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### Page 2511 of Mathematical Reviews Vol. , Issue 83f [page]

1983 Mathematical Reviews
This very important paper shows how to replace some of the intervals occurring in the interval arithmetic version of Newton’s method by reals.  ...  The problem is transformed to a nonlinear system of simultaneous equations which allows one to apply the well-known interval arithmetic version of Newton’s method. G. Alefeid (Karlsruhe)  ...

### Interval Mathematics Techniques for Control Theory Computations [chapter]

R. Baker Kearfott
1989 Computation and Control
In both, we define interval arithmetic on complex numbers by identifying a complex intervalc with an ordered pair (a, b) of real intervals; we then define complex interval arithmetic by extending the standard  ...  In particular, interval methods can find, with the rigor of a mathematical proof, even on a computer, all roots of a function in the right half of the complex plane.  ...

### Page 5649 of Mathematical Reviews Vol. , Issue 93j [page]

1993 Mathematical Reviews
Us- ing circular complex interval arithmetic, we study the dependence of the convergence rate of root-finding algorithms on errors origi- nating from the rounding as well as the approximate computations  ...  In addition to real n x n point matrices, we consider complex matrices, interval matrices and the generalized eigenvalue problem.”  ...

### Page 8727 of Mathematical Reviews Vol. , Issue 99m [page]

1999 Mathematical Reviews
Best approximation problems for some elementary problems are treated in Chapter 3.  ...  In summary, it can be said that the book belongs on the desk of every researcher who is using val- idation numerics in the complex field. G.  ...

### Review of introduction to interval computations, by Götz Alefeld and Jürgen Herzberger

L.B. Rall
1986 Linear Algebra and its Applications
of interval arithmetic available in the past.  ...  Furthermore, strides have been made in advance of the implementation of the standard in efficient support for interval arithmetic.  ...  The book covers the fundamentals of real and complex interval arithmetic, methods for finding intervals containing zeros of real functions and real and complex roots of polynomials, methods based on iteration  ...

### Why does information-based complexity use the real number model?

Henryk Woźniakowski
1999 Theoretical Computer Science
This model has been used in algebraic complexity already in the 1950s in the work of *  ...  We also indicate the possible tradeoffs between complexity and stability, and the need of using multiple or varying precision for ill-conditioned problems.  ...  Acknowledgements I thank Erich Novak, Leszek Plaskota, Joseph Traub, Grzegorz Wasilkowski and Arthur Werschulz for many valuable comments on this paper.  ...

### Page 400 of Mathematical Reviews Vol. , Issue 92a [page]

1992 Mathematical Reviews
Together with an interval residual arithmetic, some function evaluation examples are included. This is a fine paper for newcomers in the field. {For the entire collection see MR 91k:00034.} E.  ...  Summary: “Newton’s method is usually considered on R. But A. Cayley investigated a complex version in the ‘Newton-Fourier imaginary problem’.  ...

### Page 535 of Mathematical Reviews Vol. , Issue 97A [page]

1997 Mathematical Reviews
Markov, On the foundations of interval arithmetic (307-313); Yurii 1. Shokin [Yu. I.  ...  Lakeev] and Jifi Rohn, Computational complexity of interval algebraic problems: some are feasible and some are computation- ally intractable—a survey (293-306); Svetoslav M.  ...

### Midpoint-radius interval-based method to deal with uncertainty in power flow analysis

Manuel Marin, Federico Milano, David Defour
2017 Electric power systems research
This paper presents a novel method based on midpoint-radius interval arithmetic to deal with uncertainties in the power ow problem.  ...  Moreover, the proposed formulation is combined with an optimisation problem in order to prevent overestimation of the interval solution while preserving uncertainty.  ...  This can become a major issue to the application of interval arithmetic to real-world problems, especially to non-linear ones.  ...

### Page 739 of Mathematical Reviews Vol. , Issue 83b [page]

1983 Mathematical Reviews
ISBN 0-12-428650-X This book is concerned with arithmetic in the spaces of numerical computation—including real and complex numbers, intervals, complex intervals, and vectors and matrices with such components—and  ...  The author discusses the following problem: One is given a computational algorithm P which may be defined in some programming language, e.g. ALGOL 60.  ...

### "COMMONSENSE" ARITHMETIC REASONING [chapter]

Reid Simmons
Arithmetic reasoning" can range in complexity  ...  Figure 4 presents some of the operators used by Quantity Lattice in doing interval arithmetic.'  ...  Introduction "Cogito Ergo Sum" -I think, therefore I add "Arithmetic reasoning" denotes a broad class of inferences which range in complexity from simple integer arithmetic, such as "l+ 1 = 2," to complex  ...

### Page 5409 of Mathematical Reviews Vol. , Issue 88j [page]

1988 Mathematical Reviews
Interval arithmetic and interval standard functions are needed in searching for self-validating methods of numerical solution of mathematical problems.  ...  In practice, computer arithmetic (in the system S of float- ing point numbers) must be used instead of (real) arithmetic (on R).  ...

### An introduction to the scientific computing language Pascal-SC

L.B. Rall
1987 Computers and Mathematics with Applications
Its most important features are: (i) accurate floating-point arithmetic for real, complex, and interval numbers, vectors, and matrices, with controlled rounding if desired; (ii) the convenience of operator  ...  These basic features of Pascal-SC will be described, together with some applications.  ...  Acknowledgements--The author would like to thank Professor George F. Corliss for reading the manuscript carefully, and a large number of valuable suggestions.  ...

### Page 2847 of Mathematical Reviews Vol. , Issue 2003d [page]

2003 Mathematical Reviews
Summary: “In this note we give some comments on recent results concerning a simultaneous method of fourth order for finding complex zeros in circular interval arithmetic.  ...  A direct interval arithmetic evaluation of a formula for the function yields in general a superset with an error linear in the width of the interval.  ...

### Extending the Range of C-XSC: Some Tools and Applications for the Use in Parallel and Other Environments [chapter]

Markus Grimmer
2009 Lecture Notes in Computer Science
An example of this is presented for the Maple interval package intpakX.  ...  Different versions of parallel verified linear system solvers based on the package are now available.  ...  -Arithmetic Extensions -C-XSC The C-XSC Hansen Arithmetic which represents a generalized interval arithmetic has also been developed lately in the course of the development of a parametric interval linear  ...
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