## Interval Computations On The Spreadsheet (1996)

Venue: | Applications of Interval Computations |

Citations: | 6 - 1 self |

### BibTeX

@INPROCEEDINGS{Hyvönen96intervalcomputations,

author = {Eero Hyvönen and Stefano De Pascale},

title = {Interval Computations On The Spreadsheet},

booktitle = {Applications of Interval Computations},

year = {1996},

pages = {169--209},

publisher = {Kluwer}

}

### OpenURL

### Abstract

This paper reviews work on using interval arithmetic as the basis for next generation spreadsheet programs capable of dealing with rounding errors, imprecise data, and numerical constraints. A series of ever more versatile computational models for spreadsheets are presented beginning from classical interval arithmetic and ending up with interval constraint satisfaction. In order to demonstrate the ideas, an actual implementation of each model as a class library is presented and its integration with a commercial spreadsheet program is explained. 1 LIMITATIONS OF SPREADSHEET COMPUTING Spreadsheet programs, such as MS Excel, Quattro Pro, Lotus 1--2--3, etc., are among the most widely used applications of computer science. Since the pioneering days of VisiCalc and others, spreadsheet programs have been enhanced immensely with new features. However, the underlying computational paradigm of evaluating arithmetical functions by using ordinary machine arithmetic has remained the same. The wor...

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Citation Context ...used. A reliable value for U 1 cannot hence be computed by using ordinary arithmetic. However, by using interval analysis reliable bounds for U 1 can be determined. Classical interval arithmetic (IA) =-=[38]-=- alone is not necessarily applicable to this kind of problems because it treats multiple variable instances in the functions independently from each other (here variables R 2 and R 3 have two instance... |

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Citation Context ... wide, even infinite, and the problem of extra width is strongly emphasized. In order to evaluate the actual range of an interval function in the general case, global interval optimization techniques =-=[43, 13]-=- must be employed. Another limitation of classical IA is its restricted notion of interval as a closed finite continuum of values. This may suffice when dealing with rounding error bounds. However, in... |

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Citation Context ...tries to find a feasible solution (output values) by iteratively guessing argument values (input values). Problems such as (1.4) can be approached by using interval constraint satisfaction techniques =-=[8, 10, 15, 17]-=- in which function sets (or more generally equation sets) are treated as symmetric relational constraints without committing to any input--output distinction. In addition to using the trial--and--erro... |

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Citation Context ...nd tolerance (interval) propagation techniques for determining local and, especially, stricter than local interval solutions in an efficient way. Other implementations for bounding solutions, such as =-=[41, 34, 35, 45, 3, 37]-=- do not make use of algebraic or global interval techniques but rely on local numerical ones. 4.3 Interval Constraint Excel As a demonstration, a version of ICE has been integrated with MS Excel [25].... |

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Citation Context ...tries to find a feasible solution (output values) by iteratively guessing argument values (input values). Problems such as (1.4) can be approached by using interval constraint satisfaction techniques =-=[8, 10, 15, 17]-=- in which function sets (or more generally equation sets) are treated as symmetric relational constraints without committing to any input--output distinction. In addition to using the trial--and--erro... |

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Citation Context ...tries to find a feasible solution (output values) by iteratively guessing argument values (input values). Problems such as (1.4) can be approached by using interval constraint satisfaction techniques =-=[8, 10, 15, 17]-=- in which function sets (or more generally equation sets) are treated as symmetric relational constraints without committing to any input--output distinction. In addition to using the trial--and--erro... |

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Citation Context ...epresentation. By using more bits, less overflows are likely to happen but the problem still remains. The selection can also be made dynamically, like in C--XSC [30]. In many computer algebra systems =-=[9]-=-, arbitrarily large integers and precise rational numbers can be used. The price to be paid for such precision is of course increase in computational complexity. When using computationally more effici... |

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Citation Context ...ed for evaluating the actual value range of an interval function, i.e., for its global evaluation. Best--known approaches are the various numerical branch--and--bound algorithms discussed in depth in =-=[42, 43, 13]-=-. This paper considers these techniques mainly from the practical user's view point. We proceed by presenting a global function evaluator library GIA InC++ [21, 24] in whose design practical issues en... |

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Citation Context ....3) could be computed as a function of the others. This kind of "symmetric" computations where any variable can be used as input or output, can be performed by using constraint propagation t=-=echniques [47, 36]-=-. The first spreadsheet programs using constraint propagation were released already in the early 80's. TK!Solver [31] was a well--known commercial implementation. In [15, 17] four major problems of nu... |

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Citation Context ... by embedding it into programming tools that include: 1. Programming languages . Most interval arithmetic implementations such as Pascal--SC [6], Fortran--SC [5], PBasic [1], VPI [12] and BNR--Prolog =-=[41]-=- can be seen as extensions or additional libraries for various programming languages. The implementations are intended for programmers. 2. Computer algebra packages . Recently, interval extensions to ... |

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Citation Context ....3) could be computed as a function of the others. This kind of "symmetric" computations where any variable can be used as input or output, can be performed by using constraint propagation t=-=echniques [47, 36]-=-. The first spreadsheet programs using constraint propagation were released already in the early 80's. TK!Solver [31] was a well--known commercial implementation. In [15, 17] four major problems of nu... |

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Citation Context ...cuit. the first function). As a result, there is usually too much width in the results. In spite of this difficulty, classical IA can be useful when intervals are narrow estimates for rounding errors =-=[33]-=- because then the extra width accumulated into the results often remains tolerable. However, in this paper we see interval arithmetic in the first place as a general formalism for representing uncerta... |

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Citation Context ...raic optimization in GIA InC++ are discussed in some more detail. Numerical evaluation GIA's numerical evaluation is based on the numerical branch--and--bound (BB) search scheme introduced by Skelboe =-=[46]-=- and developed further by others [2, 42, 43, 40]. Roughly speaking, the idea is to consider argument intervals in ever smaller and more precise parts. Efficiency is gained by identifying and pruning i... |

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Citation Context ...ch was developed into a new generalized computational paradigm for spreadsheet computing in [16, 18]. Other approaches to interval constraint satisfaction based on related techniques are discussed in =-=[28, 3, 32]-=-. Application of interval constraints in evaluating actual function value ranges is discussed in [7]. Interval constraint spreadsheets have several attractions when compared with ordinary spreadsheet:... |

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Citation Context ...feasible regions from the variable intervals [10]. By this technique local solutions can be found. This technique is also employed in interval constraint implementations for logic programming such as =-=[41, 45]-=-. Another possibility is to generalize classical value propagation techniques [47] for interval arithmetic, i.e., to use tolerance propagation [15, 17]. In this technique, the structure of the constra... |

18 |
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Citation Context ...scussed in some more detail. Numerical evaluation GIA's numerical evaluation is based on the numerical branch--and--bound (BB) search scheme introduced by Skelboe [46] and developed further by others =-=[2, 42, 43, 40]-=-. Roughly speaking, the idea is to consider argument intervals in ever smaller and more precise parts. Efficiency is gained by identifying and pruning irrelevant subintervals from consideration as ear... |

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Citation Context ...c has usually been implemented by embedding it into programming tools that include: 1. Programming languages . Most interval arithmetic implementations such as Pascal--SC [6], Fortran--SC [5], PBasic =-=[1]-=-, VPI [12] and BNR--Prolog [41] can be seen as extensions or additional libraries for various programming languages. The implementations are intended for programmers. 2. Computer algebra packages . Re... |

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Citation Context ...nd tolerance (interval) propagation techniques for determining local and, especially, stricter than local interval solutions in an efficient way. Other implementations for bounding solutions, such as =-=[41, 34, 35, 45, 3, 37]-=- do not make use of algebraic or global interval techniques but rely on local numerical ones. 4.3 Interval Constraint Excel As a demonstration, a version of ICE has been integrated with MS Excel [25].... |

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Citation Context ...x; y [ = fa j a ! x or asyg By using complements, IA rules can be extended for some discontinuous situations not defined in classical IA, such as division by an interval containing zero. For example: =-=[3; 6]-=- [\Gamma1; 3] = ] \Gamma 3; 1[ Interval Computations on the Spreadsheet 9 Discontinuous intervals The complement is a simple case of the more general notion of discontinuous or multi--interval . A dis... |

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Citation Context ...al arithmetic has usually been implemented by embedding it into programming tools that include: 1. Programming languages . Most interval arithmetic implementations such as Pascal--SC [6], Fortran--SC =-=[5]-=-, PBasic [1], VPI [12] and BNR--Prolog [41] can be seen as extensions or additional libraries for various programming languages. The implementations are intended for programmers. 2. Computer algebra p... |

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Citation Context ...rogramming languages. The implementations are intended for programmers. 2. Computer algebra packages . Recently, interval extensions to various computer algebra packages such as Maple and Mathematica =-=[29]-=- have appeared. These implementations are intended also mainly for programmers but here in a conceptually higher level mathematical language than in (1). We argue that spreadsheets provide one more na... |

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Citation Context ...ally been implemented by embedding it into programming tools that include: 1. Programming languages . Most interval arithmetic implementations such as Pascal--SC [6], Fortran--SC [5], PBasic [1], VPI =-=[12]-=- and BNR--Prolog [41] can be seen as extensions or additional libraries for various programming languages. The implementations are intended for programmers. 2. Computer algebra packages . Recently, in... |

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Citation Context ...PREADSHEET Interval arithmetic has usually been implemented by embedding it into programming tools that include: 1. Programming languages . Most interval arithmetic implementations such as Pascal--SC =-=[6]-=-, Fortran--SC [5], PBasic [1], VPI [12] and BNR--Prolog [41] can be seen as extensions or additional libraries for various programming languages. The implementations are intended for programmers. 2. C... |

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Citation Context ...ahan [27]. Our extended IA is a full-- scale development of the idea. All interval types of table 1 can be used with interval functions. The scheme has been implemented as C++ class library LIA InC++ =-=[20]-=-. The library contains interval classes and overloads arithmetic operators and functions for them. The library covers ordinary rational, trigonometric, exponential, logarithmic and many other function... |

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Citation Context ...nd can be applied to other similar systems as well. For reasons of portability, each interval extension is based on a general purpose interval arithmetic C++ class library of the InC++ library family =-=[24]-=- that has been integrated with the spreadsheet by an interface. The functionalities of the libraries can be incorporated into other spreadsheet systems by rewriting this interface only. Following chap... |

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Citation Context ...scussed in some more detail. Numerical evaluation GIA's numerical evaluation is based on the numerical branch--and--bound (BB) search scheme introduced by Skelboe [46] and developed further by others =-=[2, 42, 43, 40]-=-. Roughly speaking, the idea is to consider argument intervals in ever smaller and more precise parts. Efficiency is gained by identifying and pruning irrelevant subintervals from consideration as ear... |

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Citation Context ...approaches to interval constraint satisfaction based on related techniques are discussed in [28, 3, 32]. Application of interval constraints in evaluating actual function value ranges is discussed in =-=[7]-=-. Interval constraint spreadsheets have several attractions when compared with ordinary spreadsheet: Interval arithmetic makes it possible to deal with inexact data and rounding errors. Infinite solut... |

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Citation Context ...erval constraint techniques can also support top--down problem solving in which a feasible solution is searched for by stepwise refining the variable intervals of the problem until a solution emerges =-=[16, 18]-=-. 1.4 Outline of research In this paper it is shown how interval techniques can be used for extending the usage of ordinary spreadsheets into the three directions discussed above. As a demonstration o... |

5 | InC++: A library for interval constraint equations
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Citation Context ...roblem should not be underconInterval Computations on the Spreadsheet 31 strained. In our work, local and some global tolerance propagation algorithms have been implemented as a C++ library ICE InC++ =-=[23]-=-. By default, this system uses local propagation but individual constraints are solved globally. If desired, additional global solution functions can be generated. ICE is designed to be used as a gene... |

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Citation Context ...lgorithms discussed in depth in [42, 43, 13]. This paper considers these techniques mainly from the practical user's view point. We proceed by presenting a global function evaluator library GIA InC++ =-=[21, 24]-=- in whose design practical issues encountered in spreadsheet--like computations have been the main concern. Figure 3 Architecture of function objects (IFunction) in GIA InC++ library. GIA provides the... |

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Citation Context ...tion but individual constraints are solved globally. If desired, additional global solution functions can be generated. ICE is designed to be used as a general purpose ICSP library in C++ programming =-=[26]-=-. From the programmer 's view point, it is implemented as a C++ class Ice. Objects of this class correspond to ICSPs. Communication with Ice--objects is based on simple member functions that define IC... |

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Citation Context ...s input or output, can be performed by using constraint propagation techniques [47, 36]. The first spreadsheet programs using constraint propagation were released already in the early 80's. TK!Solver =-=[31]-=- was a well--known commercial implementation. In [15, 17] four major problems of numerical constraint propagation based on ordinary arithmetic were discussed: The systems cannot deal with imprecise da... |

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Citation Context ...erval constraint techniques can also support top--down problem solving in which a feasible solution is searched for by stepwise refining the variable intervals of the problem until a solution emerges =-=[16, 18]-=-. 1.4 Outline of research In this paper it is shown how interval techniques can be used for extending the usage of ordinary spreadsheets into the three directions discussed above. As a demonstration o... |

3 | Evaluation of Cascaded Interval Function Constraints
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Citation Context ...nternal state on. Such reuse of earlier work often dramatically shortens needed computational time. Interval Computations on the Spreadsheet 23 3.2 Global evaluation of an interval function system In =-=[19]-=- a technique is developed for evaluating a function system globally. The hearth of the technique is the simple algorithm Local--Function--Evaluation below that implements cascaded function evaluation ... |

3 | CIF InC++: A library for cascaded interval function evaluation
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Citation Context ...e local evaluation is sufficient. 3.3 Implementation on a spreadsheet The global cascaded evaluation scheme has been implemented as a C++ library called CIF InC++ (Cascaded Interval Functions In C++) =-=[22]. The libr-=-ary provides the user with a class "Cif" for defining, maintaining, parametrizing and evaluating a function system. The main input of Cif--objects is functions and argument interval values i... |

2 |
Interval constraint spreadsheets: An implementation for MS Excel
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Citation Context ..., 37] do not make use of algebraic or global interval techniques but rely on local numerical ones. 4.3 Interval Constraint Excel As a demonstration, a version of ICE has been integrated with MS Excel =-=[25]-=-. This chapter discusses modifications that need to be made in a spreadsheet program such as MS Excel when extending it with interval constraints. Important issues are reviewed mainly from the user's ... |

2 |
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Citation Context ...te results the growth of constituent number in practise usually gets smaller. The idea of extending IA rules for the discontinuous case has been suggested probably first [13] by Hanson [14] and Kahan =-=[27]-=-. Our extended IA is a full-- scale development of the idea. All interval types of table 1 can be used with interval functions. The scheme has been implemented as C++ class library LIA InC++ [20]. The... |

2 |
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Citation Context ...nd tolerance (interval) propagation techniques for determining local and, especially, stricter than local interval solutions in an efficient way. Other implementations for bounding solutions, such as =-=[41, 34, 35, 45, 3, 37]-=- do not make use of algebraic or global interval techniques but rely on local numerical ones. 4.3 Interval Constraint Excel As a demonstration, a version of ICE has been integrated with MS Excel [25].... |

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1 |
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Citation Context ...by using the macro languages and other programming devices provided by current spreadsheet products. In this chapter, an interval extension called Extended Interval Arithmetic on MS Excel (EIA Excel) =-=[11]-=- is presented. The primary goal of EIA Excel is to provide the user with a notion of interval arithmetic in as versatile form as possible. For this reason, we first review properties of the extended I... |

1 |
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Citation Context ...ch was developed into a new generalized computational paradigm for spreadsheet computing in [16, 18]. Other approaches to interval constraint satisfaction based on related techniques are discussed in =-=[28, 3, 32]-=-. Application of interval constraints in evaluating actual function value ranges is discussed in [7]. Interval constraint spreadsheets have several attractions when compared with ordinary spreadsheet:... |

1 |
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(Show Context)
Citation Context ... wide, even infinite, and the problem of extra width is strongly emphasized. In order to evaluate the actual range of an interval function in the general case, global interval optimization techniques =-=[43, 13]-=- must be employed. Another limitation of classical IA is its restricted notion of interval as a closed finite continuum of values. This may suffice when dealing with rounding error bounds. However, in... |

1 |
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Citation Context ...spreadsheet situation: A1 = 10864 B1 = 18817 (1.2) C1 = 9sA14 \Gamma B14 + 2B12 When the system is evaluated by using MS Excel spreadsheet program, value C1 = 2 is obtained although the correct value =-=[44]-=- should be C1 = 1 (!). The Interval Computations on the Spreadsheet 3 problem is due to the rounding errors generated by finite precision machine arithmetic (double precision) used by the system. This... |

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