The authors' survey paper is devoted to the present state of computational methods in linear algebra. Questions discussed are the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, the inverse eigenvalue problem, and more traditional questions such as algebraic eigenvalue problems and the solution of systems with a square matrix (by direct and iterative methods).

Computational methods of linear algebra

A unifying local–semilocal convergence analysis and applications for two-point Newton-like methods in Banach space

There are a wide arrayof smoothing methods available for finding structure in data. A general framework is developed which shows that manyof these can be viewed as a projection of the data, with respect to appropriate norms. The underlying vector space is an unusually large product space, which allows inclusion of a wide range of smoothers in our setup (including manymethods not typicallyconsidered to be projec- tions). We give several applications of this simple geometric interpreta- tion of smoothing. A major payoff is the natural and computationally frugal incorporation of constraints. Our point of view also motivates new estimates and helps understand the finite sample and asymptotic behavior of these estimates.

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A General Projection Framework for Constrained Smoothing

The Lagrangian and Hamiltonian for nonlinear gravity waves in a cylindrical basin are constructed in terms of the generalized co-ordinates of the free-surface displacement, {qn(t)} ≡ q, thereby reducing the continuum-mechanics problem to one in classical mechanics. This requires a preliminary description, in terms of q, of the fluid motion beneath the free surface, which kinematical boundary-value problem is solved through a variational formulation and the truncation and inversion of an infinite matrix. The results are applied to weakly coupled oscillations, using the time-averaged Lagrangian, and to resonantly coupled oscillations, using Poincare's action—angle formulation. The general formulation provides for excitation through either horizontal or vertical translation of the basin and for dissipation. Detailed results are given for free and forced oscillations of two, resonantly coupled degrees of freedom.

Nonlinear surface waves in closed basins

This survey traces the development of extrapolation processes in numerical analysis, dealing mainly with those based on polynomial or rational functions. The more important results are presented in a uniform notation and interconnections between work in different fields are brought out. An extensive bibliography is appended.

Survey of Extrapolation Processes in Numerical Analysis

A criterion for the positivity of a cubic polynomial on a given interval is derived. By means of this result a necessary and sufficient condition is given under which cubicC
1-spline interpolants are nonnegative. Further, since such interpolants are not uniquely determined, for selecting one of them the geometric curvature is minimized. The arising optimization problem is solved numerically via dualization.

Positivity of cubic polynomials on intervals and positive spline interpolation

Die Regula Falsi wird auf Operatoren im Banach-Raum erweitert. Der dazu erforderliche Begriff der Steigung eines Operators ist durch die Forderungen (2.3) bis (2.5) festgelegt. Es folgen Konvergenzsatze, die zum Teil auch Existenzaussagen enthalten und Fehlerabschatzungen ermoglichen. Insbesondere wird gezeigt (s. Satz 3.3), das die Regula Falsi fur hinreichend gute Anfangsnaherungen stets konvergiert, wenn die Inverse der Frechet-Ableitung an der Nullstelle existiert. Die Konvergenzgeschwindigkeit ist dann uberlinear.



Die Anwendung der Untersuchungen auf nichtlineare Gleichungssysteme und numerische Beispiele dazu enthalt der Teil II.



The regula falsi is extended to operators in Banach spaces. The concept “Steigung” of an operator required for this purpose is defined by the postulates (2.3)–(2.5). Convergence theorems are derived some of which imply statements on existence and the possibility of error estimation. In particular, it is shown that the regula falsi always converges for sufficiently good first approximations provided the inverse of the Frechet derivative, taken in the zero, exists. The speed of convergence is then better than linear.



Applications to systems of nonlinear equations as well as numerical examples will be given in Part II.

Eine Übertragung der Regula Falsi auf Gleichungen in Banachräumen I

A necessary and sufficient criterion is presented under which the property of positivity carry over from the data set to rational quadratic spline interpolants. The criterion can always be satisfied if the occuring parameters are properly chosen.

Positive interpolation with rational quadratic splines

Im Zusammenhang mit den Konvergenzsatzen des Teiles I werden hier zwei verwandte Verfahren ((2.10) und (3.1)) zur Auflosung nichtlinearer Gleichungssysteme betrachtet. Beide konnen als Erweiterung der Regula Falsi angesehen werden. Sie konvergieren im Falle nichtsingularer Funktionalmatrix fur die Losung, wenn die Anfangswerte hinreichend nahe bei der Losung liegen. Ihre Ordnung ist gleich (1 + √5)/2, d.h. ihre Konvergenzgeschwindigkeit ist groser als eins. Im numerischen Teil werden auf der Regula Falsi basierende Verfahren zur Tschebyscheff-Approximation und zur Bestimmung komplexer Nullstellen bei Polynomen angegeben und Beispiele dazu vorgefuhrt.



Schlieslich wird in einem kurzen Abschnitt gezeigt, wie die Regula Falsi auf Integralgleichungen ubertragen werden kann.



In connection with the convergence theorems of Part I two related methods ((2.10) and (3.1)) are considered for solving systems of nonlinear equations. Either of them can be regarded as a generalization of the regula falsi. They both converge if the functional matrix is nonsingular and if, moreover, the initial values approximate the solution sufficiently well. Their order equals (1 + √5)/2, i. e., their speed of convergence is greater than 1. In the numerical section the regula falsi is used as a basis for Tschebyscheff approximations and for determining complex zeros of polynomials, illustrated by examples. In a final short section it is shown how the regula falsi can be extended to integral equations.

Eine Übertragung der Regula Falsi auf Gleichungen in Banachräumen II Nichtlineare Gleichungssysteme

The construction of range restricted bivariateC1 interpolants to scattered data is considered. In particular, we deal with quadratic spline interpolation on a Powell-Sabin refinement of a triangulation of the data sites subject to piecewise constant lower and upper bounds on the values of the interpolant. The derived sufficient conditions for the fulfillment of the range restrictions result in a solvable system of linear inequalities for the gradients as parameters, which is separated with respect to the data sites. Since there exists an infinite number of spline interpolants meeting the constraints, the selection of a visually pleasant solution is based on the minimum norm modification of a suitable initial interpolant or on the minimization of the thin plate functional. While the first proposal reduces to the solution of independent local quadratic programs, the second proposal results in a global quadratic optimization problem.

Jochen W. Schmidt

Papers

Positivity of cubic polynomials on intervals and positive spline interpolation

Eine Übertragung der Regula Falsi auf Gleichungen in Banachräumen I

Positive interpolation with rational quadratic splines

Eine Übertragung der Regula Falsi auf Gleichungen in Banachräumen II Nichtlineare Gleichungssysteme

Powell-Sabin splines in range restricted interpolation of scattered data