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

We compare the Ostrowski efficiency of some methods for solving systems of nonlinear equations without explicitly using derivatives. The methods considered include the discrete Newton method, Shamanskii’s method, the two-point secant method, and Brown’s methods. We introduce a class of secant methods and a class of methods related to Brown’s methods, but using orthogonal rather than stabilized elementary transformations. The idea of these methods is to avoid finding a new approximation to the Jacobian matrix of the system at each step, and thus increase the efficiency. Local convergence theorems are proved, and the efficiencies of the methods are calculated. Numerical results are given, and some possible extensions are mentioned.

https://maths-people.anu.edu.au/~brent/pd/rpb014a.pdf

Some Efficient Algorithms for Solving Systems of Nonlinear Equations

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

This paper presents a smooth spline interpolation technique for five-axis machining of sculptured surfaces. The tool tip and orientation locations generated by the CAM system are first fitted to quintic splines independently to achieve geometric jerk continuity while decoupling the relative changes in position and orientation of the cutter along the curved path. The non-linear relationship between spline parameters and displacements along the path is approximated with ninth order and seventh order feed correction splines for position and orientation, respectively. The high order feed correction splines allow minimum deviation from the reference axis commands while preserving continuous jerk on three translational and two rotary drives. The proposed method has been experimentally demonstrated to show improvements in reducing the excitation of inertial vibrations while improving tracking accuracy in five-axis machining of curved paths found in dies, molds and aerospace parts.

Smooth trajectory generation for five-axis machine tools

A consistency test for α-functions of cubic equations of state

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

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

Fur Gleichungen in mehrdimensionalen Raumen werden Klassen von ableitungsfreien Verfahren angegeben, welche in der Durchfuhrung allein Steigungen erster Ordnung benotigen. Ein Iterationsschritt besteht ausk Stufen, wobei die einzelnen Stufen die Anwendung der vereinfachten Regula falsi bedeuten. Es ergibt sich, das sich das Maximum des Wirkungsgrades mit wachsender Dimension zu den groserenk-Werten hin verschiebt. Die Konvergenz der Verfahren wird fur einfache Nullstellen und hinreichend gute Startwerte nachgewiesen.

Ableitungsfreie Verfahren mit höherer Konvergenzgeschwindigkeit

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.

Powell-Sabin splines in range restricted interpolation of scattered data

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