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.

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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.

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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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