# Hend Dawood

## Senior Lecturer of Computational Mathematics

Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt. (email)

Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt. (email)

Showing results in 'Publications'. Show all posts

InCLosure (Interval enCLosure) is a Language and Environment for Reliable Scientific Computing. Interval computations are radically different from traditional numerical approximation methods and reliable computing under uncertainty is a key focus for modern research in mathematics, computer science, physics, and engineering. InCLosure is a system for carrying out reliable and self-validated computations in arbitrary precision. From its name, InCLosure (abbreviated as "InCL") focuses on "computing guaranteed interval enclosures", that is, "enclosing the exact real result in an interval".

InCLosure is powerful enough to carry out computations ranging from simple real and interval arithmetic, through symbolic and numeric differentiation to an arbitrary order, real automatic differentiation, and interval automatic differentiation, up to and including interval enclosures of integrals and Taylor model computations. No matter how complicated the problem under consideration is, InCLosure provides arbitrary precisions and reliable interval results that can be as narrow as possible by the computational power of the hosting machine.

InCLosure supports arbitrary precision in both real and interval computations. In real arithmetic, the precision is arbitrary in the sense that it is governed only by the computational power of the machine (default is 20 significant digits). The user can change the default precision according to the requirements of the application under consideration. Since interval arithmetic is defined in terms of real arithmetic, interval computations inherit the arbitrary precision of real arithmetic with an added property that the interval subdivision method is provided with an arbitrary number of subdivisions which is also governed only by the computational power of the machine. The user can get tighter and tighter guaranteed interval enclosures by setting the desired number of subdivisions to cope with the problem at hand.

All the computations defined in terms of real and interval arithmetic (e.g., real and interval automatic differentiation, Taylor model computations, and so forth) inherit the same arbitrary precision.

InCLosure is designed to support both interactive and batch modes. In the InCLosure interactive interface, the user can input an InCL command and see its result before moving on to the next command. InCLosure can also be used in batch mode in which case sequences of InCL commands can be given to InCLosure via InCL input files with the results saved in simple and intuitively formatted output files.

InCLosure is coded entirely in Lisp, arguably the fastest and most powerful language for scientific computations. InCLosure provides a friendly and easy-to-use user interface, a simple and intuitive language, a detailed documentation, and clear and fast results. Anyone can compute with InCLosure.

Interval arithmetic is a fundamental and reliable mathematical machinery for scientific computing and for addressing uncertainty in general. In order to apply interval mathematics to real life uncertainty problems, one needs a computerized (machine) version thereof, and so, this article is devoted to some mathematical notions concerning the algebraic system of machine interval arithmetic. After formalizing some purely mathematical ingredients of particular importance for the purpose at hand, we give formal characterizations of the algebras of real intervals and machine intervals along with describing the need for interval computations to cope with uncertainty problems. Thereupon, we prove some algebraic and order-theoretic results concerning the structure of machine intervals.

keywords: Interval mathematics, Machine interval arithmetic, Outward rounding, Floating-point arithmetic, Machine monotonicity, Dense orders, Orderability of intervals, Symmetricity, Singletonicity, Subdistributive semiring, S-semiring.

InCLosure (Interval enCLosure) is a Language and Environment for Reliable Scientific Computing. InCLosure, provides rigorous and reliable results in arbitrary precision. From its name, InCLosure (abbreviated as "InCL") focuses on "enclosing the exact real result in an interval". The interval result is reliable and can be as narrow as possible.

InCLosure supports arbitrary precision in both real and interval computations. In real arithmetic, the precision is arbitrary in the sense that it is governed only by the computational power of the machine (default is 20 significant digits). The user can change the default precision according to the requirements of the application under consideration. Since interval arithmetic is defined in terms of real arithmetic, interval computations inherit the arbitrary precision of real arithmetic with an added property that the interval subdivision method is provided with an arbitrary number of subdivisions which is also governed only by the computational power of the machine. The user can get tighter and tighter guaranteed interval enclosures by setting the desired number of subdivisions to cope with the problem at hand.

All the computations defined in terms of real and interval arithmetic (e.g., real and interval automatic differentiation) inherit the same arbitrary precision.

InCLosure is written in Lisp, the most powerful and fast language in scientific computations. InCLosure provides easy user interface, detailed documentation, clear and fast results. Anyone can compute with InCLosure.

- Abstract algebra (2)
- Algebra (2)
- Axiomatics (9)
- Book (2)
- Bounding Error (2)
- Cardinality (3)
- Categoricity (4)
- Chapter (1)
- CLisp (2)
- Common Lisp (2)
- Complex analysis (1)
- Computer Algebra (3)
- Computer Science (11)
- Consistency (4)
- Dense Orders (1)
- Dependence logic (2)
- Dissertation (1)
- Errors (1)
- Formal logic (9)
- Hend Dawood (6)
- InCL (5)
- InCLosure (7)
- InCLosure (Interval enCLosure) (3)
- InCLosure v1.0 (1)
- Interval Analysis (16)
- Interval arithmetic (13)
- Interval lattice (1)
- Interval Mathematics (14)
- Lisp (2)
- Mathematics (11)
- Metalogic (9)
- Metamathematics (11)
- Modal intervals (3)
- Model theory (7)
- Order theory (8)
- Ordinal power (2)
- Outward Rounding (1)
- PDF (3)
- Proof theory (7)
- Publication (3)
- Ray tracing (1)
- Real Analysis (9)
- Real Functions (4)
- Reliability (6)
- Rounding Error (2)
- S-Field (1)
- S-field algebra (3)
- S-Semiring (4)
- Semantics (9)
- Set theory (2)
- Singletonicity (1)
- Skolemization (4)
- Software (2)
- Symmetricity (1)
- Thesis (1)
- Total order (3)
- Uncertainty (8)
- Well order (2)