updates
v43
New features that have been added to Maple for version 4.3
----------------------------------------------------------
Development of improved user interfaces
---------------------------------------
The user interface, known as Iris, is programmed separately from the
Maple "algebra engine". Availability of particular features is workstation-
dependent. All implementations of Maple include a new version of the pretty-
printer (see below) and a new lexical scanner. The lexical scanner now pro-
vides improved error messages, with a caret symbol pointing to the most likely
place where a syntax error has occurred and the line number at which it
occurred is also specified (for input from a file).
Versions of Iris which exploit workstation features such as windows,
mouse selection, session history, and editing have been developed. An Iris
facility with these advanced features is available for the Macintosh and will
soon be released for X-Windows and SunView.
3-D plotting
------------
A 3-D plotting facility has been developed. Availability of this facil-
ity is workstation-dependent. The 3-D plotting facility will soon be released
for X-Windows, SunView, and the Macintosh. Both 2-D and 3-D plotting exploit
the new `evalhf` function (see below) to achieve improved efficiency of point
generation.
New prettyprinter
-----------------
The generic Iris facility available on all host systems now uses a new
prettyprinter. The new prettyprinter supports the following:
1) break on interrupt -- the new prettyprinter is interruptible;
2) prettyprinting of Maple procedures;
3) columns of matrices are aligned where possible;
4) automatic labelling of common subexpressions in large output
as %1, %2, ... ;
5) the labelled subexpressions can be used for subsequent input.
Improved help facility
----------------------
The help facility has been improved so that the information is displayed
one screenfull at a time, when such a facility is supported by the host system.
In addition, when the help system fails to find information for the word speci-
fied, a list of "nearby names" is displayed. The definition of "nearby names"
is all names known to help which match the given name in the first three char-
acters. For example, help(f) will display all names known to help which begin
with the letter f. A facility to generate examples of Maple expressions is
also available.
Macro and alias facilities
--------------------------
A macro facility and an alias facility provide the user with a mechanism
for macros and abbreviations, respectively. For example, a user may state:
alias(J=BesselJ);
This allows the user to use the abbreviation J on input, Maple receives the
value BesselJ which it understands, and the abbreviation J will be used on out-
put.
Rule-based facility for function definition
-------------------------------------------
A new facility allows functions to be defined via the specification of a
sequence of rules which will be applied based on pattern and type matching.
For example, the following defines a linear operator:
define( L, forall( [x,y], L(x+y) = L(x)+L(y) ),
forall( [constant(n),x], L(n*x) = n*L(x) ) );
Operators may also be defined in terms of properties such as commutative, asso-
ciative, antisymmetric, etc. See help(define) and its subtopics.
Internal support for polynomials over the integers modulo n
-----------------------------------------------------------
An internal routine called 'modp1' provides support for arithmetic in
Zn[x], i.e. univariate polynomials over the integers modulo n. Modp1 greatly
enhances the performance of many operations including greatest common divisors,
resultants, and factorization of univariate polynomials over the integers.
Modp1 is designed to compute with both small primes (word-size primes) and
large primes. All the basic arithmetic operations are supported including Quo,
Rem, Eval, Interp, Gcd, Gcdex, Resultant, Discrim, Irreduc, Roots, and Factors.
Modp1 also makes possible an efficient implementation of Galois fields (see GF
and evalgf).
Hardware floating-point evaluation
----------------------------------
A new function 'evalhf' is available. Like 'evalf' its purpose is to
evaluate an expression in floating-point mode but the new function evaluates
using the hardware floating-point capabilities (if available) of the host sys-
tem. In particular, it uses the fixed-precision numerical accuracy of the
"double precision" hardware floats. The argument may be an expression involv-
ing the invocation of built-in functions or Maple procedures but all data must
evaluate to numeric values (unlike evalf which allows mixed symbolic and
numeric data). Evalhf achieves much faster numeric evaluation than evalf which
uses a software floating-point facility.
Numerical computation facilities
--------------------------------
The numerical integration facility `evalf/int` exploits the new evalhf
function to achieve improved speed of execution (at moderate precision require-
ments). There have also been functionality improvements in singularity-
handling and infinite-range capabilities. The inert function name Int is known
to evalf and therefore the invocation evalf(Int(f,x)) can be used to apply
numerical integration without first invoking symbolic integration.
New numerical computation facilities include Eigenvals (eigenvalue compu-
tation) and Svd (singular value decomposition). The floating-point solver
(fsolve) has been enhanced to handle a wider class of problems, including the
computation of all complex roots of a polynomial.
New mathematical functions and constants known to Maple
-------------------------------------------------------
- the Bessel functions BesselJ, BesselY, BesselI, BesselK
- the Beta function Beta
- the generalized Gamma constants gamma(1), gamma(2), ... (replacing the
former notation gamma1, gamma2, ...)
- the Fresnel integrals C and S have been renamed FresnelC and FresnelS
- the evalf knowledge of the Psi function has been enhanced to handle the n-th
derivative Psi(n,x)
- the incomplete GAMMA function
GAMMA(a,x) = int( exp(-t)*t^(a-1), t = x..infinity)
Enhanced integration facilities
-------------------------------
The Risch integration algorithm now handles some cases of radical func-
tions. The Risch-Norman method is now applied prior to the standard Risch
algorithm, providing a fast heuristic method which deals with trigonometric
functions in real form rather than in complex exponential form.
New capabilities for definite integration have been added, particularly
for infinite ranges when the integrand involves exponential, logarithmic, and
trigonometric functions.
New simplification routines
---------------------------
A facility for simplification with respect to side relations (rewrite
rules) has been installed. For example,
e := -1/3*x^5*y + x^4*y^2 + 1/3*x*y^3 + 1;
simplify( e, {x^3 = x*y, y^2 = x+1} );
causes the rewrite rules (specified as a set of equations) to be applied to the
expression e via an algorithm which uses a Groebner basis, resulting in the sim-
plified expression:
x^2*y + x^2 + x + 1 .
See help(simplify,siderels) for further information.
The following special functions are now known to simplify -- see
help(simplify) for the complete list of simplification procedures:
BesselJ,BesselY Bessel functions
BesselI,BesselK Modified Bessel functions
Ei Exponential integral
dilog Dilogarithm function
Other simplification enhancements:
- simplify/GAMMA extended; it now knows Gauss's multiplication formula, the
reflection formula, and it handles quotients of GAMMA functions
- simplify/power now does a^x*(-a)^y ==> (-1)^y*a^(x+y)
- simplify/hypergeom now includes all summation theorems given in Appendix III
of L.J.Slater's Generalized Hypergeometric Functions
- simplify(..,infinity) simplifies expressions which contain infinity when
there are no ambiguities
Combine function
----------------
The new function "combine" provides another class of transformations
which are a form of simplification. Its purpose is to combine terms of
specific forms, in a sense which is generally the inverse of expand. It han-
dles trigonometric functions, exponentials, logarithms, powers, limits, sums,
and integrals. For example:
combine( sin(x)*cos(x), trig ); ==> 1/2*sin(2*x)
expand( 1/2*sin(2*x) ); ==> sin(x)*cos(x)
New features of fsolve
- incorporation of the option "complex" to compute complex roots of polynomials
- new options and new formats (ranges can use "infinity" now, and a single
range is used for all variables)
- a new algorithm for a Newton iteration on systems of equations
- a new algorithm for systems of equations based on reducing the system (with
the symbolic solver) for all the "simple" substitutions, and then solving
numerically for what is left
- a new algorithm for polynomials works in three stages: a heuristic method
based on a Newton-type iteration complemented with bisection, a complex New-
ton iteration to find complex quadratic factors, and a routine for refining
the roots found; the refinement of roots is done for each root over the ori-
ginal polynomial before deflating and hence relative errors should be signi-
ficantly reduced
- for scalar equations, the scalar zero finder has been made faster by recog-
nizing the case of slow convergence and accelerating the steps in such a
case.
Type expressions and structured types
-------------------------------------
The notion of a type (as defined by the type function) has been extended
to include "structured types". A structured type is a Maple expression which
is interpreted as a composite type; for example, list(algebraic),
name=integer..integer . Structured types allow for more concise and complete
argument checking and type analysis. For example,
type( expr, name=algebraic..algebraic );
returns true iff expr is an equation whose left-hand side is a name and whose
right-hand side is a range of algebraic's. See help(type,structured) for
details. Additionally, the following new types are known:
type/RootOf a RootOf expression
type/anything matches any type
type/even an even integer
type/odd an odd integer
type/negative a negative number
type/nonneg a non-negative number
type/positive a positive number
type/square a perfect square
type/type a valid Maple type expression
The interface function
The interface function allows one to set or inquire about values relating
to the user interface. This provides a unique mechanism for communication
between the Iris interface and the Maple "algebra engine". The variables
relating to the interface are:
variable type/value description
=============================================================================
echo 0,1,2,3,4 for echoing of input
indentamount integer indentation for multiline expressions/procedures
labelling true/false labelling of expressions on output
labelwidth integer minimum size to be labelled
prettyprint true/false for two dimensional output
printfile string system command for displaying a file
prompt string user prompt
quiet true/false suppress auxiliary printing
screenheight integer screen height in characters
screenwidth integer screen width in characters
plotdevice string name of the plotting device
plotoutput string file name for plot output
preplot list(integer) terminal escape sequence
postplot list(integer) terminal escape sequence
terminal string terminal setup for plotting
verboseproc 0,1,2 controls printing of procedure bodies
version string name of the interface
Miscellaneous new features in version 4.3
-----------------------------------------
- series expansion of unknown functions now uses the D operator
- enhanced computation of generalized series, particularly for the case of tri-
gonometric functions with non-analytic function arguments
- asymptotic series with respect to expressions other than names; e.g.
asympt( sin(1/ln(n)), ln(n) )
- gcd now has special code for the case of a very small against a very large
argument; it factors the small one and does trial divisions
- a facility for modifying or creating Maple procedures is provided via two
functions: procbody and procmake; a neutralized language is used to
represent as expressions the <statement sequence> which makes up the body of
a procedure
- the optimize code includes a utility routine which allows a Maple procedure
to be dynamically constructed from a computation sequence; see
help(optimize,makeproc) for details
- match has been extended to handle various cases with missing powers which
were not previously handled
- definite and indefinite summation have improved handling of logarithms, and
of Psi(x) and its derivatives
- d in the difforms package does a better job of expanding a scalar w.r.t. a
basis of one-forms; for example, d(f, [d(x),d(y),d(z)]) gives
fx*d(x)+fy*d(y)+fz*d(z)
- dsolve, laplace, and powseries[powsolve] use the D operator to express ini-
tial conditions; for example, D(x)(0)=5 instead of xp(0)=5
- dsolve(..,..,series) now uses the method of Frobenius for linear differential
equations at regular singular points
- dsolve now solves systems of linear ODEs (previously handled only via
laplace transforms or series)
- convert(....,rational,exact) with the new option "exact" performs exact
conversion of a floating-point number to a rational number (rather than the
more intelligent and more expensive conversion via continued fractions)
- the function symmpoly has been renamed recipoly (computes the "reciprocal
polynomial")
- three-valued logic: true, false, or FAIL, is now used when evaluating in a
boolean context; FAIL is treated like false for branching decisions and for
evaluating boolean expressions
- internal format files (".m" files) are now identified by version so that
Maple will not attempt to read incorrect ".m" files; furthermore, Maple
understands how to read ".m" files whether they have been generated by a
VAX-type integer ordering or its reverse
- sort is now an internal function; it uses a variation of the basic merge
sort algorithm, consequently sorting is much faster
- cat is now an internal function
- the evalgf evaluator has been added for computing over non-prime finite
fields (i.e. GF(p^k) == algebraic extension of degree k over Z mod p for a
prime p); multiple extensions are supported
- various drivers have been added to the plotting routine; the present list of
drivers is:
plotdevice description
_______________________________________________________________
char character plot
i300 imagen 300 laser printer
ln03 DEC LN03 laser printer
mac Macintosh plot
pic output file for use with the troff pic preprocessor
postscript PostScript-compatible printer
regis terminals with Regis graphics
tek Tektronix graphics terminals
1 output file for use with the Unix plot command
vt100 VT100 line graphics
- when loading a package via "with", the list of functions available in the
package is now sorted
- the memory usage messages now report in bytes, rather than words
- diff can now operate over equations, lists, and sets
- taylor now checks that the resulting series is a true Taylor series and
returns an error otherwise; the series function will compute a generalized
series expansion
- `if` function incorporated into Maple. E.g.
absx := `if`( x<0, -x, x );
- asymptotic and generalized series expansions can be computed for unevaluated
recurrences
- generalized series, asymptotic series, and limits can be computed for
unevaluated integrals
- limit now handles the GAMMA and erf functions for all cases
- the "asympt" function now can develop asymptotic series with essential singu-
larities; these have the same power as the generalized hierarchical series
used in limit computation
- collect now supports an option to allow any Maple function (procedure) to be
applied to the coefficients
- computation over algebraic number fields has been improved via the use of
fast heuristic methods and improved factorization over the rationals
New functions
-------------
(use "help(f);" in a Maple session for information about f)
alias Define an abbreviation or denotation
allvalues Evaluate expressions involving RootOf's and return all values
bernstein Bernstein polynomial approximating a function
combine Combine terms into a single term
cost Operation evaluation count
Det Inert determinant function for use with mod and modp1
Eigenvals Inert eigenvalues function, for use with evalf
Eval Inert evaluation function, for use with the mod operator
evalgf Finite field arithmetic (including polynomial factorization)
evalhf Evaluate using hardware floating-point arithmetic
example Generate a random example for a specified function
Expand Inert expand function, for use with evala, evalgf, and mod
extrema Find relative extrema of given expression
Factors Inert factors function, for use with mod and modp1
factors Generate a list of factors and multiplicities
Functions Maple function information
GF Galois field (finite field) arithmetic
ifactors Generate a list factors and multiplicities for an integer
interface Setting and inquiry of interface parameters
Irreduc Inert irreducibility test function, for mod and modp1
isqrfree Integer square-free factorization
lhs The left-hand-side of an equation, relation, or range
macro Define a macro (input abbreviation)
modp1 Efficient polynomial arithmetic over the integers mod n
modpol Expression evaluation in a quotient field
norm Compute the norm of a polynomial
Normal Inert normal form function, for evala and evalgf
optimize Generate an optimized computation sequence
plot3d Generate a 3-dimensional surface plot
plotsetup Parameter setting for plotting
Power Inert power function, for use with mod and modp1
Powmod Inert powering modulo a polynomial, for use with mod and modp1
procbody Maple procedure created in a "neutralized form"
procmake Maple procedure creation
recipoly Reciprocal polynomial
rhs The right-hand-side of an equation, relation, or range
Roots Inert roots function, for use with mod and modp1
Sqrfree Inert square-free factorization function, for evala, mod, modp1
Svd Inert singular value decomposition function, for evalf
series Generalized series expansion (replaces taylor)
taylor Now computes a pure Taylor series, else returns an Error
unassign Unassign a sequence of names
convert/hypergeom Convert summations to hypergeometrics
convert/mod2 Reduce modulo 2
New functions in packages
-------------------------
(use "help(<packagename>,<funcname>);" for help)
grobner
finduni Find smallest univariate polynomial in an ideal
finite Determine if system has finitely many solutions
gsolve Reduced Groebner basis (for solving a system of polynomials)
solvable Determine if system is solvable
linalg
basis Basis of a vector space
charmat Characteristic matrix
charpoly Characteristic polynomial
col Pick off the j'th column of a matrix as a vector
copyinto Copy one matrix into another
eigenvects Eigenvectors of a matrix of rationals
equal Test for matrix equivalence
exponential Matrix exponential
extend Extend a matrix by n rows and m columns
fibonacci Fibonacci matrix
gaussjord Gauss-Jordan form (row echelon form) over a field
hermite Hermite normal form (row echelon form) for polynomials
htranspose Hermitian transpose
ihermite Hermite normal form (row echelon form) for integers
intbasis Basis for intersection of spaces
iszero Test for zero matrix
mdet removed -- use Det(..) mod p
minpoly Minimal polynomial
randmatrix Generate random matrix
row Pick off the i'th row of a matrix as a vector
sumbasis Basis for sum of spaces
swapcol Changed to run in place (destructive)
swaprow Changed to run in place (destructive)
numtheory
B Synonym for bernoulli
binomial Moved to combinat
E Synonym for euler
fermat Fermat primes
fibonacci Moved to combinat
issqrfree Test for square-free (formerly named isqrfree)
mersenne Mersenne primes
nthpow Greatest integer power which divides n
sigma Sum of the positive divisors of an integer
student
leftbox Graph of an approximation to an integral
leftsum Numerical approximation to an integral using rectangles
middlesum Numerical approximation to an integral using rectangles
rightbox Graph of an approximation to an integral
rightsum Numerical approximation to an integral using rectangles
showtangent Plot of a tangent to a function
simpson Numerical approximation to an integral using Simpson's rule
trapezoid Numerical approximation to an integral using trapezoids
value Evaluate inert forms (formerly named Eval)
New packages
(use "help(<packagename>);" for information about a specific package)
combinat
A combinatorial functions package has been added. It is named "combinat"
and it includes the following functions:
bell Bell numbers B(n)
binomial Binomial coefficients C(n,k)
combine Construct the combinations of size m from n objects
combinations Count the combinations
fibonacci Fibonacci numbers F(k)
multinomial Multinomial coefficients (n; k1, k2, ..., km)
partition Construct the partitions of an integer
partitions Count the partitions
kpartition Construct the k-partitions of an integer
kpartitions Count the k-partitions
permute Construct the permutations of a list of objects
permutations Count the permutations
powerset Generate the power set for a set or multiset
randperm Generate random permutation of n objects
stirling1 Stirling numbers of the first kind
stirling2 Stirling numbers of the second kind
geometry
A package dealing with two-dimensional Euclidean geometry has been added.
The package is named "geometry" and the following functions are avail-
able:
Appolonius Eulercircle Eulerline Gergonnepoint
Nagelpoint Simsonline altitude are_collinear
are_concurrent are_harmonic are_orthogonal are_parallel
are_perpendicular are_similar are_tangent area
bisector center centroid circle
circumcircle conic convexhull coordinates
detailf diameter distance ellipse
excircle find_angle harmonic incircle
inter inversion is_equilateral is_right
line make_square median midpoint
on_circle on_line onsegment orthocenter
parallel perpen_bisector perpendicular point
polar_point pole_line powerpc projection
rad_axis rad_center radius reflect
rotate sides similitude square
symmetric tangent tangentpc triangle
projgeom
A package dealing with projective geometry has been added. This package,
named "projgeom", includes the following functions:
collinear concur conic conjugate ctangent
fpconic harmonic inter join lccutc
lccutr lccutr2p line linemeet midpoint
onsegment point polarp poleline ptangent
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