a 'sleeper', receiving only modest attention for 50 years before emerging as a cornerstone of communications engineering in more recent times. Roughly one in six of Walsh's 281 publications are included, photographically reproduced. Reproduction is excellent except for one paper from 1918. The book also reproduces three brief papers about Walsh and his work, by W. E. Sewell, D. V. Widder and Morris Marden. The first two were written for a special issue of the SIAM Journal celebrating Walsh's 70th birthday; the third is an obituary.

Matrix iterative analysis (2nd edn), by Richard S. Varga. Springer Series in Computational Mathematics 27. Pp. 358. £55. 2000. ISBN 3 540 66321 5 (Springer Verlag).

Computer Solution of Linear Algebraic Systems. By G. Forsythe and C. B. Moler. Pp. xi, 148. 1967. (Prentice-Hall.)

The smoothing Newton method for solving a system of nonsmooth equations F(x) = 0, which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the kth step, the nonsmooth function F is approximated by a smooth function f(.,∈ κ ), and the derivative of f(.,∈ κ ) at x k is used as the Newton iterative matrix. The merits of smoothing methods and smoothing Newton methods are global convergence and convenience in handling. In this paper, we show that the smoothing Newton method is also superlinearly convergent if F is semismooth at the solution and f satisfies a Jacobian consistency property. We show that most common smooth functions, such as the Gabriel-More function, have this property. As an application, we show that for box constrained variational inequalities if the involved function is P- uniform, the iteration sequence generated by the smoothing Newton method will converge to the unique solution of the problem globally and superlinearly (quadratically).

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Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities

Geometric constructions of iterative functions to solve nonlinear equations

Let X, Y be Banach spaces £1 an C.d F X : —* Y be a nonlinear twice Frechetdifferentiable operator in an open convex domain (1Q C fl. Let us assume thatF'(xo) G C(Y,X) exists at some x € fio, where C(Y, X) is the set of boundedlinear operators from Y into X .Among the third-order methods for solving the equation(1) F{x) = 0we have:• Chebyshev's method [3, 5, 15],'(zn)-

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A family of Chebyshev-Halley type methods in Banach spaces

This paper gives a method for finding sharpa posteriori error bounds for Newton's method under the assumptions of Kantorovich's theorem. On the basis of this method, new error bounds are derived, and comparison is made among the known bounds of Dennis [2], Doring [4], Gragg-Tapia [5], Kantorovich [6, 7], Kornstaedt [9], Lancaster [10], Miel [11–13], Moret [14], Ostrowski [17, 18], Potra [19], and Potra-Ptak [20].

A method for finding sharp error bounds for newton's method under the kantorovic assumptions

Historical developments in convergence analysis for Newton's and Newton-like methods

On the basis of the results obtained in a series of papers [25]–[28], a convergence theorem for Newton’s method in Banach spaces is given, which improves the theorems of Kantorovich [4], Lancaster [8] and Ostrowski [10]. The error bounds obtained improve the recent results of Potra [17].

A convergence theorem for Newton’s method in Banach spaces

In this paper, we consider the iteration for solving the equation where f and g are operators in Banach spaces, f is Fechet differentiable, and A(x) is an approximation for f'(x), while the differentiability of g is not assumed. Under Zabrejko-Nguen type hypotheses, we determine a domain Ω such that starting from any point of Ω the method converges to a solution of the equation

Tetsuro Yamamoto

Papers

A method for finding sharp error bounds for newton's method under the kantorovic assumptions

Historical developments in convergence analysis for Newton's and Newton-like methods

A convergence theorem for Newton’s method in Banach spaces

Convergence domains of certain iterative methods for solving nonlinear equations

Inversion of band matrices