deﬁning the new approximation to a Jacobian or a Hessian, in the transpose form d > B0 D y > ; (3) where d = x0 x, y = g 0 g.

From this relation, if m = n, one obtains the following semi-explicit factorization of the inverse, with P = Pn , V = V n , L = Ln A 1 D PL 1 V > : (10) For several choices of the matrix V the matrix L is diagonal, hence formula (10) gives a fully explicit factorization of the inverse as a byproduct of the ABS solution of a linear system, a property that does not hold for the classical solvers. It can also be shown that all possible factorizations of the form (10) can be obtained by proper parameter choices in the scaled ABS class, another completeness result.