Kulisch, U. and Miranker, W. L.: Computer Arithmetic in Theory and Practice. Academic Press, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....feasible [12] In both cases we deal with interval uncertainty, i.e. we do not know exact values of parameters, only intervals where the values of these parameters belong to. A number of methods to define operations on intervals that produce guaranteed precision have been developed in [21] 22] [18], and [1] among others. In many real life cases there is also some noise which does not allow direct measurement of parameters. To get rid of this noise it is necessary to subtract its value from the result of measurement. The noise can be considered as an undesirable effect to the evaluation of ....

Kulisch U., Miranker W., Computer Arithmetic in Theory and Practice, Academic Press, 1981.

....errors and it is better to control or even to try to avoid them. For this purpose, several ways have been explored: correction of the arithmetic [202] p adic arithmetic [121, 122] multiprecision arithmetic, and interval arithmetic with guaranteed and sharp inclusion regions for the exact result [161]. There is another notion related to the nite precision of the computer: it is the concept of condition number which measures the sensitivity of the exact result of a mathematical problem to variations in the data. This concept should not be confused with the numerical stability of an algorithm ....

U.W. Kulisch, W.L. Miranker, Computer Arithmetic in Theory and Practice, Academic press, New York, 1981.

....to be deterministic if no stochastic hypothesis is assumed; otherwise we consider it to be stochastic. According to this characterization, we brie y review the main automatic rounding error methods. Introduced in 1966 by Moore [26] and widely developed and automated by Kulisch and his colleagues [16], interval analysis is a forward deterministic approach. The propagation of rounding errors is simulated using interval arithmetic. Each interval contains the exact solution and its range quanti es the e ect of rounding errors propagation. The classical drawback is the growth of the interval ....

U. W. Kulisch and W. L. Miranker, Computer Arithmetic in Theory and in Practice, Academic Press, New York, 1981.

....b are two intervals and o 6 , is an interval arithmetic operation, then the machine arithmetic counter part will give a result a b which satisfies ab Daob. With the rounding modes available in the IEEE Standard 754 , the result a b can in addition be made closest possible to aob, see [16] or [3, Ch. 4] We again distinguish between the theoretical algorithm and the machine algorithm. Due to the containment property, it is possible to identify crucial relations which, when observed for some output of the machine algorithm, necessarily also hold for the corresponding output of the ....

U. Kulisch and W. Miranker. Computer Arithmetic in Theory and Practice. Academic Press, New York, 1981.

....numbers on a computer and F = f[a IR j a 2 Fg is the set of generalised intervals over F. Denote by 5, resp. 4 the floating point directed roundings (5; 4 : R Gamma F) towards Gamma1, resp. 1, defined in the IEEE binary (or radix independent) floating point standard or in [4]. Outward ( Sigma) and inward (fl) roundings Sigma; fl : I IR Gamma I F are defined for generalised intervals by a semimorphism the same way as for normal intervals [4] For [a] a IR, Sigma[a] 5a ] a] fl[a] 4a ] a] If ffi 2 f ; Gamma; Theta; g is an ....

....R Gamma F) towards Gamma1, resp. 1, defined in the IEEE binary (or radix independent) floating point standard or in [4] Outward ( Sigma) and inward (fl) roundings Sigma; fl : I IR Gamma I F are defined for generalised intervals by a semimorphism the same way as for normal intervals [4]. For [a] a IR, Sigma[a] 5a ] a] fl[a] 4a ] a] If ffi 2 f ; Gamma; Theta; g is an arithmetic operation in I IR, the corresponding computer operations Sigma ; fl ffi : I F Theta I F Gamma I F are defined by [a] Sigma [b] Sigma ( a] ....

Kulisch, U., Miranker, W. L.: Computer Arithmetic in Theory and Practice. Academic Press, New York 1981.

.... [ARITH] Related collections of research papers: Ada93, Alb93] Collected algorithms: Ham93, Ham95, Kra96] ffl Computer arithmetic for verification methods (interval operations, exact dot product) introduction [Loh92a, Hof93, Wol90a, Mei87] foundations [Kul75, Kul76, Kul76a, Kul77, Kul77a, Kul81, Kul83a, Kul84] software implementation [Apo67, Wip68, Boh78, Gru79, Gru80, Boh82, Boh82a, Boh83, Boe85, Rum85, Sue86, Pfa89, Pri91, Teu92, Kob94, Wol94] library for FORTRAN 8x [Boh84, Die85, Ull85] library for Ada [Fis85, Kla85, Kla86, Kla86a, Ull87, Erl88] portable implementation in C ....

Kulisch, U.; Miranker, W. L.: Computer Arithmetic in Theory and Practice. Academic Press, New York, 1981 (ISBN 0-12-428650-x).

....in our algorithm are performed in interval arithmetic. Remark: On a computer, a guaranteed and optimal enclosure of the real triangular form of the polynomial system can be obtained by using an exact machine interval arithmetic with optimal outwardly directed rounding (see [10] 14] and [15] for details) 2.3.2 Interval Versions of S polynomial Construction and M reduction Now we develop an interval version of Buchberger s algorithm in order to get a guaranteed enclosure of the exact triangular system and later guaranteed enclosures of all zeros of the system. We use the same ....

Kulisch, U., Miranker, W. L.: Computer Arithmetic in Theory and Practice. Academic Press, New York, 1981.

....with directed roundings which are directly accessable in the language. Further arithmetic operations for intervals and complex numbers and even vector matrix operations provided by precompiled arithmetical modules are defined with maximum accuracy according to the rules of semimorphism (see [25]) 2 The Language PASCAL XSC PASCAL XSC is an eXtension of the programming language PASCAL for Scientific Computation. A first approach to such an extension (PASCAL SC) has been available since 1980. The specification of the extensions has been continuously improved in recent years by means ....

....with the dimension of the dynamic arrays as a transfer parameter. 2. 6 Accurate Expressions The implementation of enclosure algorithms with automatic result verification or validation (see [17] 24] 28] 33] makes extensive use of the accurate evaluation of dot products with the property (see [25]) RG) a K b : fl n X i=1 a i Delta b i ; fl 2 f2; 4; 5 g; n 2 IN : To evaluate this kind of expression the new datatype dotprecision was introduced. This datatype accomodates the full floating point range with double exponents (see [25] 24] Based upon this type, so called accurate ....

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Kulisch, U. and Miranker, W. L.: Computer Arithmetic in Theory and Practice. Academic Press, New York, 1981.

....with x 2 IR n , X IR n , f : IR n IR, and F : IR n IR the machine interval evaluation of f . The machine interval arithmetic used in our implementation guarantees highest accuracy for the elementary operations and function evaluations due to their definition by semimorphism (Kulisch [7]) In addition to the general rules for the interval operations, our algorithm deals with cases in which an expression a Gamma A=B occurs, with a 2 IR, A; B IR, and 0 2 B. For this special application, an extension of the interval arithmetic operations is introduced (Kaucher [4] Moore [8] ....

Kulisch, U. and Miranker, W. L.: Computer Arithmetic in Theory and Practice. Academic Press, 1981.

.... according to the definitions of computer operations on directed intervals considered in [Durst 1975] and [Kaucher 1973] Symbols 5;4 denote the corresponding monotone downwardly and monotone upwardly directed roundings 5;4 : R Gamma SR, where SR is the set of computer representable real numbers [Kulisch and Miranker 1981], ANSI IEEE 1985] Present specification provides complete information about interval operations on operands involving NaNs (Not a Number) and or signed zero as part of IEEE floating point systems [ANSI IEEE 1985] ANSI IEEE 1987] Some interval arithmetic operations involve algorithmic and ....

Kulisch, U., Miranker, W. L.: "Computer Arithmetic in Theory and Practice"; Academic Press, New York (1981).

.... according to the definitions of computer operations on directed intervals considered in [Durst 1975] and [Kaucher 1973] Symbols 5;4 denote the corresponding monotone downwardly and monotone upwardly directed roundings 5;4 : R Gamma SR, where SR is the set of computer representable real numbers [Kulisch and Miranker 1981], ANSI IEEE 1985] Present specification provides complete information about interval operations on operands involving NaNs (Not a Number) and or signed zero as part of IEEE floating point systems [ANSI IEEE 1985] ANSI IEEE 1987] Some interval arithmetic operations involve algorithmic and ....

Kulisch, U., Miranker, W. L.: "Computer Arithmetic in Theory and Practice"; Academic Press, New York (1981).

....two values for each result [44] 36] 33] The two values correspond to the endpoints of an interval such that true result is guaranteed to be lie on the interval. Scientific programming languages [68] support special instructions and data types for interval arithmetic and exact dot products [49]. Techniques for variable precision arithmetic and interval arithmetic have been combined in the languages PASCAL XSC [40] C XSC [41] ACRITH XSC [70] and VPI [23] These extended scientific languages provide data types and special instructions for variable precision numbers, intervals, complex ....

....accuracy, and reliability of scientific computations. These coprocessors also provide special instructions and hardware support for vectors, matrices, and complex numbers. 3. 1 Dot Product Coprocessors Several hardware designs for accurate, self validating arithmetic support exact dot products [49], in which all calculations are mathematically exact and only a single rounding is performed at the very end. The exact dot product of two vectors X and Y , with n floating point elements is defined as X Delta Y = 2( n X i=1 x i y i ) Techniques for Accurate, Self Validating ....

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U. W. Kulisch and W. L. Miranker. Computer Arithmetic in Theory and in Practice. Academic Press, Inc., 1981. 22 Chapter 1

....to be deterministic if no stochastic hypothesis is assumed; otherwise we consider it to be stochastic. According to this characterization, we brie y review the main automatic rounding error methods. Introduced in 1966 by Moore [26] and widely developed and automated by Kulisch and his colleagues [16], interval analysis is a forward deterministic approach. The propagation of rounding errors is simulated using interval arithmetic. Each interval contains the exact solution and its range quanti es the e ect of rounding errors propagation. The classical drawback is the growth of the interval ....

U. W. Kulisch and W. L. Miranker, Computer Arithmetic in Theory and in Practice, Academic Press, New York, 1981.

....reliable arithmetic have been developed. One approach for improving the accuracy of numerical computations is through the use of coprocessors that support exact dot products. In an exact dot product, all calculations are mathematically exact and only a single rounding is performed at the very end [1 39]. Exact dot products greatly reduce numerical errors due to roundoff and catastrophic cancellation. To facilitate the use of interval arithmetic, dot Michael Schulte product coprocessors typically provide the four rounding modes specified in the IEEE 754 floating point standard [1 40] Exact dot ....

U. W. Kulisch and W. L. Miranker, Computer Arithmetic in Theory and in Practice. Academic Press, 1981.

....both end points of a real interval are not representable (which is often the case) then they must be rounded outward to the closest representable floating point numbers. Interval arithmetic is often called a machine, or rounded, interval arithmetic. A discussion of its properties can be found in [41]. 2.3 Interval Valued Functions Let f : R n R be a continuous function on D R n . We consider functions whose representations contain only a finite number of constants, variables, arithmetic operations, and standard functions (sin, cos, log, exp, etc. We define the range of f over an ....

Ulrich W. Kulisch and Willard L. Miranker. Computer Arithmetic in Theory and Practice. Academic Press, New York, 1981. Bibliography 145

....using the nite precision solution of a given problem. This can be done by means of interval arithmetic. Interval arithmetic can be implemented in a reliable way (the result of an interval arithmetic computation is guaranteed to include the true solution) References to interval arithmetic include [1, 2, 3, 10, 11, 12, 13, 14, 15]. There exist many implementations of interval arithmetic. Some are extensions of existing languages [4, 5, 7] but even new languages including interval arithmetic have been proposed, see for instance [6] A very fast implementation of interval arithmetic is given with the ANSI C library BIAS by ....

U. Kulisch and W.L. Miranker. Computer Arithmetic in Theory and Practice. Academic Press, New York, 1981.

....one product but the problem then is reduced to the computation of a sum. We show that it suffices to compute 5 partial products for a binary system. The algorithms which are introduced in the next chapter have been developped in the framework of the Karlsruhe accurate arithmetic approach [Kul 76, Kul 81] see also [Boh 90] for an overview. We then compare these algorithms and a recently published method [Kob 94] with respect to time and space complexity. We also investigate the requirements for computing accumulated dotproducts, i.e. an addition of a value to an already computed dotproduct ....

Kulisch, U., Miranker, W. L. : Computer Arithmetic in Theory and Practice Academic Press, Orlando, 1981.

....computer algebra deal with the question how to choose a suitable abstract algorithm for a given requirement specification, it is an open problem how to deal in a formal way with the step from an abstract algorithm to a concrete algorithm. For the latter [Moo88] presents some solutions. Based on [KM81] we provide some ideas for a relation between a floating point system specified in Casl and the specification BasicReal in [RSM00] Har99] discusses a different approach to this problem in more detail. 4 Relating BasicReal with Other Theories of IR The standard approaches to the specification of ....

Ulrich W. Kulisch and Willard L. Miranker. Computer Arithmetic in Theory and Practice. Academic Press, 1981.

.... extended by four inner interval operations [7] The obtained extended interval arithmetic structure is suitable for the effective computation of functional ranges reducing their overestimation with ordinary interval arithmetic [2] 4] Additionally, interval arithmetic, with directed roundings [6], can provide mathematically rigorous results from floating point operations on computers. Although the arithmetic operations with directed roundings, specified by IEEE floating point standards [3] are sufficient for the implementation of conventional interval operations with 1 ulp (unit in the ....

.... structure (IR; Gamma ; Theta; Theta Gamma ; its relations to other interval arithmetic extensions and a large list of references can be found in [8] WIDTH AND SYMMETRY BASED IMPLEMENTATION The theory of computer arithmetic defines computer interval arithmetic by semimorphism [6]. Let F be a floating point symmetric screen over R and F = F (b; p; emin; emax) is defined by its base b, its precision p, and its minimal and maximal exponent, emin and emax. If denote by IF = f[a Gamma ; a ] 2 IR j a Gamma ; a 2 Fg the set of computer representable intervals, ....

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Kulisch, U., Miranker, W. L.: Computer Arithmetic in Theory and Practice. Academic Press, New York (1981).

....= F (a) fi F F (b) it is well known that the equalities can only be satisfied approximately, and this is the source of the errors in practical computations. It is important to note that these errors can be accounted for in a mathematically rigorous way by using interval methods (see for example [6]) A way to treat functions that is conceptually similar to the truncation of numbers to n digits is to retain the first n orders of their Taylor expansion. For the field of particle optics, this approach is particularly useful because in most cases, it is not the map M proper that is needed, but ....

U. W. Kulisch and W. F. Miranker. Computer Arithmetic in Theory and Practice. Academic Press, New York, 1981.

....are not representable, then they must be rounded out ward to the closest representable floating point numbers. Moore [30, pp. 1 13] describes how interval arithmetic can be implemented. It is often called a machine, or rounded, interval arithmetic. A discussion of its properties can be found in [23]. 6 N. S. Nedialkov, K. R. Jackson and G. F. Corliss 2.3. Interval Valued Functions. Let f : R n Rbe a continuous function on D R n . We consider functions whose representations contain only a finite number of constants, variables, arithmetic operations, and standard functions (sin, ....

U. W. Kulisch and W. L. Miranker, Computer Arithmetic in Theory and Practice, Academic Press, New York, 1981.

....evaluating arithmetic expressions, accuracy plays a decisive role in many numerical algorithms. Even if all arithmetic operators and standard functions are of maximum accuracy, expressions composed of several operators and functions do not necessarily deliver results with maximum accuracy (see [7]) Therefore, methods have been developed for evaluating numerical expressions with high and mathematically guaranteed accuracy. A special kind of such expressions are called dot product expressions, which are defined as sums of simple expressions. A simple expression is either a variable, a ....

Kulisch, U.: Computer Arithmetic in Theory and Practice. Academic Press, New York, 1983.

....computer arithmetic was developed. It was realized in this context that it is appropriate to provide, apart from the four basic floating point operations, a fifth floating point operation for the computation of scalar products with maximum accuracy according to the principle of semimorphism [19] [20]. The first solution methods that have been proposed were of an algorithmic nature. About 20 years ago it was discovered that the problem can be solved in software and in hardware in an elegant manner if the products of the vector components are accumulated into a fixed point register that covers ....

....range reduction when computing elementary functions, multiple precision arithmetic and many other places. By means of an optimal dot product all operations of the usual product spaces of numerical analysis can be provided with maximum accuracy according to the principle of semimorphism [19] [20]. Among these spaces are the real and complex floatingpoint numbers, the real and complex floating point intervals as well as the vectors and matrices over these four basic data types. By semimorphism the real and complex vector spaces and the corresponding interval spaces are mapped to their ....

Kulisch, U., Miranker, W.L.: Computer Arithmetic in Theory and Practice, Academic Press, New York, 1981.

....with directed roundings which are directly accessible in the language. Further arithmetic operations for intervals and complex numbers and even vector matrix operations provided by precompiled arithmetic modules are defined with maximum accuracy according to the rules of semimorphism (see [25]) The development of PASCAL XSC programs is supported by the PASCAL XSC development system [3] consisting of the PASCAL XSC compiler [2] and the PASCAL XSC runtime system [12] which are both written in ANSI C [5] Instead of implementing a large variety of native code generators for ....

....then free (v) read(f, length) allocate(v, 1. length) for i: lb(v) to ub(v) do read(f, v[i] end; A detailed description of syntax and semantic for the concept of dynamic and flexible arrays is given in Appendix B. 2. 7 Accurate Expressions The theory of computer arithmetic (see [25]) requires the implementation of the dot product with only one rounding according to the following definition (see [6] Given two vectors x and y with n floating point components each, and a prescribed rounding mode 2, the floating point result s of the dot product operation (applied to x and y) ....

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Kulisch, U. and Miranker, W. L.: Computer Arithmetic in Theory and Practice. Academic Press, New York, 1981.

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Kulisch, U. and Miranker, W. L.: Computer Arithmetic in Theory and Practice. Academic Press, 1981.

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