Keith O. Geddes

Maple V: programming guide

1. Introduction.- 1.1 Getting Started.- Locals and Globals.- Inputs, Parameters, Arguments.- 1.2 Basic Programming Constructs.- The Assignment Statement.- The for Loop.- The Conditional Statement.- The while Loop.- Modularization.- Recursive Procedures.- Exercise.- 1.3 Basic Data Structures.- Exercise.- Exercise.- A MEMBER Procedure.- Exercise.- Binary Search.- Exercises.- Plotting the Roots of a Polynomial.- 1.4 Computing with Formulae.- The Height of a Polynomial.- Exercise.- The Chebyshev Polynomials, Tn(x).- Exercise.- Integration by Parts.- Exercise.- Computing with Symbolic Parameters.- Exercise.- 2. Fundamentals.- 2.1 Evaluation Rules.- Parameters.- Local Variables.- Global Variables.- Exceptions.- 2.2 Nested Procedures.- Local or Global?.- The Quick-Sort Algorithm.- Creating a Uniform Random Number Generator.- 2.3 Types.- Types that Modify Evaluation Rules.- Structured Types.- Type Matching.- 2.4 Choosing a Data Structure: Connected Graphs.- Exercises.- 2.5 Remember Tables.- The remember Option.- Adding Entries Explicitly.- Removing Entries from a Remember Table.- 2.6 Conclusion.- 3. Advanced Programming.- 3.1 Procedures Which Return Procedures.- Creating a Newton Iteration.- A Shift Operator.- 3.2 When Local Variables Leave Home.- Creating the Cartesian Product of a Sequence of Sets.- Exercises.- 3.3 Interactive Input.- Reading Strings from the Terminal.- Reading Expressions from the Terminal.- Converting Strings to Expressions.- 3.4 Extending Maple.- Defining New Types.- Exercises.- Neutral Operators.- Exercise.- Extending Certain Commands.- 3.5 Writing Your Own Packages.- Package Initialization.- Making Your Own Library.- 3.6 Conclusion.- 4. The Maple Language.- 4.1 Language Elements.- The Character Set.- Tokens.- Token Separators.- 4.2 Escape Characters.- 4.3 Statements.- The Assignment Statement.- Unassignment: Clearing a Name.- The Selection Statement.- The Repetition Statement.- The read and save Statements.- 4.4 Expressions.- Expression Trees: Internal Representation.- The Types and Operands of Integers, Strings, Indexed Names, and Concatenations.- Fractions and Rational Numbers.- Floating-Point (Decimal) Numbers.- Complex Numerical Constants.- Labels.- Sequences.- Sets and Lists.- Functions.- The Arithmetic Operators.- Non-Commutative Multiplication.- The Composition Operators.- The Ditto Operators.- The Factorial Operator.- The mod Operator.- The Neutral Operators.- Relations and Logical Operators.- Arrays and Tables.- Series.- Ranges.- Unevaluated Expressions.- Constants.- Structured Types.- 4.5 Useful Looping Constructs.- The map, select, and remove Commands.- The zip Command.- The seq, add, and mul Commands.- 4.6 Substitution.- 4.7 Conclusion.- 5. Procedures.- 5.1 Procedure Definitions.- Mapping Notation.- Unnamed Procedures and Their Combinations.- Procedure Simplification.- 5.2 Parameter Passing.- Declared Parameters.- The Sequence of Arguments.- 5.3 Local and Global Variables.- Evaluation of Local Variables.- 5.4 Procedure Options and the Description Field.- Options.- The Description Field.- 5.5 The Value Returned by a Procedure.- Assigning Values to Parameters.- Explicit Returns.- Error Returns.- Trapping Errors.- Returning Unevaluated.- Exercise.- 5.6 The Procedure Object.- Last Name Evaluation.- The Type and Operands of a Procedure.- Saving and Retrieving Procedures.- 5.7 Explorations.- Exercises.- 5.8 Conclusion.- 6. Debugging Maple Programs.- 6.1 A Tutorial Example.- 6.2 Invoking the Debugger.- Displaying the Statements of a Procedure.- Breakpoints.- Watchpoints.- Error Watchpoints.- 6.3 Examining and Changing the State of the System.- 6.4 Controlling Execution.- 6.5 Restrictions.- 7. Numerical Programming in Maple.- 7.1 The Basics of evalf.- 7.2 Hardware Floating-Point Numbers.- Newton Iterations.- Computing with Arrays of Numbers.- 7.3 Floating-Point Models in Maple.- Software Floats.- Hardware Floats.- Roundoff Error.- 7.4 Extending the evalf Command.- Defining Your Own Constants.- Defining Your Own Functions.- 7.5 Using the Matlab Package.- 7.6 Conclusion.- 8. Programming with Maple Graphics.- 8.1 Basic Plot Functions.- 8.2 Programming with Plotting Library Functions.- Plotting a Loop.- A Ribbon Plot Procedure.- 8.3 Maples Plotting Data Structures.- The PLOT Data Structure.- A Sum Plot.- The PLOT3D Data Structure.- 8.4 Programming with Plot Data Structures.- Writing Graphic Primitives.- Plotting Gears.- Polygon Meshes.- 8.5 Programming with the plottools Package.- A Pie Chart.- A Dropshadow Procedure.- Creating a Tiling.- A Smith Chart.- Modifying Polygon Meshes.- 8.6 Example: Vector Field Plots.- 8.7 Generating Grids of Points.- 8.8 Animation.- 8.9 Programming with Color.- Generating Color Tables.- Adding Color Information to Plots.- Creating A Chess Board Plot.- 8.10 Conclusion.- 9. Input and Output.- 9.1 A Tutorial Example.- 9.2 File Types and Modes.- Buffered Files versus Unbuffered Files.- Text Files versus Binary Files.- Read Mode versus Write Mode.- The default and terminal Files.- 9.3 File Descriptors versus File Names.- 9.4 File Manipulation Commands.- Opening and Closing Files.- Position Determination and Adjustment.- Detecting the End of a File.- Determining File Status.- Removing Files.- 9.5 Input Commands.- Reading Text Lines from a File.- Reading Arbitrary Bytes from a File.- Formatted Input.- Reading Maple Statements.- Reading Tabular Data.- 9.6 Output Commands.- Configuring Output Parameters using the interface Command.- One-Dimensional Expression Output.- Two-Dimensional Expression Output.- Writing Maple Strings to a File.- Writing Arbitrary Bytes to a File.- Formatted Output.- Writing Tabular Data.- Flushing a Buffered File.- Redirecting the default Output Stream.- 9.7 Conversion Commands.- C or FORTRAN Generation.- LATEX or eqn Generation.- Conversion between Strings and Lists of Integers.- Parsing Maple Expressions and Statements.- Formatted Conversion to and from Strings.- 9.8 A Detailed Example.- 9.9 Notes to C Programmers.- 9.10 Conclusion.

A tutorial introduction to Maple

The Maple computer algebra system is described. Brief sample sessions show the user syntax and the mathematical power of the system for performing arithmetic, factoring, simplification, differentiation, integration, summation, solving algebraic equations, solving differential equations, series expansions, and matrix manipulations. Time and space statistics for each sample session show that the Maple system is very efficient in memory space utilisation, as well as in time. The Maple programming language is presented by describing the most commonly used features, using some non-trivial computations to illustrate the language features.

Polynomial Approximation by Projections on the Unit Circle

In the space

The design of maple: A compact, portable and powerful computer algebra system

A(C)

GCDHEU: Heuristic polynomial GCD algorithm based on integer GCD computation

of functions continuous on the closed unit disc and analytic in interior points normed in the minimax sense, it is proved that the projection

Evaluation of classes of definite integrals involving elementary functions via differentiation of special functions

T_n

Near-Minimax Polynomial Approximation in an Elliptical Region

onto truncated Taylor series is a minimal projection onto polynomials. Moreover by computing a bound for

On the derivatives of the Bessel and Struve functions with respect to the order

\| {T_n } \|

Convergence behavior of the Newton iteration for first order differential equations

it is shown that

Exploiting fast hardware floating point in high precision computation

T_n f