Hearn, A.C., REDUCE user's manual (version 3.0), The Rand Corporation, Santa Monica, 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....5 Computer Algebra Aspects Computer algebra systems have allowed us to perform tedious algebra during our analysis. Simple tasks, which we needed, as simpli cation, substitution or solution of a system of linear equations are supported by any general computer algebra system (we have used Reduce [8] and Maple [9] More complicated is the treatment of inequalities which are not supported well in general computer algebra systems yet. Many problems involving inequalities as equivalence of inequalities or proving inequalities can be easily stated as quanti er elimination (QE) problems ....

A. C. Hearn, REDUCE user's manual, version 3.7, Tech. rep., RAND, Santa Monica (1999).

....important to model and which are unlikely to affect accuracy. Then a large degree of mathematical dexterity is required to combine the com ponents of the model and solve the resulting equations. Some aspects of this system might be automatable using algebraic manipulation software such as REDUCE [Hearn, 1973], but generally it requires the time of expensive, highly skilled experts. The mathematical models require enormous computational power. This greatly increases the expense of calculating real time dynamics and inverse dynamics for tasks such as trajectory tracking. However, increasingly powerful ....

A. C. Hearn. REDUCE 2 User's Manual. Technical report, University of Utah, March 1973.

....REAL,DIMENSION(4) x REAL,DIMENSION(5) y S10=I 1 S9=I 1 X(S10) A(S10,S9) B(I) Y(S9) A(S9,S10) B(I) # 6 GENERATION OF DECLARATIONS 77 6.1 Coe#cient Arithmetic and Precision Handling REDUCE knows a variety of coe#cient domains, as presented in subsection 9.11 of the REDUCE 3. 6 manual [17], entitled Polynomial Coe#cient Arithmetic. As stated in subsection 3.1 SCOPE supports integer and real coe#cients. By turning ON the switch ROUNDED we introduce float arithmetic for coe#cients. The operator PRECISION can be applied to change the default, machine dependent precision. Internally, ....

A.C. Hearn. REDUCE User's Manual, Version 3.6. Rand Corporation, Publication CP78(Rev. 7/95), Santa Monica, Cal., 1995.

.... whose limit is obviously 0: load limits; limit(x 7 e x,x,infinity) 7 x limit( x,infinity) x e This particular problem arises, because L Hopital s rule for the computation of indefinite forms (such as 0 0, or # # ) can only be applied in a CAS a finite number of times, and in REDUCE [Red36], this number is 3. Applied 7 times to the above problem would have yielded the correct answer 0. The new algorithm solves this particular problem, and enables the computation of many more limit calculations in REDUCE. We first define the domain in which we work, and then give a statement of the ....

Hearn, Anthony C. and Fitch, John F. REDUCE User's Manual 3.6, RAND Corporation, 1995 Limits

....this does not guarantee that the transform will be included in T syntax . The symbol must also be computable as an output of the transform. This condition may not be satisfied in one of two cases. First, the equation may not be mathematically solvable for the symbol. SIGMA uses the REDUCE (Hearn, 1991) computer algebra system to symbolically manipulate each equation formula to isolate the target symbol on one 6 With respect to SIGMA s user interface, note that the user sees only two sets of transforms in the transform selection window (Figure 6) Applicable Transforms and Near Miss ....

A. C. Hearn (1991). REDUCE User's Manual, Version 3.4 No. CP78). RAND Corporation, Santa Monica, CA.

....n Gamma 1; m m 2) It is easy to see that functional F = 3n m reduces at least by 1 after each iteration. Therefore, the final result is obtained after 3n iterations [15] 5 Implementation of calculation in computer algebra system Reduce We have used the Computer Algebra System (CAS) Reduce [16] extended by our package for the calculations in chiral meson theories [17] Reduce suits well for our purposes since it is a universal and open system which can be easily extended by the users. It is widely used for our calculations of amplitudes of various meson processes and for the derivation ....

A.C.Hearn, Reduce User's Manual, version 3.5, RAND Publication CP78 (Rev.10/93), Santa Monica, October 1993.

....we also get integrals with some of the powers of the denominators equal to two, see eq. 2.3) However, with the help of the integration by parts technique [27] these integrals can be algebraically reduced to the above basic set (see ref. 28] While performing the calculations, the REDUCE system [29] was heavily employed. Before presenting the results, let us define two totally symmetric combinations of the invariants formed from the external momenta, Q # (p 1 p 2 ) p 1 p 3 ) p 2 p 3 ) 1 2 (p 2 1 p 2 2 p 2 3 ) 3.1) K # p 2 1 p 2 2 (p 1 p 2 ) 2 = p 2 1 p 2 3 ....

A.C. Hearn, REDUCE User's Manual (version 3.5), RAND publication CP78 (Santa Monica, 1993);

....33C45, 11B37, 34A30, 33C20, 68Q40. 1. Basics. Modern computer algebra systems use heuristics and algorithms for the fast computation with mathematical formulas. For instance, general purpose computer algebra systems like Axiom [14] Derive [22] Macsyma [19] Maple [6] Mathematica [25] or Reduce [13] handle integrals surprisingly well. Even a small system like Derive computes all explicitly given integrals of Bronshtein and Semedyayev s integral table [4] The purpose of this paper is to familiarize the reader with some of the underlying techniques which permit computer algebra systems to ....

A. C. HEARN, Reduce User's Manual, Version 3.6, RAND Co., Santa Monica, CA, 1995.

....in an expanded form, up to 0 , i.e. keeping the poles and the finite terms. 4 Two loop results for the three gluon vertex Below, we list the unrenormalized two loop contributions to the functions U i (p 2 ) occurring in the three gluon vertex. They were calculated using a set of REDUCE [33] programs based on the algorithm described in the previous section. The two loop diagrams contributing to the three gluon vertex are shown in Fig. 1 of ref. 15] The results are expanded in up to the finite terms. The factor j is defined (and its expansion in is given) in eq. 2.19) The ....

A.C. Hearn and J.P. Fitch (ed.), REDUCE User's Manual (version 3.6), RAND publication CP78 (Santa Monica, 1995).

....A d fffi fl ff fl fl fi fl 5 oe (6 k 1 Gamma m g ) Theta ae A ay ae fl fl ae fl 5 A by ae fl ae fl fl 5 A cy fl fl 5 A fy ae fl ae fl 5 A dy aeoe fl oe fl fl ae fl 5 oe (6 k 2 m g ) # ; 4. 3) and evaluate the trace using computer algebra [25, 26]. By summing over the eight gluino colours, and integrating over the solid angle, we find that the cross section is proportional to the square of the sum of two partial amplitudes, corresponding to the contributions of the two diagrams (a) and (b) This is possible, since by general arguments ....

A. C. Hearn, REDUCE user's manual, RAND publication CP78 (Santa Monica, 1987).

....we will construct a polynomial algebra in several variables and an involution of the Riemann surface is found. Finally, the case of F 4 is more di#cult from a purely computational point of view, and calculations have to be done using a computer. We did calculations in the symbolic languages REDUCE [13] and MAPLE [14] 2.1 The polynomial ring and ideals Due to the cubic and square roots in (2.1) the # i = #W #u i have terms containing # q # q 2 4p 3 # 2 3 which are certainly not polynomial in x. For classical gauge groups this problem does not occur and the # i are ....

A. Hearn, REDUCE user's manual (version 3.6) (The Rand corporation, Santa Monica, 1983).

....Like before we put 1 i 1 j c k; X m;n;p ;q w m;n;p ;q 1 m 1 n b p ;q X m m f m .b X m m c m . b : From this the linear constraints and the expressions for the goal functions L = P m;n;p ;q w m;n;p ;q were automatically derived with the symbolic program Reduce [Hea91]. Finally C 1 continuity was again proved using numerical software for 1 4 # 3 4 . Having established convergence, it is now proved that it the resulting derivative functions are continuous, in other words that the limiting function f # is of class C 1 . Theorem 4.4. Given a ....

....with the algorithm in the previous section, it turns out that the regularity of the grid comes into play. Consider figure 18 and put km : sin # k cos # sin # m ; km : sin k cos sin m ; # k; m = 1;2;3 : 5. 1) BIVARIATE HERMITE SUBDIVISION 21 The result found with the aid of Reduce [Hea91], was that (see figure 18 for definitions) K 1 = 2J 1 ; K 2 = 1 4 . 321 321 2 J 1 .2 321 321 J 2 .2 321 .J 5 J 6 .2 321 .J 7 J 8 ; K 3 = 1 4 .2 321 312 .J 1 J 2 132 .J 3 J 4 .1 2 321 .J ....

A.C. Hearn. REDUCE User's manual 3.4. Rand, 1991.

....1 1 1 Macaulay Theta Theta Theta Theta Theta Theta 1 Maple 1 1 Theta Theta 1 1 Theta Theta Mathematica 1 Theta Theta Theta Theta 1 Theta Theta MAS 1 1 1 1 1 1 1 1 GB 1 11112 1 17043 1 1.4 Conclusion Description of the CAS used for these tests: ffl Reduce Version 3. 5[Fit85, Hea87] We notice that Reduce 2 is faster with small homogeneous systems but cannot compute C 6 because The Bucheberger Algorithm is not implemented with the sugar strat 1 The size of the output file is 31 Megas bytes and 460156 lines 2 This is the standard package which has been used for this ....

Anthony C. Hearn. Reduce user's manual, Version 3.3. The RAND Corporation, report cp 78 edition, July 1987.

.... in a similar treatment, Chyzak tried to derive this type of result with a Maple implementation, without success ( 5] x4.1) Using the Reduce implementation [21] any of these calculations (for the Jacobi polynomials) needs about five minutes on a DEC Alpha workstation (using the ezgcd switch, see [10], x9.3) Note that also with a pure lexicographic term order, the Grobner bases are derived in a similar time. For all other families considered, the calculation needs only seconds. As an example, we consider the Whittaker functions M n;m (x) with respect to the parameter m for which no ....

Hearn, A. C.:. Reduce User's Manual, Version 3.5. RAND Co., Santa Monica, CA, 1993.

....for comparing Computer Algebra Systems in a specific domain. We do not intend to give an overall rating as for example in [8] 1 The Contestants We compare six major Computer Algebra Systems. Axiom 2.0 [5] Derive 3.06 [6] Macsyma 418.1 [7] Maple V R4 [2] Mathematica 2.2 [9] and Reduce 3. 6 [4]. When available, we tried to use the latest shipping version of each system. 2 The Problem Set The following table presents the set of 82 problems that we used to evaluate the different solvers. It also gives the solution that is necessary to get full credit for each problem. If not stated ....

Hearn, A. C. REDUCE user's manual, Version 3.4. Report CP 78, RAND, July 1991.

....else go back to Step 3. It is worth to note that Algorithm PtoA for transformation from P to A s yields the moments (and also the cumulants) of r represented in the population cumulants; see section 3.1.4. Algorithms PtoA are implemented on computers as Lisp programs with interface to REDUCE [7], a computer algebra system used worldwide. 3 Asymptotic distributions of R We now turn to derive an asymptotic distribution of r. Assume that the bivariate population F has finite cumulants and product cumulants 10 , 01 , 20 , 11 , 02 , Delta Delta Delta up to total order eight. ....

A. C. Hearn. REDUCE User's Manual. Santa Monica, version 3.3 edition, 1987.

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Hearn, A.C., REDUCE user's manual (version 3.0), The Rand Corporation, Santa Monica, 1983.

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A. C. HEARN, REDUCE User's Manual, Version 3.3, The Rand Corporation, Santa Monica, Calif., 1987.

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A. C. Hearn, REDUCE User's Manual, version 3.5, RAND Publication, CP78(Rev. 7/94), 1994.

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Anthony C. Hearn. REDUCE user's manual, version 3.6. Technical Report CP 78 (Rev. 7/95), The RAND Corporation, Santa Monica, 1995.

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Paris VI. Hearn, A.C. (1995). REDUCE User's Manual, Version 3.6. RAND Co., Santa Monica, CA.

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A.C.Hearn, Reduce User's Manual, version 3.5, RAND Publication CP78 (Rev.10/93), Santa Monica, October 1993.

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Anthony C. Hearn. REDUCE user's manual, version 3.6. Technical Report CP 78 (Rev. 7/95), The RAND Corporation, Santa Monica, 1995.

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A. C. Hearn, REDUCE User's Manual, RAND publication CP78 (Santa Monica, 1987).

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A.C. Hearn. REDUCE User's Manual, Version 3:5. The RAND Corp., Santa Monica, U.S.A. 1993.

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