# zbMATH — the first resource for mathematics

Waterloo Maple Software. Maple V student version. Release 3. Notes. Macintosh version 6.07, 7.01, 7.1x. (English) Zbl 0817.68060
Berlin: Springer-Verlag. 3 disc with instructions: vi, 56 p. (1994).
[For a review of the tutorial book (1992) see Zbl 0758.68037.]
Systems for the manipulation and evaluation of symbolic expressions have gained in the most recent years a proper niche in the world of scientific software.
Their support is, by today, invaluable to the work of the mathematicians, pure and applied, both in research and in teaching.
Among the several systems now available, Maple is one of the best and this new release confirms its high quality and dependability.
The Student Version, release 3, for Macintosh requires 4MB of RAM and 14- 16MB of hard disk space. The use of FPU is optional but highly recomandable if any complex computation should be carried out. It works with version 6.07 or higher of the Macintosh Operating System.
The Student version differs from the complete version only for the size of the computations it is able to carry out: max stack size 32Kb, max user memory allocated 4Mb, max array size 5000, versus unlimited capacity of the full system and max floating-point digits 100 versus the 500,000 of the full version. Moreover the share library is not shipped with the student version but instructors could easily acces to it over the network and make portions of it available on an “as needed” basis.
The documentation for the Student version includes a “Getting Started” booklet, a “Release 3 notes” booklet and a hardcover book titled “First Leaves: A Tutorial Introduction To Maple V”.
The “Getting Started” bookled explains how to install the software and gives a fast tour of the Maple features. The “Release 3 Notes” give an overview of the features added, or changed, in this new release.
The tutorial book is unchanged with respect to the previous releases. It has been already reviewed in the Zentralblatt.
The installation of the software is easy, althought a more modular structure would have be convenient. For example, the user has to download on the hard disk the Quick Time utility even if it is already installed.
The “Release 3 Notes” after a short introduction enumerate, in successive sections, the new features about the Interface, the Mathematics, the Graphics and the System and Language. A brief description of the Student Version and an Index conclude the document.
The more relevant news about the Interface are the possibility to save entire worksheets in Postscript or LaTeX format, a hierarchical help browser, new very convenient help commands to get, selectively, the first line, or the calling sequence section, or the examples sections, or the ‘see also’ section of the help page and, finally, an on-line, 14 chapters, customizable tutorial.
The list of new or enhanced Mathematics features is too long to be analyzed here in any detail. The following is only a short summary of them.
For Definite and Indefinite Integration Maple has now better capacity to use, manipulate and integrate special functions.
The definition of signum has changed so that signum(0) is now undefined by default. The value of signum(0) can however be fixed by the user.
The Laplace functions have been extended to handle convolutions.
In the new version the function $$\text{Root} Of(a(x) = x,x,z)$$ specifies the complex root of $$a(x)$$ which is near the numerical approximation $$z$$.
The definition of minimize now supports the option infinite which allows to minimize over the closed real interval $$[\infty,\infty]$$. The default is to minimize over $$(-\infty,\infty)$$. For polynomials, if the minimum cannot be expressed in terms of radicals, it is now expressed in the RootOf notation.
Release 3 has better capacity for solving large systems which have algebraic functions solutions. There is also added power in solving systems of trigonometric equations.
Complex numerical evaluation of the Riemann Zeta function and its derivatives has been implemented.
Maple V Release 3 can factor univariate polynomials over $$GF (p^ k)$$ and multivariate polynomials over finite fields.
Some more tools for manipulating unevaluated sums, integrals, products and limits are now available.
Several new tools are available in Maple for the manipulation of Differential Equations. First of all, a new structure, DESol, has been introduced for representing the solutions of differential equations. The purpose of DESol is to allow Maple to represent and manipulate the solution of a DE symbolically without first computing it’s solution in closed form – which often is not possible. The DESol structure then is rather like the RootOf structure which represents a solution to an algebraic equation. Presently Maple knows how to differentiate, integrate, generate a series expansion, and numerically evaluate a DESol.
For ODE’s where the characteristic polynomial cannot be factored, Maple now expresses the solution in terms of the roots of the characteristic polynomial using a RootOf. Moreover Maple now implements the exponential linear ODE algorithm of Manuel Bronstein. This algorithm finds solutions whose logarithmic derivative is in the coefficient field, i.e. $$y'(x)/y(x)$$ is a rational function.
An alternative output form is available in terms of a basis. Alternatively, the output can be a list of plottable procedures. This is done implementing a new numerical algorithm called dverk78 of W. Enright which guarantees a preset accuracy.
The Radical Simplifications has been changed to avoid the, sometime, inconsistent simplifications of previous Releases. The transformations made by the sqrt, simplify, and combine functions to square roots and radicals which are not correct for all values of the arguments are no longer made. Only provable correct transformations are made as determined by the signum, csgn, and Re and Im functions.
Since transformations that might be incorrect are no longer made, there needs to be mechanisms for the user to make these transformations when they are known to be correct. The user has two options. Using the symbolic option, the user can force Maple to make the transformation. Alternatively, the user can use the assume facility to tell Maple that expressions are real, real and positive, real and negative etc.
Nested square roots of the form $$\sqrt {r + s \sqrt {(n)}}$$ where $$n$$ is an integer, $$r$$ and $$s$$ are rationals are denested where possible by the sqrt and simplify functions. A more powerful facility for denesting and simplifying radicals is available with the radnormal command. Presently this facility only works for algebraic numbers, not algebraic functions.
A new command, rationalize, rationalizes an expression containing radicals.
Several System and Language facilities have been enhanced. Procedures may now include an optional global statement for variables.
Moreover a protect facility has been added to prevent the user from accidentally assigning to Maple system variables or other, user specified, variable.
Mapple versions now come with a library archive facility called march. This tool can be used to create, insert and update “.m” files in a Maple library archive. A new utility, makehelp, that takes as input the name of a file which contains plain text and makes a Maple TEXT object out of a file for use with the online help has also been provided.
The fortran function now accepts an optional argument mode $$= \langle \text{modtype} \rangle$$ where the mode type must be one of single (the default), double, complex or generic which specifies how function names are to be translated.
Maple strings now have no limitation on their length. Previously the length limitation was 499 characters.
The package mechanism has been extended to support subpackages. Subpackages are used by the new statistical package.
It is not possible in this review to examinate the contents and improvements of all the packages supplied with Maple. Significative enhancement have been done on the number theory package, the linear and vector algebra package, the Gaussian integer package. The Statistics Package, stats, has been completely redesigned. Since the data structure for the arguments to the functions has changed programs using the previous version of the stats package will need to be updated. Moreover the package has been partitioned in several subpackages for descriptive statistics, transforms, statistical distributions, regression and plotting.
Reviewer: G.Gallo (Catania)

##### MSC:
 68N99 Theory of software 68-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to computer science 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 68W30 Symbolic computation and algebraic computation
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