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Division by zero can lead to separation of distinct zeros, though the separation may not be complete; it can be complemented by the bisection method. The various interval methods deliver conservative results as dependencies between the sizes of different intervals extensions are not taken into account. However the dependency problem becomes less significant for narrower intervals. With very wide intervals, it can be helpful to split all intervals into several subintervals with a constant and smaller width, a method known as mincing.
This then avoids the calculations for intermediate bisection steps.
Both methods are only suitable for problems of low dimension. Interval arithmetic can be used in various areas such as set inversion , motion planning , set estimation or stability analysis to treat estimates with no exact numerical value.
Exploring the Extension of Natural Operations on Intervals, Matrices and Complex Numbers
Interval arithmetic is used with error analysis, to control rounding errors arising from each calculation. The advantage of interval arithmetic is that after each operation there is an interval that reliably includes the true result. The distance between the interval boundaries gives the current calculation of rounding errors directly:. Interval analysis adds to rather than substituting for traditional methods for error reduction, such as pivoting.
Interval arithmetic - Wikipedia
Parameters for which no exact figures can be allocated often arise during the simulation of technical and physical processes. The production process of technical components allows certain tolerances, so some parameters fluctuate within intervals. In addition, many fundamental constants are not known precisely.
This provides an alternative to traditional propagation of error analysis.
Unlike point methods, such as Monte Carlo simulation , interval arithmetic methodology ensures that no part of the solution area can be overlooked. However, the result is always a worst-case analysis for the distribution of error, as other probability-based distributions are not considered. Interval arithmetic can also be used with affiliation functions for fuzzy quantities as they are used in fuzzy logic. A distribution function assigns uncertainty, which can be understood as a further interval.
The form of such a distribution for an indistinct value can then represented by a sequence of intervals. Warwick Tucker used interval arithmetic in order to solve the 14th of Smale's problems , that is, to show that the Lorenz attractor is a strange attractor . Thomas Hales used interval arithmetic in order to solve the Kepler conjecture. Interval arithmetic is not a completely new phenomenon in mathematics; it has appeared several times under different names in the course of history.
Actual calculation with intervals has neither been as popular as other numerical techniques nor been completely forgotten. Rules for calculating with intervals and other subsets of the real numbers were published in a work by Rosalind Cicely Young, a doctoral candidate at the University of Cambridge . Arithmetic work on range numbers to improve the reliability of digital systems were then published in a textbook on linear algebra by Paul Dwyer University of Michigan  ; intervals were used to measure rounding errors associated with floating-point numbers.
A comprehensive paper on interval algebra in numerical analysis was published by Teruo Sunaga The birth of modern interval arithmetic was marked by the appearance of the book Interval Analysis by Ramon E. Moore in Independently in , Mieczyslaw Warmus suggested formulae for calculations with intervals,  though Moore found the first non-trivial applications. In the s, Eldon R. Hansen dealt with interval extensions for linear equations and then provided crucial contributions to global optimisation, including what is now known as Hansen's method, perhaps the most widely used interval algorithm.
In , Rudolf Lohner developed Fortran -based software for reliable solutions for initial value problems using ordinary differential equations. The journal Reliable Computing originally Interval Computations has been published since the s, dedicated to the reliability of computer-aided computations. As lead editor, R. Baker Kearfott, in addition to his work on global optimisation, has contributed significantly to the unification of notation and terminology used in interval arithmetic Web : Kearfott.
There are many software packages that permit the development of numerical applications using interval arithmetic. There followed in , Pascal-SC , a Pascal variant on a Zilog Z80 that it made possible to create fast, complicated routines for automated result verification. At the beginning of C-XSC 2. It emphasized the efficient use of hardware, portability and independence of a particular presentation of intervals. The Frink programming language has an implementation of interval arithmetic that handles arbitrary-precision numbers. Programs written in Frink can use intervals without rewriting or recompilation.
The Julia programming language  has an implementation of interval arithmetics along with high-level features, such as root-finding for both real and complex-valued functions and interval constraint programming , via the ValidatedNumerics. A library for the functional language OCaml was written in assembly language and C. A standard for interval arithmetic has been approved in June It should be easier to implement and may speed production of implementations . Several international conferences or workshop take place every year in the world.
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Wissenschaftliches Rechnen mit Ergebnisverifikation. Wiesbaden: Vieweg-Verlag. Aachen, Germany: Shaker Verlag. William; Hansen, Eldon Robert Global Optimization using Interval Analysis 2nd ed. Applied Interval Analysis. Berlin: Springer. The Lorenz attractor exists. The algebra of many-valued quantities. Mathematische Annalen, 1 , Linear computations.
Oxford, England: Wiley. Theory of interval algebra and its application to numerical analysis. RAAG Memoirs. Interval Analysis. Baker Introduction to Interval Analysis.
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University of Louisiana at Lafayette Press. Retrieved Reihe Informatik in German. In Laugwitz, Detlef ed. Mannheim, Germany: Bibliographisches Institut. The near- future IEEE standard for interval arithmetic. IEEE Standard. Data types. Address physical virtual Reference. Character String null-terminated. See also platform-dependent and independent units of information.