R. P. Brent, Algorithm 524: MP, a Fortran multiple-precision arithmetic package, ACM Trans. on Mathematical Software 4 (1978), 71--81.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....operations are outward rounded to preserve the correctness of the computation. However, we also assume that the largest computing error when computing a bound of a variable of the initial constraint system is always smaller than one float. This hypothesis may require the use of big floats [4] when computing intermediate results. Definition 3. Interval Extension [20, 10] ffl f : I I is an interval extension of f : R R iff 8x 1 ; xn 2 I : f( x 1 ; xn ) 2 f (x 1 ; xn ) ffl c : I Bool is an interval extension of c : R Bool iff 8 x 1 ; ....

R.P. Brent. A FORTRAN multiple-precision arithmetic package ACM Trans. on Math. Software, 4, no 1, 57-70, 1978.

....nd methods which are simple to adapt in practical large scale coding. 4 Multi Precision Arithmetic The above xed point arithmetic is a simple example of a class of multi precision arithmetic software packages which carry out numerical calculations at a pre speci ed precision [15] Brent s BMP [3] is the rst complete Fortran package on the multi precision arithmetics o ering a complete set of arithmetic operations as well as the evaluation of some constants and special functions. Bailey s MPFUN [2] is a more sophisticated and more eciently implemented complete package. He used this ....

R. P. Brent. A Fortran Multiple Precision Arithmetic Package. ACM Transactions on Mathematical Software, 4:57-70, 1978.

.... Gol91, Swa91, Her94c, Sch95, Sch95a] conversion of decimal data [Coo81] increasing precision [Kah65, Moe65, Dek71, Yoh73, Lin74, Lin81] linear systems with increased precision [Gru76] interval arithmetic [Apo67, Chr68, Cle83] multiprecision arithmetic (for real numbers and intervals) Lor71, Bre78, Sas79, Bre80, Bre81, Hul85, Kru86, Abe88, Abe91, Ely90, Ely91, Rat91, Moo91a, Kra88b, Shi89, Wal90a, Kra92b, Kra93c, Loh93] determination of properties of arithmetic [Kah83, Cod88a] comparison of IEEE coprocessors [Juf93] test matrices [Gre69, Duf89] ffl Standard functions with maximum ....

Brent, R. P.: A FORTRAN Multiple Precision Arithmetic Package. ACM Trans. Math. Software 4, pp. 57-70, 1978.

....most of these problems have subsequently been recti ed. 2. A Comparison of MPFUN with Other Multiprecision Systems Several software packages are available for multiprecision computation. One that has been around for a while is the Brent MP multiprecision package, authored by R. P. Brent [11]. This package has the advantage of being freely available either from the author or from various other sources. It is very complete, including detailed numerical controls and many special functions. Recently Smith [30] described a similar package that features improved performance for certain ....

Brent, R. P., \A Fortran Multiple Precision Arithmetic Package", ACM Transactions on Mathematical Software, vol. 4 (1978), p. 57 - 70.

....abort with little information to guide the programmer. As a result of these diculties, few serious scienti c programs have been manually converted to use the MPFUN routines. Similar diculties have plagued programmers who have attempted to use other multiprecision systems, such as Brent s package [4]. To facilitate such conversions, the author has developed a translator program that accepts as input a conventional Fortran 77 program to which has been added certain special comments that declare the desired level of precision and specify which variables in each subprogram are to be treated as ....

R. P. Brent, \A Fortran Multiple Precision Arithmetic Package", ACM Transactions on Mathematical Software, vol. 4 (1978), p. 57 - 70.

....They can be divided into two groups based on the way precision numbers are represented. Some libraries store numbers in a multiple digit format, with a sequence of digits coupled with a single exponent, such as the symbolic computation package Mathematica, Bailey s MPFUN [2] Brent s MP [4] and GNU MP [8] An alternative approach is to store numbers in a multiple component format, where a number is expressed as unevaluated sums of ordinary floating point words, each with its own significand and exponent. Examples of this format include [7, 10, 11] The multiple digit approach can ....

R. Brent. A Fortran multiple precision arithmetic package. ACM Trans. Math. Soft., 4:57--70, 1978.

....They can be divided into two groups based on the way precision numbers are represented. Some libraries store numbers in a multiple digit format, with a sequence of digits coupled with a single exponent, such as the symbolic computation package Mathematica, Bailey s MPFUN [2] Brent s MP [4] and GNU MP [8] An alternative approach is to store numbers in a multiple component format, where a number is expressed as unevaluated sums of ordinary oating point words, each with its own signi cand and exponent. Examples of this format include [7, 10, 11] The multiple digit approach can ....

R. Brent. A Fortran multiple precision arithmetic package. ACM Trans. Math. Soft., 4:57-70, 1978.

....They can be divided into two groups based on the way precision numbers are represented. Some libraries store numbers in a multiple digit format, with a sequence of digits coupled with a single exponent, such as the symbolic computation package Mathematica, Bailey s MPFUN [2] Brent s MP [5] and GNU MP [8] An alternative approach is to store numbers in a multiple component format, where a number is expressed as unevaluated sums of ordinary floating point words, each with its own significand and exponent. Examples of this format include [7, 11, 12] The multiple digit approach can ....

R. Brent. A Fortran multiple precision arithmetic package. ACM Trans. Math. Soft., 4:57--70, 1978.

....Some groups have developed multiple precision packages based on the error free integer arithmetic to compute some very precise quantities on a computer. Some of the most successful packages available today for a fast accurate scientific computation are GMP (Gnu Multiple Precision) Brent s MP [3] and Bailey s package 1 [1] Expansions were introduced by Priest [12] based on some earlier compound operations defined to reduce the rounding error of a long expression [11, 10] or to compute an accurate rounding [2] Actually, Dekker was the first in the past to propose this technique but he ....

R. P. Brent. Algorithm 524. MP, a fortran multiple precision arithmetic package. ACM Transactions on Mathemetical Software, 4(1):71--81, 1978.

....versions 0 with accuracy Gamma59. For all real numbers, the sine is known to be between 1 and 1. To not know the sine to within Sigma10 59 is very pessimistic. There are several packages for computing arbitrary precision transcendental functions reasonably correctly in the open literature [7], 55] Much of the manipulation of the big O asymptotic notation used in Series was problematical in version 1.1. For example Sin[O[x]4] never returned. Probably the issue of combining significantly distinct not entirely algebraic datatypes is difficult to handle in the Mathematica scheme. Even ....

....In[3] a Out[3] 0. Wow, just 55 iterations got a zero, not a 1.111. with some error. Furthermore that zero is a black hole: In[4] a 1 Out[4] 0. Everything works fine if we use a machine float instead of a big float: In[5] b=1. 111111111; In[6] Do[b=2b b, 1000] Out[7]= 1.11111 In[8] b 5 Out[8] 0. The problem is that each iteration ratchets down the precision by one until it eventually gets to zero. Although it is less serious, there are also constructs that ratchet up the precision, which, by the way, indicates that the precision calculation cannot be ....

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R. P. Brent. "A Fortran Multiple-Precision Arithmetic Package," ACM Trans. on Math. Softw. 4 no. 1, (March, 1978), 57-70.

....or algebraic structure. One such extension that has repeatedly appeared as userimplemented programs is that of a multiple precision floatingpoint number system. A design for such a system in Fortran goes back at least as far as Wyatt [17] A well distributed version is the one by Brent [5]. An early implementation in Lisp is described by Fateman [8] Other implementations have been written through the last several decades. We note a few more: Sasaki [14] wrote programs in Lisp for the Reduce system; for the SAC I system, Pinkert wrote a system [12] most other programs seem to have ....

Richard P. Brent. A Fortran Multiple Precision Arithmetic Package, ACM Trans. Math. Softw. 4 (March, 1978), p. 57 -- 70.

....The leading mantissa element must be a nonzero block. The constant 0 can be identified by a zero value for the mantissa length, and no exponent or mantissa units need to be allocated. Algorithms to perform the basic four arithmetic operations on such representations are described in Brent [Bre78] and Knuth [Knu81] Each of these operations can be performed only by specifying an upper bound on the number of result mantissa elements. Such a bound is clearly needed for the division operation, that can possibly result in a nonterminating result mantissa. Even though the other operations can ....

R. Brent. A fortran multiple precision arithmetic package. ACM Transactions on Mathematical Software, 4:57--70, 1978.

.... big digits each stored as an element of an array of integers or integer valued doubles. This kind of software simulated floating point is built into automated algebra systems like Macsyma [28] Mathematica [51] and Maple [40] and provided to other programmers by packages like Richard Brent s MP [10] and David Bailey s MPFUN [5] BigFloats are optimized for very high precision, and run so much slower than Double double or Quadruple when delivering comparable precisions that BigFloats shall not be treated further in this document. Unfortunately, our three names Double extended , Quadruple ....

R. Brent. A Fortran multiple precision arithmetic package. ACM Trans. Math. Soft., 4:57--70, 1978.

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R. P. Brent, Algorithm 524: MP, a Fortran multiple-precision arithmetic package, ACM Trans. on Mathematical Software 4 (1978), 71--81.

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R. P. Brent, Algorithm 524: MP, a Fortran multiple-precision arithmetic package, ACM Trans. on Mathematical Software 4 (1978), 71--81.

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R. P. Brent, A Fortran multiple-precision arithmetic package, ACM Transactions on Mathematical Software 4 (1978), 57--70.

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R. P. Brent, A Fortran multiple-precision arithmetic package, ACM Transactions on Mathematical Software 4 (1978), 57--70.

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R. Brent. A Fortran multiple precision arithmetic package. ACM Trans. Math. Soft., 4:57-70, 1978.

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R. P. Brent. A Fortran multiple-precision arithmetic package. ACM TOMS, 4:57--70, March 1978.

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Brent, R.P. A Fortran Multiple-Precision Arithmetic Package. ACM Trans. Math. Software 4, 1 (March 1978), 57-70.

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R. Brent. A Fortran multiple precision arithmetic package. ACM Trans. Math. Soft., 4:57-70, 1978.

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R. Brent. A Fortran multiple precision arithmetic package. ACM Trans. Math. Soft., 4:57-70, 1978.

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R.P. Brent, "A Fortran multiple-precision arithmetic package ... MP, Algorithm 524." ACM Transactions on Mathematical Software, 4 (1975), pp. 57-81.

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Brent, R.P. 1978. A Fortran Multiple-Precision Arithmetic Package. ACM Trans. Math. Softw. 4, 1 (March), 57--70.

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R. P. Brent, \A Fortran Multiple Precision Arithmetic Package," ACM Transactions on Mathematical Software, vol. 4 (1978), p. 57-70.

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