Essential Maple 7 (eBook)

What’s in This Book 7
Acknowledgements 9
Contents 11
List of Figures 13
1 Basics 16
1.1 Getting Started 16
1.1.1 Basic Command Syntax 18
1.1.2 Use of Context-Sensitive Menus to Execute Maple Commands 19
1.1.3 Sample Maple Sessions 19
1.1.4 Arithmetic 36
1.1.5 Interrupting a Maple Computation 39
1.1.6 Saving Work 39
1.2 Some Things to Watch Out For 41
1.2.1 Good Worksheet Hygiene 41
1.2.2 Common Syntax Errors 42
1.2.3 Assigning Values to Variables 43
1.2.4 Removing Values from Variables 44
1.2.5 sign versus signum versus csgn 45
1.2.6 Accidental Creation of a Remember Table 46
1.2.7 Fences: Parentheses ( ) versus Braces { } versus Brackets [ ] versus Angle Brackets 47
1.2.8 Quotation marks: Left versus Right versus String 47
1.2.9 Precedence of Operators 49
1.2.10 Protected and Reserved Names 50
1.2.11 Having Different Assumptions about Domains 52
1.3 Documenting YourWork 52
1.4 The Three Levels of Maple “Black Boxes” 57
1.5 No Nontrivial Software Package is Bug-Free 58
1.6 Evaluation Rules 59
1.6.1 Working With Complex Numbers and Expressions 62
1.6.2 Inert Functions 63
1.7 The assume Facility 67
2 Useful One-Word Commands 71
2.1 Simplification 71
2.1.1 normal 72
2.1.2 collect 73
2.1.3 factor 83
2.1.4 expand 89
2.1.5 combine 90
2.1.6 simplify 91
2.2 Solving Equations 94
2.2.1 solve 94
2.2.2 fsolve 97
2.2.3 dsolve 101
2.2.4 rsolve 113
2.2.5 Linear Equations 114
2.2.6 Other Solvers 117
2.2.7 Systems of Polynomial Equations 117
2.3 Manipulations from Calculus 125
2.3.1 diff 125
2.3.2 int 127
2.3.3 limit 135
2.3.4 series 135
2.4 Adding Terms versus the Finite-Difference Calculus 139
2.5 Floating-Point Evaluation 144
2.5.1 Using evalhf 146
2.5.2 Signed Zero 150
2.6 The Most Helpful Maple Utilities 150
2.6.1 I/O Utilities 151
2.6.2 alias and macro 152
2.6.3 Interacting with the Operating System and External Calls 152
2.6.4 Mapping Functions Onto Compound Objects 152
2.6.5 Code Generation 154
2.7 Plotting in Maple 157
2.7.1 Two-Dimensional Plots 157
2.7.2 Three-Dimensional Plots 170
2.7.3 Contour Plots and Other Plots 174
2.7.4 Common Errors 183
2.7.5 Getting Hard Copy of your Plots 185
2.8 Packages in Maple 188
2.8.1 The MATLAB Link 189
2.8.2 numapprox 193
2.8.3 Units 194
2.8.4 MathML 199
3 Programming in Maple 201
3.1 Procedures 202
3.1.1 Structured Types 205
3.1.2 Example: Modified Gram–Schmidt 206
3.2 Operators and Modules 211
3.2.1 A Module for Finite-Difference Operators 216
3.2.2 Remarks on Mathematical Operators 219
3.3 Data Structures 220
3.4 Local versus Global versus Environment Variables 226
3.4.1 Exporting Local Variables 226
3.4.2 Global Variables 227
3.4.3 Environment Variables 227
3.4.4 Nested Lexical Scopes 230
3.5 Recursion and option remember 230
3.6 Variable Number or Type of Arguments 239
3.7 Returning More Than One Result 241
3.8 Debugging Maple Programs 243
3.9 Sample Maple Programs 250
3.9.1 Parametric Solution of Algebraic Equations 250
3.9.2 Path Following in p(x, y) = 0 254
3.9.3 Large Expression Management, Revisited 260
3.9.4 Fourier Sine Series, Revisited 260
3.9.5 Solution of y (t) = ay(t - 1) 266
Appendix A A Primer on Complex Variables 273
A.1 Polar Coordinates and the Two-Argument Arctan Function 274
A.2 The Exponential Function 275
A.3 The Natural Logarithm 277
A.4 Trig Functions and Hyperbolic Functions 279
A.5 Inverse Trigs and Hyperbolics 279
Bibliography 286
Index 290
More eBooks at www.ciando.com 0

3 Programming in Maple (p.186-187)

[Lady Fiorinda was] in the theoric of these matters liberally grounded    through daily sage expositions and informations by Doctor Vandermast, who had these four years past been to her for instructor and tutor. To try her paces and put in practice the doctor’s principles and her own most will-o’-the-wisp and unexperimental embroiderings upon them, ready means lay to hand. . .
- E. R. Eddison, The Mezentian Gate, Book VI.


Maple is useful as a collection of "black boxes," but it is more useful still as a very high-level programming language. Since most of the tasks undertaken by Maple are "one-off" calculations (as opposed to "batch" calculations, which require many executions of the same program), it makes sense that Maple is an interpreted, rather than compiled, language. This is true even for the Maple library, because for large problems the cost will be dominated by the manipulation of large objects. Some crucial operations, though, are performed by kernel routines, which are compiled for efficiency.

With the external calling features, new to Maple 6 and improved for Maple 7, it is now possible to "tune" these efficiency tradeoffs more closely for any particular application. With the external calling features, new to Maple 6 and improved for Maple 7, it is now possible to "tune" these efficiency tradeoffs more closely for any particular application.

Maple procedures can be divided loosely into two types: operators, and more general procedures. Procedures and related structures may be bundled together into a module. An operator is meant to imitate a mathematical operator, both in notation (insofar as this is possible in ASCII) and in action. The .rst section of this chapter deals with general procedures and their uses. These can do essentially anything computable. Since Maple is a high-level language, you can express these actions in many ways. The section following that looks at operators.

For more in-depth information on how to program in Maple, see [44] and the detailed examples in the directory samples/ch06, which can be found on Windows systems in the directory Program Files/Maple 7. For an extended example of revising a program for efficiency, see [50]. For examples of useful programs, see [21].

3.1 Procedures

A Maple procedure always returns a value. It is the value of the last statement executed in the procedure before returning, or else the value of an explicit return statement. See ?return. This value may be NULL, which does not print anything on output. The distinction between NULL and "no value" is academic. One important consequence of a procedure returning NULL is that the environment variables %, %%, and %%% are not changed. The procedures print and lprint use this deliberately. [We will discuss environment variables in more detail later.] The procedures solve, fsolve, and dsolve, for example, will return NULL if they .nd no solution, and sometimes this takes special handling in programs. One simple way to deal with this with solve is to enclose the results from solve in set braces ( { } ), converting a possible NULL value to the empty set (and incidentally removing multiplicity; use list brackets if you wish to preserve multiplicity).