B. W. Char, K. O. Geddes, G. H. Gonnet, M. B. Monagan, and S. M. Watt. Maple Reference Manual, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the full problem (1.1) numerically using the finite element method described in with approximately 1001 elements. The crosses in these figures represent the asymptotic results obtained from solving the relevant di#erential equation of numerically using a standard solver routine from Maple [6]. We take the parameter values V (x) 0, x) 1, D = 1, and # = 0.03. The initial condition used for the finite element solution, as obtained from (2.3c) 2.15a) and (2.15b) is 3H 2 h(x, 0) H where from (2.14b) H = 6G(x 0 ; x 0 ) 1 . 5.1c) In this case, the initial ....

....(x 0 # 1 ) dy 0 (y 0 # 2 ) 6.2) Here # = # 1 , # 2 ) In Fig. 9a we show the close agreement between the full numerical result for the distance of the spike to the origin and the corresponding asymptotic result obtained from integrating (6. 2) numerically using Maple [6]. In Fig. 9b we show that the trajectory of the spike is approximately orthogonal to the level curves of the potential W as predicted by (3.27) 7 Discussion In a one dimensional domain, we have given a complete characterization of the dynamics of a onespike solution to (1.1) allowing for ....

B. W. Char et al. Maple 5 Reference Manual, Springer-Verlag, (1991).

....solution. 2.3 Example Consider the toy example of Figure 3, with p = 4 processors. If we feed the values w i and c i into the linear program, and compute the solution using a tool like the Maple 6 3 1 P 1 (master) P 3 3 3 Figure 3: An example with four processors. simplex package [7], we obtain the optimal throughput n task (G) 4 . This means that the whole platform is equivalent to a single processor with processing capability w = 7 , i.e. capable of processing 7 tasks every 4 seconds. With the values of i , s ij and r ij returned in the solution of the linear ....

B.W. Char, K.O. Geddes, G.H. Gonnet, M.B. Monagan, and S.M. Watt. Maple Reference Manual, 1988.

....which the resolvents are Swinnerton Dyer polynomials. On the other hand the coe#cients of the resolvents become that big, that the break even point for a polynomial time factoring algorithm [36] is yet beyond the runtime for the classical approach. The algorithm of [45] is implemented in Maple [7] up to degree 7. An extension to degree 8 for polynomials over Q(t) is described in [38] It is again implemented in Maple. Tabulating further data for linear resolvents and using also factorization over algebraic extensions to determine orbits of subgroups, this approach has been implemented by ....

Char, B. W., Geddes, K. O., Gonnet, G. H., Monagan, M. B., Watt, S. M.: MAPLE -- Reference Manual, 5th edition. University of Waterloo, (1988)

....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 been written in C such as the desk calculator (dc) in the UNIX system [15] and components of Maple [6], Mathematica [16] PARI [4] and Saclib [11] Viewed from the perspective of a subroutine library for applications, it is clear that the authors of the Fortranextension packages were sensitive to usability issues, and that a raw library interface was thought to be too hard to use. Therefore ....

B. Char et al. Maple Reference Manual, Springer-Verlag, 1992.

.... constrained discrete optimization algorithm searching for the optimal solution for the number of stages, N , and the discrete sizes, represented byvector (w) For the full custom problem, instead, the sizes are unrestricted and the algorithm uses the MapleV Lagrange multipliers optimization method [5] 4 Application example The solution of the proposed design automation problem for different sized lines using ES2 standardcell and full custom design environments is considered in this section. Figure 4 shows the optimal solution, in terms of total consumed power, for differentvalues of line ....

Bruce W.Char et al. Maple Reference Manual. Ontario, 1988.

....phenomenon may be due to the lack of precision in the floating point representation of numbers. Let us try to use a more powerful floating point arithmetic that allows the programmer to fix the number of significant digits used during computation (as in the symbolic computation system Maple (Char et al. 1988)) Here are the values obtained for an with various precisions: precision a10 a20 a30 a40 a50 10 147:26 100 100 100 100 20 5:86 99:7 100 100 100 30 5:86 5:97 100 100 100 40 5:86 5:97 5:994 100 100 50 5:86 5:97 5:996 5:92 100 The sequence stubbornly converges to 100.0. We conjecture that with p ....

Char, B., Geddes, K., Gonnet, G., Monagan, M., Watt, S. (1988). MAPLE: Reference Manual. Symbolic Computation Group, Department of Computer Science, University of Waterloo. 5th Edition.

....second order linear difference equations with constant coefficients, difference equations from divide and conquer paradigm and a special class of difference equations derivable from clauses with the size measure term size. Apart from this, we have incorporated the Maple Symbolic Computation System [4] into CASLOG, so that the system can resort to Maple when the difference equations encountered cannot be handled by its difference equation solver. If neither our difference equation solver nor Maple can deal with the difference equations encountered, a conservative upper bound, x:1, is returned. ....

B. Char, K. Geddes, G. Gonnet, M. Monagan and S. Watt, MAPLE: Reference Manual, University of Waterloo, 1988.

....checking of the pattern matching process deployed in this function checks that all program forms are handled by the function. The matrix manipulation routines which are required to solve the generator matrix either symbolically or numerically are provided by the Maple computer algebra package 1 [13]. For the prototype 1 Maple is a registered trademark of Waterloo Maple Software. 14 it was judged to be simpler to use the existing Maple routines rather than re implement these in Standard ML. Thus we have implemented the functionality to allow a workbench user to call Maple from the ....

B.W. Char, K.O. Geddes, G.H. Gonnet, M.B. Monagan, and S.M.Watt. Maple Reference Manual. Springer-Verlag, 1988.

....: S i3 od: S; end; 31 The local statement makes the variables i and S private to the function cubes. This way they can t interfere with variables of the same name used outside the definition of cubes. Once the definition is made, we test it: cubes(1) cubes(2) cubes(3) map( cubes, [1, 2, 3, 4, 5]) The function map is used to apply a function or procedure, in this case cubes to a list of elements. It then returns a list of values. There is something we can learn from the experiment just done. It looks like cubes is returning numbers which are perfect squares One way to check this is to ....

....integer n is prime or not using the ideas outlined at the end of the section on functions. Remarks More examples with map and lists: 1. Sometimes we need to apply functions to lists of things. For this we use map, as in the following example: sqr : x x x; # squaring function map( sqr, [1, 2, 3, 4, 5] ) # make list of squares A list is a sequence of elements separated by commas and enclosed by brackets. The map function takes two arguments: a function, and a list to which the function is applied element by element. 2. Another way to generate the list of square above is to use i : i : # ....

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Bruce W. Char et al., Maple Reference Manual, 5th Edition, WATCOM Publications 1988, pp. 403. 48

....the full problem (1.1) numerically using the nite element method described in x4 with approximately 1001 elements. The crosses in these gures represent the asymptotic results obtained from solving the relevant di erential equation of x2 numerically using a standard solver routine from Maple [4]. 17 5.1 Experiment 1 We take the parameter values V (x) 0, x) 1, D = 1, and = 0:03. The initial condition used for the nite element solution, as obtained from (2.3c) 2.15a) and (2.15b) is a(x; 0) 3H 2 sech 2 x x 0 2 ; 5.1a) h(x; 0) H G(x; x 0 ) G(x 0 ; x 0 ) ....

....jx 0 j ; dy 0 dt 4 2 (y 0 2 ) 4 jx 0 j : 6.2) Here = 1 ; 2 ) In Fig. 11a we show the close agreement between the full numerical result for the distance of the spike to the origin and the corresponding asymptotic result obtained from integrating (6. 2) numerically using Maple [4]. In Fig. 11b we show that the trajectory of the spike is approximately orthogonal to the level curves of the potential W as predicted by (3.26) 6.3 Experiment 3: Several Initial Spikes In this experiment we examine numerically the behavior of the solution to (1.3) at two di erent values of D ....

B. W. Char et al. Maple 5 Reference Manual, Springer-Verlag, (1991).

....imperative programming constructs. 2 Language Design In this section we explain the design motivations for the syntax and semantics of the high level language. 2. 1 Language Syntax We have made an attempt to combine the best features among the language constructs present in the SACs maple [5], mathematica [18] reduce [6] and matlab [15] The maple, mathematica, and reduce systems are general purpose symbolic and algebraic computing systems (SACs) while matlab is a matrix based system for scientific and engineering calculations. Each of these systems adopts a kind of ....

.... a(x) j i y b(y) j ; i x c(x) j i y d(y) ji In[1] S : 1, 2, 3, 4 In[2] T : 1, 1, 2, 1 In[3] MatrixPower[S, 2] 2 T Out[3] 9, 12, 19, 20 In[4] Sin[S] T Out[4] Sin[1] 2 Sin[2] Sin[1] Sin[2] Sin[3] 2 Sin[4] Sin[3] Sin[4] In[5]: S IdentityMatrix[2] a, b . U Out[5] 2 b, 3 a 3 b . U In[6] MatrixPower[S, 0] Out[6] 1, 0, 0, 1 In[7] Transpose[ 1, 2] 3, 4 Out[7] 11 In[8] S 1, 2 Out[8] 2, 3, 5, 6 In[9] Map[Div, a, b, c, d] Out[9] Div[ a, b] Div[ c, d] reduce matlab 1: ....

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B.W. Char, K.O. Geddes, G.H. Gonnet, M.B. Monagan, and S.M. Watt, MAPLE Reference Manual, Waterloo, 1988.

....As suggested by Sakellariou (Sakellariou, 1996; Sakellariou, 1997) a computer algebra system can be exploited off line to solve the equations paper.tex; 23 09 1999; 14:38; p.21 22 Healy, Sjodin, Rustagi, Whalley and van Engelen of summations. However, computer algebra systems, such as Maple (Char et al. 1988), give inaccurate results when the bounds restriction on the summation is violated in Equation 2. In general, every loop iteration count problem that is cast as a summation should evaluate to zero if the lower bound is greater than the upper bound. However, it is not always possible to evaluate ....

Char, B., K. Geddes, G. Gonnet, M. Monagan, and S. Watt: 1988, `MAPLE Reference Manual'.

....algorithms for constructing locally supported basis functions and for knot insertion. As a result of our development we obtain a generalization of the polar form of a B spline to geometrically continuous spline curves. The given algorithms have been implemented using the linalg package of Maple [14], and concrete examples illustrate the approach. Although all algorithms in this paper have been specifically designed for arbitrary degree n, all examples are chosen to be cubic: This makes the output much more manageable and should help the reader to follow some of the given computations by ....

.... we obtain r 0 = 0; r 1 = Gamma1; r 2 = 2 (109) and s 1 = 2; s 2 = Gamma1; s 3 = 0; 110) and d 2 is given as d 2 = 0 Delta b 3;0 Gamma b 3;1 2 Delta b 3;2 (111) 0; Gamma1; 2; 0; 0; 0; 0; 0) 2 IR 8 : 112) The results below have been obtained using the linalg package of Maple [14]. The control points d 0 , d 7 are given by the rows of the following table d 0 = 1 0 0 0 0 0 0 0) d 1 = 0 1 0 0 0 0 0 0) d 2 = 0 Gamma1 2 0 0 0 0 0) d 3 = 0 3 Gamma10 8 0 0 0 0) d 4 = 0 Gamma17 62 Gamma56 12 0 0 0) d 5 = 0 29 Gamma106 96 Gamma21 3 0 0) d 6 = 0 ....

B.W. Char et al. Maple Reference Manual. Watcom Publ. Ltd., Waterloo, fifth edition, 1988.

....former approach suffers from being tedious and prone to errors, while the latter can produce large errors if the size of the perturbation is not carefully chosen; even in the best case, half of the significant digits will be lost. For problems of limited size, symbolic manipulators, such as Maple [10] and Mathematica [29] are available. These programs can simplify the task of deriving an expression for derivatives and converting this expression into code, but they are typically unable to handle functions that are large or contain branches, loops, or subroutines. An alternative to these ....

B. W. Char, K. O. Geddes, G. H. Gonnet, M. B. Monagan, and S. M. Watt, MAPLE Reference Manual, Watcom Publications, Waterloo, Ontario Canada, 1988.

....checking of the well formedness of the PEPA model through the creation of the state transition diagrams to the calculation of performance measures based on the infinitesimal generator matrix. The tool is implemented in Standard ML [2] and provides an interface to the Maple Symbolic Algebra package [3] for the solution of matrix equations. 1 Introduction Formal descriptions of computer systems are amenable to analysis by a range of formal techniques. At the simplest level, they may be checked for conformance with the syntax, grammar and type correctness rules of the formal language used. More ....

....state PEPA models whose derivation graph is strongly connected, ergodic Markov process) the equilibrium distribution of the model, Pi, is found by solving the matrix equation PiQ = 0 (2.1) subject to the normalisation condition X Pi(C i ) 1 (2. 2) The computer algebra package Maple 2 [3] is used to find Pi. The equations 2.1 and 2.2 are combined by replacing a column of Q by a column of 1s and placing a 1 in the corresponding row of 0. Moreover, since Maple deals with row vectors instead of column vectors, this modified Q is transposed. 2 Maple is a registered trademark of ....

B.W. Char, K.O. Geddes, G.H. Gonnet, M.B. Monagan, and S.M.Watt. Maple Reference Manual. 1988.

....t 3=2 e Gamma( x 2 y 2 2oe 2 ) 1 t Gamma( 2 x 2 y 2oe 2 z )t dt = 1 2 r p fl e 1 fl ( x x yy) Gamma p 1=fl 2 x =fl 2 2 y =fl 2 r : B. 9) The integral involved in the third step of the above equation was solved using the symbolic mathematical program Maple (Char, 1988). Interestingly, if x = y = 0, that is, if there is no wind directionality, it follows from (B.9) and (A.5) that the radial distance RjZ = 0 will be exponentially distributed with parameter 1= p fl, that is, although the underlying process is different, the generated distribution is equal ....

Char, B. W., 1988. Maple Reference Manual. Waterloo, Ont.

....languages. It offers type constrained matching, conditional rules, associative, commutative and identity matching, pattern alternatives, repeated patterns etc. On the other hand there are also relatively efficient systems that do not support this declarative kind of programming (e.g. Maple 4 [5]) This gives the Research supported by the Swiss National Science Foundation 1 Mathematica is a registered trademark of Wolfram Research, Inc. 2 Axiom is a registered trademark of The Numerical Algorithms Group Ltd. 3 Miranda is a registered trademark of Research Software Ltd. 4 Maple ....

....level the command Unify[expr 1 , expr 2 ] The system selects then the right unifier for composite expressions and gives the substitution list: fvar 1 val 1 , var n val n g. With unification a non ground expression can rewrite either to a ground term, such as: In[1] Unify[f[x, 2, g[3, 5]] f[a, 2, g[y, z] Out[1] z 5, y 3, x a or to a non ground term: In[2] Unify[f[x, g[a, y] f[1, g[a, h[z2] Out[2] x 1, y h[z2] The unification algorithm alone (i.e. without the resolution mechanism) gives the possibility to program in the following way: Find a ....

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B.W. Char, K.O. Geddes, G.H. Gonnet, M.B. Monagan, and S.M. Watt. MAPLE: Reference Manual. University of Waterloo, 1988. 5th edition.

....of the combination and integration of software tools to form a PSE. 2.4 Symbolic Numeric Computing in PSEs Much attention has been directed towards the symbolic numeric interface for the PSEs in scientific computing. Such systems utilise the capabilities of symbolic packages, such as Maple [22] and MACSYMA [60] to generate numerical information. The use of symbolic systems to generate FORTRAN code which can utilise numerical libraries is discussed by Davenport in [25] and more recently using the AXIOM 17 system to drive the NAG libraries, see Section 2.3.4. When looking at the ....

....the L A T E X system that will be used here. It is L A T E X that has also been used to format this thesis. 2.6.4 Computer Algebra Systems There are six main computer algebra systems widely available in the scientific computing area. These six systems are REDUCE [44] MACSYMA [60] Maple [22], Mathematica [92] the Derive system based on the earlier muMATH system [93] and AXIOM [26] a recent newcomer to this area. Each system is a general purpose interactive package that provides similar operations. Much literature is devoted to the study of one particular package or an overview of ....

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B W Char et al. MAPLE Reference Manual. WATCOM Publications Ltd, Waterloo, Canada, 5th edition, 1988.

....former approach suffers from being tedious and prone to errors, while the latter can produce large errors if the size of the perturbation is not carefully chosen; even in the best case, half of the significant digits will be lost. For problems of limited size, symbolic manipulators, such as Maple [7], are available. These programs can simplify the task of deriving an expression for derivatives and converting this expression into code, but they are typically unable to handle functions that are large or contain branches, loops, or subroutines. An alternative to these techniques is automatic ....

Bruce W. Char, Keith O. Geddes, Gaston H. Gonnet, Michael B. Monagan, and Stephen M. Watt. MAPLE Reference Manual. Watcom Publications, Waterloo, Ontario Canada, 1988.

....F 2 B P 2 N Z 1 Xr (r;m) C(x)h(x)dx We then substitute the three cost components in the cost expression and a random access cost expression is derived, as a function of the middle of the P 1 area. In order to simplify the expression we set P 2 = 1. Using the Maple symbolic mathematics package ([9]) we obtain the expression for the cost metric. Cost(k; f; a; b; P 1 ; r; B;F 1 ; F 2 ; m) Gamma C 1 120 (P 1 r 1 Gamma r) 2 (2 km 2 Gamma k) 5 (16) where C 1 is an expression output by Maple which is ten pages long and therefore we omitted. The optimal value for m will be derived ....

B.C. Char, K.O.Geddes, M.B. Monagan, S.M. Watt, "MAPLE Reference Manual", 5 th Edition, University of Waterloo, Waterloo, Ontario, March 1988.

.... capabilities for solving elementary equations (linear equations; algebraic equations; simple differential equations) As of 1990, the programme has over 10,000 instructions, partly in a high level functional language CAML, a dialect of ML [84] and partly in a computer algebra language Maple [15]. In the present paper, we have used the Upsilon Omega system in order to produce what is called automatic theorems. In principle, an automatic theorem is a statement that is derived automatically from formal specifications by the logical framework exposed in this paper. We have however decided ....

Char, B., Geddes, K., Gonnet, G., Monagan, M., and Watt, S. MAPLE: Reference Manual. University of Waterloo, 1988. 5th edition.

.... = GammaX : Gammaa 2 b 2 = Gammaa; Gammac 2 d 2 = Gammab X 1 0 : a 1 0; b 1 0; c 1 0; d 1 0 X 2 0 : a 2 0; b 2 0; c 2 0; d 2 0 N Theta min (X 1 X 2 )b : N Theta (b 1 b 2 d 1 d 2 ) To solve this problem, we can use the simplex method provided by packages such as Maple [2]. We obtain the solution: X = 7; 1) X 1 = 0; 7; 0; 1) and X 2 = 7; 0; 1; 0) The total execution time is T linear = N Theta (7 0 1 0) 8N . 5 Adding a scheduling constant to each statement We have seen in the previous section how to compute the optimal scheduling vector for a ....

B.W. Char, K.O. Geddes, G.H. Gonnet, M.B. Monagan, and S.M. Watt. Maple Reference Manual, 1988.

....determinant of a matrix whose entries are polynomials in x and y. The ill conditioned nature of the coefficient computations leads to large errors in the implicit form. Such errors do not necessarily distort the curve F (x; y) 0 a lot, but in this case they often do. For instance, using MAPLE [1] to implement the resultant method on the curve in Figure 1 with five digit floating point arithmetic leads to an implicit curve F (x; y) 0 that never gets within 100 units of the point (519; 285) The maximum error is reduced to 12 units when seven digits are used and 0.48 units with nine ....

B. W. Char, K. O. Geddes, G. H. Gonnet, M. B. Monagan, and S. M. Watt. MAPLE Reference Manual. WATCOM, Waterloo, Ontario, 1988.

....Furthermore if is calculated using the current solution in some way, then obtaining this dependence analytically may be impossible. This evaluation difficulty can be overcome (at least when is independent of u) by using one of the computer algebra packages now available, such as Maple [44] or Mathematica [127] They will symbolically evaluate the Jacobian, given a residual function, and generate source code which can be directly inserted into a Fortran subroutine. Even this is not a straightforward task however, when the problem being solved consists of a system of p.d.e. s, with ....

....2 i 1 4a i 1 hm i Gamma4a i 1 hm i 1 Gamma 2h 2 m i m i 1 h 2 m 2 i 1 ) 1 24 ( Gamma6a 2 i Gamma a i hm i a i hm i 1 6a 2 i 1 a i 1 hm i Gamma a i 1 hm i 1 ) 1 C C C C C C C C C C C C C C C C C C C C C C C C A : 7. 78) Using the computer algebra package Maple [44] to perform the complicated algebraic manipulation, we make use of (7.56) and rewrite a i Gamma2 ; a i 2 in terms of a i to obtain a comparison of the interpolants, as previously. At node x i At new node x i h Interpolated residual 2ha i a 0 i O(h 3 ) 2(ha i a 0 i a i h 2 a 00 ....

B. W. Char et al. MAPLE Reference Manual. WATCOM Publications Ltd., Waterloo, Canada, 5th edition, 1988.

....fi i = 1; fi i 0; i 2 m: 55) This could be done by using symbolic math software. For example, for the two disk problem, 0 P 3 1 : min q2A0 max i22 n ka i q 0 b i k 2 1 o ; 56) The similar result to that in (Holohan and Safonov, 1992) can be obtained by solving (53 55) via Maple (Char, et al. 1988). Corollary 1: Assume that a i q 0 b i = 0; i 2 2) have no solution in A 0 : If the two disk H1 problem 0 P 3 1 has a solution b q 2 A 0 ; then either (1) max n ja i b q 0 b i j 2 0 e j 1 ; i 2 2 o = const: 8 2 T; or (2) if for each i 2 2; a i 0 e j ....

Char, B. W., et al. (1988). Maple Reference Manual, WATCOM Publications Limited.

....While a problem can be conveniently specified in a compact form using Ctadel s high level language, the underlying numerical complexity of the problem can be huge. Much of this complexity can be reduced by exploiting algebraic simplifiers of computer algebra packages such as reduce [14] Maple [6], and Mathematica [24] However, most algebraic simplifiers of these existing computer algebra packages produce expressions in canonical forms which are easy to read but not very economical with respect to arithmetic complexity. Furthermore, a large collection of rewriting rules should be added ....

B.W. Char, K.O. Geddes, G.H. Gonnet, M.B. Monagan, and S.M. Watt, MAPLE Reference Manual, Waterloo, 1988.

....imitates the method a scientist uses and at the same time offers a natural and easy way for the specification of mathematical formulas. On the other hand, there are also relatively efficient symbolic computation systems that do not support this declarative kind of programming (e.g. Maple [4]) This gives the indication that a procedural language is more efficient. The main reason for this is the relatively bad efficiency of pattern matching algorithms used in existing declarative symbolic computation interpreters, since the pattern matcher is at the core of a rule based system. ....

B. W. Char, K. O. Geddes, G. H. Gonnet, M. B. Monagan, and S. M. Watt. MAPLE: Reference Manual. University of Waterloo, 5th edition, 1988.

....code by some compilers. Unification itself, which is at the heart of logic programming systems is usually implemented using sharing in order to avoid the combinatorial explosion that would arise from repeated duplications of subtrees. In a related context, a symbolic manipulation system like Maple [8, 7] manages storage using pointers and hashing in such a way that subexpressions exist uniquely in main memory. With this representation, the effect of applications of expansion rules that cause subtree duplications, like distributivity or symbolic differentiation, is somewhat decreased: For instance ....

Char, B., Geddes, K., Gonnet, G., Monagan, M., and Watt, S. MAPLE: Reference Manual. University of Waterloo, 1988. 5th edition.

....explanation is the following: ffl 8p 2 Iter; Xp = X 1 Ap X 1 b ffl 8q 2 Iter; GammaX q = X 2 Aq X 2 b ffl the execution time is max p;q2Iter (bXpc Gamma bXqc) 1 X 1 b X 2 b Example 1 We are already given matrices A and D, and vector b. For solving ( we can use packages like Maple [2] if there are no parameters, or a parameterized simplex algorithm like PIP [15] otherwise. The reason why we can use PIP is because parameters only occur in the objective function, but not in the constraints of the optimization problem 5 . We find that the best scheduling vector is = 1; 1; 1) ....

B.W. Char, K.O. Geddes, G.H. Gonnet, M.B. Monagan, and S.M. Watt. Maple Reference Manual, 1988.

....(Lambda Upsilon Omega) which takes as inputs specifications of combinatorial structures and characteristic parameters, and produces (in a number of cases) automatically the expected values of the parameters. The system makes extensive use of resources of the computer algebra system Maple [5]. Such an approach proves useful when analyzing complex models. A description of the current state of the system is given in [15] and it will only be illustrated by treating a sensitivity analysis problem due to Mich ele Soria who discusses systematically such phenomena in her thesis [42] The ....

B.W. Char, K.O. Geddes, G.H. Gonnet, M.B. Monagan, and S.M. Watt. MAPLE: Reference Manual. University of Waterloo, 1988. 5th edition.

....the values of the constant fl b , see Table 1. It may be of interest to note that Table 1 does not result from straight numerical integration, which would be conducive to various numerical difficulties. Its derivation was first obtained instead by symbolic integration performed by the Maple system [3]. For values 1 10 and 15(5)50, the computation took a little over 600 seconds of CPU time (on a Sun3 machine performing 3 Delta 10 6 instructions per second and equipped with 12 Delta 10 6 bytes of memory) For instance, we have for b = 50, the verbatim form of fl b , ....

....and possibly optimize various allocation strategies. In this spirit, the paper concludes with a brief survey of analytical results available for index trees of various sorts. To a large extent our Theorem 2 owes its existence to the integration capabilities of the Maple system for computer algebra [3] which revealed the unsuspected occurrence of closed form expressions involving dilogarithms and made it possible to carry out easily rather intensive computations. 2 Paged Quadtrees Our data model assumes data in random order. Without loss of generality, we take them independently and uniformly ....

Char, B. W., Geddes, K. O., Gonnet, G. H., Monagan, M. B., and Watt, S. M. MAPLE: Reference Manual. University of Waterloo, 1988. 5th edition.

....equation dx dt = x Gamma x 2 ; x(0) 1=2: Let x(t) 1 2 t 4 Gamma t 3 96 be an approximate solution on 0 t h. The approximate solution can be generated by a number of methods. Here, we introduced a deliberate error into a 4 term Taylor series. From Equation (2) using Maple [3]) ffi(t) dx dt Gamma f(x; t) 1 4 Gamma t 2 32 Gamma x x 2 = 1 32 t 2 1 Gamma 1 6 t 2 1 288 t 4 : The defect ffi(t) is a polynomial in t. There is no remaining evidence of the ODE. The function x(t) is the true solution of the equation dx dt = f(x; t) ....

B. W. Char, K. O. Geddes, G. H. Gonnet, M. B. Monagan, and S. M. Watt, MAPLE Reference Manual, Watcom Publications, Waterloo, Ontario, Canada, 1988.

....of surfaces, such as lighting, shading and color. The view reference point may also be adjusted. 5 Related Work There exist a number of systems and research areas which in some way are related to the ObjectMath programming environment. Some of these are: Computer algebra systems such as Maple [2] or Mathematica [11] Systems for matrix computations, e.g. MATLAB [7] Symbolic and numerical hybrid systems. An example is the FINGER package [9] a hybrid system supporting finite element analysis. An exhaustive survey can be found in [3] 6 Conclusions There is a strong need for efficient ....

Char, Geddes, Gonnet, Monagan, and Watt. Maple Reference Manual. WATCOM Publications, 5th edition, 1988.

....given with the array declarations. For example, p: real on i=1. n by k=1. nlev declares a two dimensional floating point array p that is implicitly indexed with the tuple (i,k) and dimensional ranges 1. m and 1. nlev. Furthermore, the Ctadel language adopts a similar mathematical style as maple [3] for the syntax of reduction operators. For Production Semantic Rule E const E:c = true E:i = false E:d = false E:m = true E var E:c = true E:i = false E:d = false E:m = true E index E:c = false E:i = true E:d = false E:m = true E id(E1 ) E:c = E1 :c E:i = id:i E1 :i) ....

B.W. Char, K.O. Geddes, G.H. Gonnet, M.B. Monagan, and S.M. Watt, MAPLE Reference Manual, Waterloo, 1988.

....of the problem can be vast. Much of this complexity can be reduced by application of algebraic simplification. To this end, the decision was made to build a dedicated algebraic simplifier within the Ctadel system. Soon it was realized that existing symbolic algebra packages such as Maple [4] and Mathematica tm [24] were not very suitable for this task. Although these packages are powerful tools with respect to, for example, symbolic differentiation, they lack a suitable algebraic simplifier that can be easily extended. Furthermore, most existing algebraic simplifiers produce ....

B.W. Char, K.O. Geddes, G.H. Gonnet, M.B. Monagan, and S.M. Watt, MAPLE Reference Manual, Waterloo, 1988.

....and expert systems help (which this community of users needs particularly badly) was being developed and introduced. 3.3 Component Areas 3.3. 1 Symbolic and Algebraic Computing Examples of symbolic and algebraic computing systems (SACs) are MACSYMA [70] 132] REDUCE [97] 98] Maple [37], Scratchpad II [115] DERIVE [181] and Mathematica [1] 197] Reference [99] provides several interesting observations on the development of SAC systems in relation to other areas of computer science. A recent important report summarizes a wealth of information about current and future ....

B. W. Char, K. O Geddes, G. H. Gonnet, M. B. Monagan, and S. M. Watt. MAPLE Reference Manual. Waterloo, 1988.

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B. W. Char, K. O. Geddes, G. H. Gonnet, M. B. Monagan, and S. M. Watt. Maple Reference Manual, 1988.

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B.W. Char, K.O. Geddes, G.H. Gonnet, M.B. Monagan, and S.M. Watt. MAPLE: Reference Manual. University of Waterloo, 1988. 5th edition.

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B. W. Char, K. O. Geddes, G. H. Gonnet, M. B. Monagan, and S. M. Watt. Maple Reference Manual, 1988.

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B. W. CHAir, K. O. GEDDES, G. H. GONNET, M. B. MONAGAN, AND S. M. WATT, MAPLE Reference Manual, Watcom Publications, Waterloo, Ontario, Canada, 1988.

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Char et al., 1988 Char, B. W., Geddes, K. O., Gonnet, G. H., Monagan, M. B., Watt, S. M. (1988). Maple Reference Manual, 5th Edition. Watcom Publications Limited, Waterloo, Ontario, Canada.

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B. W. Char, K. O. Geddes, G. H. Gonnet, M. B. Monagan, and S. M. Watt. Maple Reference Manual. WATCOM Press, 1988.

No context found.

B.W. Char, K.O. Geddes, G.H. Gonnet, M.B. Monagan, and S.M. Watt. MAPLE: Reference Manual. University of Waterloo, 1988. 5th edition.

No context found.

Char, B. W., Geddes, K. O., Gonnet, G. H., Monagan, M. B., and Watt, S. M., Maple Reference Manual, 5th edition, Watcom, 1988.

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Char, B. W., Geddes, K. O., Gonnet, G. H., Monagan, M. B., Watt, S. M. (1988). Maple Reference Manual, 5th Edition. Watcom Publications Limited, Waterloo, Ontario, Canada.

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B.W. Char, K.O. Geddes, G.H. Gonnet, M.B. Monagan, and S.M. Watt. MAPLE: Reference Manual. University of Waterloo, 1988. 5th edition.

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