INTLAB is a toolbox for Matlab supporting real and complex intervals, and vectors, full matrices and sparse matrices over those. It is designed to be very fast. In fact, it is not much slower than the fastest pure floating point algorithms using the fastest compilers available (the latter, of course, without verification of the result). Beside the basic arithmetical operations, rigorous input and output, rigorous standard functions, gradients, slopes and multiple precision arithmetic is included in INTLAB. Portability is assured by implementing all algorithms in Matlab itself with exception of exactly one routine for switching the rounding downwards, upwards and to nearest. Timing comparisons show that the used concept achieves the anticipated speed with identical code on a variety of computers, ranging from PC’s to parallel computers. INTLAB is freeware and may be copied from our home page.

INTLAB — INTerval LABoratory

A classical mathematical proof is constructed using pencil and paper. However, there are many ways in which computers may be used in a mathematical proof. But ‘proof by computer’, or even the use of computers in the course of a proof, is not so readily accepted (the December 2008 issue of the Notices of the American Mathematical Society is devoted to formal proofs by computer).In the following we introduce verification methods and discuss how they can assist in achieving a mathematically rigorous result. In particular we emphasize how floating-point arithmetic is used.

Verification methods: Rigorous results using floating-point arithmetic

https://www.kelidestan.com/forum/attachment.php?aid=735

Interval Monte Carlo methods for structural reliability

This paper discusses important hybrid aspects of chemical processing plants. It is outlined that discrete phenomena occur both on the physical level and in the control of these plants. As the dynamics of the transformations of energy and material are predominantly continuous, large and complex hybrid systems arise. We focus on three different aspects of dealing with such systems: (1) Modeling and simulation of hybrid systems for the design and optimization of plants, controllers and operating strategies. We present powerful simulation environments that have been developed in recent years. (2) Validation of plant instrumentation and discrete controllers. These systems are largely responsible for the safe and economic operation of chemical plants and the protection of the workforce, and the environment. Techniques for the verification of discrete controllers for continuous processes are discussed, which are based on a discrete approximation of the continuous dynamics. (3) Scheduling of batch plants. For plants that are operated in a discontinuous fashion, the timing and sequencing of the operations are very important for the efficient use of the equipment. This leads to large mixed-integer optimization problems. For a typical example, we show how the process and the constraints can be modeled and describe an efficient solution algorithm.

Continuous-discrete interactions in chemical processing plants

Since hybrid embedded systems are pervasive and often safetycritical, guarantees about their correct performance are desirable. The hybrid systems model checker HyTech provides such guarantees and has successfully verified some systems. However, HyTech severely restricts the continuous dynamics of the system being analyzed and, therefore, often forces the use of prohibitively expensive discrete and polyhedral abstractions. We have designed a new algorithm, which is capable of directly verifying hybrid systems with general continuous dynamics, such as linear and nonlinear differential equations. The new algorithm conservatively overapproximates the reachable states of a hybrid automaton by using interval numerical methods. Interval numerical methods return sets of points that enclose the true result of numerical computation and, thus, avoid distortions due to the accumulation of round-off errors. We have implemented the new algorithm in a successor tool to HyTech called HyperTech. We consider three examples: a thermostat with delay, a two-tank water system, and an air-traffic collision avoidance protocol. HyperTech enables the direct, fully automatic analysis of these systems, which is also more accurate than the use of polyhedral abstractions.

/pdf/beyond-hytech-hybrid-systems-analysis-using-interval-2c8ulpiupz.pdf

Beyond HYTECH: Hybrid Systems Analysis Using Interval Numerical Methods

The interval data type is currently not supported in common programming languages. Therefore the implementation of algorithms using interval arithmetic requires special programming environments or at least special libraries. In this paper we present the C++ class library PROFIL which provides a user friendly environment for implementing interval algorithms. The main goals in the design of PROFIL were speed and portability. Therefore all interval operations in PROFIL use BIAS (Basic Interval Arithmetic Subroutines) [16]. BIAS defines a concise and portable interface for the basic scalar, vector, and matrix operations. The interface is independent of a specific interval representation or computation but permits machine specific and fast implementations. Based on this general specification we present an implementation in C using a lower/upper bound representation of intervals and directed roundings. By using few assembler instructions for switching the rounding modes and avoiding sign tests and rounding mode switches wherever possible, the computational costs of the interval operations were reduced significantly. This is especially important for RISC machines, where floating point instructions can be executed in few machine cycles. Comparisons with other interval arithmetic packages show an improvement in speed of about one order of magnitude.

O. Knüppel

Papers

PROFIL/BIAS—A fast interval library