The notion of least-change secant updates is extended to apply to nonsquare matrices in a way app... more The notion of least-change secant updates is extended to apply to nonsquare matrices in a way appropriate for quasi-Newton methods used to solve systems of nonlinear equations that depend on parameters. Extensions of the widely used least-change secant updates for square matrices are given. A local convergence analysis for certain paradigm iterations is outlined as motivation for the use of these updates, and numerical experiments involving these iterations are discussed. Key words, least-change secant updates, quasi-Newton updates, parameter-dependent systems AMS(MOS) subject classification. 65H10 1. Introduction. Quasi-Newton methods are very widely used iterative methods for solving systems of nonlinear algebraic equations. The basic form of a quasi-Newton method for solving F(x) 0, F R n-. Rn, is (1.1) Xk+: =xk-B:F(xk), in which Bk ,. F(xk) E Rnn, the Jacobian (matrix) of F at xk. For practical success, it is usually essential to augment this basic form with procedures for modifying the step-B:F(xk) to ensure progress from bad starting points, but we need not consider such procedures here. For a general reference on all aspects of quasi-Newton methods, see Dennis and Schnabel [11]. The most effective quasi-Newton methods are those in which each successive Bk+: is determined as a least-change secant update of its predecessor Bk. As the name suggests, Bk+l is determined as a least-change secant update of Bk by making the least possible change in Bk (as measured by a suitable matrix norm) which incorporates current secant information (usually expressed in terms of successive x-and F-values) and other available information about the structure of F. There are also notable updates which, strictly speaking, are least-change inverse secant updates obtained in an analogous way by making the least possible change in B-:. When speaking generically of least-change secant updates, we intend to include these. In [10], Dennis and Schnabel precisely formalize the notion of a least-change secant update and show how the most widely used updates can be derived as least-change secant updates. In [12], Dennis and Walker show that least-change secant update methods, i.e., quasi-Newton methods which use least-change secant updates, have desirable convergence properties in general.
The notion of least-change secant updates is extended to apply to nonsquare matrices in a way app... more The notion of least-change secant updates is extended to apply to nonsquare matrices in a way appropriate for quasi-Newton methods used to solve systems of nonlinear equations that depend on parameters. Extensions of the widely used least-change secant updates for square matrices are given. A local convergence analysis for certain paradigm iterations is outlined as motivation for the use of these updates, and numerical experiments involving these iterations are discussed. Key words, least-change secant updates, quasi-Newton updates, parameter-dependent systems AMS(MOS) subject classification. 65H10 1. Introduction. Quasi-Newton methods are very widely used iterative methods for solving systems of nonlinear algebraic equations. The basic form of a quasi-Newton method for solving F(x) 0, F R n-. Rn, is (1.1) Xk+: =xk-B:F(xk), in which Bk ,. F(xk) E Rnn, the Jacobian (matrix) of F at xk. For practical success, it is usually essential to augment this basic form with procedures for modifying the step-B:F(xk) to ensure progress from bad starting points, but we need not consider such procedures here. For a general reference on all aspects of quasi-Newton methods, see Dennis and Schnabel [11]. The most effective quasi-Newton methods are those in which each successive Bk+: is determined as a least-change secant update of its predecessor Bk. As the name suggests, Bk+l is determined as a least-change secant update of Bk by making the least possible change in Bk (as measured by a suitable matrix norm) which incorporates current secant information (usually expressed in terms of successive x-and F-values) and other available information about the structure of F. There are also notable updates which, strictly speaking, are least-change inverse secant updates obtained in an analogous way by making the least possible change in B-:. When speaking generically of least-change secant updates, we intend to include these. In [10], Dennis and Schnabel precisely formalize the notion of a least-change secant update and show how the most widely used updates can be derived as least-change secant updates. In [12], Dennis and Walker show that least-change secant update methods, i.e., quasi-Newton methods which use least-change secant updates, have desirable convergence properties in general.
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