 Maple 4.0 - Maple Help

 New features that have been added to Maple for version 4.0 ---------------------------------------------------------- Changes that require modifications to old maple source codes ------------------------------------------------------------ $:$ replaces maple's seq function as the repetition operator. It can be used in the forms: expr $k=1..n replaces seq(expr, k=1..n) expr$ n          replaces  seq(expr, n) $m..n replaces seq(m..n) or $(expr, k=1..n) directly corresponds to the former seq function invocation, yielding a "quick fix" for old code using seq:  simply assign  seq := $. mod: mod is now an operator (it is a keyword like and, or, not) so function calls of the form mod(expr,p) must all be changed to expr mod p ( or to mod(expr,p) ). The functions modp and mods remain as before, and by default mod := modp. Note also that expr mod p remains unevaluated if p is not a numerical constant. union, intersect, minus: The operators used with sets are now "union" (formerly +), "intersect" (formerly *), and "minus" (formerly -). These three operator names are keywords in the language. The symbols +, *, - are now reserved for arithmetic expressions only. underscore-names: Previously, names beginning with @ were valid in Maple. Now, @ is an operator (used in the operator algebra), and the special character used for forming system names is the underscore _ . It is recommended that users refrain from using names beginning with an underscore since the system creates and uses such names globally. internal-format files: Internal-format (.m) files created by a previous version of maple cannot be read by a newer version. Such .m files must be recreated from source. --------------------------------------------------------------------------- Subscripted (indexed) names become valid Maple names. Previous versions of Maple allowed subscripts (table references) to be used only in some places where Maple expects a variable name. Now, subscripted names are fully valid names in Maple. Examples: divide(a,b,q); and member(a,b,q); irem(a,b,q); and iquo(a,b,r); for t do ... od; and seq(..., t = 1..n); taylor(..., x); and diff(..., x); map(f,a); save a, a, file; The definition of type "name" has changed. What used to be type "name" is now type "string". Type "name" is now defined to be either "string" or "indexed". Thus x, x1, x_y, t, and t[k] are all of type name. Unevaluated functions. Some functions return unevaluated. For example, if A is an array, then the array reference A[k] does not issue an error if k does not have a value. This allows, for example: sum(A[k],k=1..6) without having to quote the A[k] to prevent it from evaluating prematurely. This feature applies for any subscript (except for a sparse table). Also, it applies for the following functions: coeff(p,x,k) ==> coeff(p,x,k) op(k,p) ==> op(k,p) substring(s,1..n) ==> substring(s,1..n) a.(1..n) ==> a.(1..n) a mod p ==> a mod p Neutral Operators (&-operators). A neutral operator symbol is any string of two or more characters that starts with ampersand (&) and may contain any character except ()[]{};:'& or blank. They may be used as unary prefix operators, or infix binary operators, or as function calls. They generate function calls, with the name of the function the same as the name of the neutral operator. New evaluation rules for Locals. The evaluation rules for local variables have changed to "one-level" evaluation instead of full evaluation. This means that evaluation of local variables is the same as evaluation of formal parameters, but different than that of global variables. For example, suppose the input to a Maple session is a := b; b := 1; a; Then what does last statement yield? In previous versions of Maple a; evaluated to 1 . In this version, if these statements are entered in an interactive session (a and b are global variables) then a; still evaluates to 1 . However, if a and b are local variables and these statements appear inside a Maple procedure, then a; evaluates to b . Users should not notice any differences in normal usage of Maple. If, however, it is desired to have full evaluation or one-level evaluation explicitly, the eval function (see below) provides this functionality. Explicit remember without option remember. Previously, when the remember option was specified in a procedure, all computed values were automatically remembered. In addition, it was possible to remember a particular result by an explicit call of the form remember( f(x) = y ); but ONLY if the function f had option remember. In this version of Maple, this can be done if f does not have option remember. This is possible because unlike earlier versions, every procedure now has its own remember table. This feature allows one to remember some values but not others, thus providing greater flexibility. For example, suppose we want to write a procedure for sin(x) that automatically evaluates sin of a float to a float but does not remember any of these calculations because they would require too much space. On the other hand, we also want to remember that sin(Pi) = 0 etc. One would do: sin := proc(x) if type(x,float) then RETURN(evalf('sin(x)')) fi; 'sin(x)' end; remember(sin(0)=0); remember(sin(Pi)=0); remember(sin(Pi/6)=1/2); New evaluation function eval for one-level or full evaluation. The normal mode of evaluation is "full recursive evaluation". For example, in a := b; b := 1; a; The variable "a" evaluates to 1. Sometimes one needs to obtain a's "value", that is the object that a points to, namely "b". The eval function via eval(a,1) provides this functionality. We call this "one-level evaluation". It was previously used by save when saving in external format and for the evaluation of parameters, and is now also used for local variables. The eval function works for names, subscripted names, local variables, and formal parameters. Other expressions are returned unevaluated. In contrast, in some places one wants to evaluate locals and parameters fully, in the same mode as global variables. The eval function again provides the desired functionality via simply eval(expr). New global variable "status". This variable returns an expression sequence of length seven, which contains (words used, words allocated, time, words used increment, gc increment, words collected at last gc, words available at last gc), respectively. The words used message now prints the first three status values after every 20000 words used. The following previously-existing functions are now automatically loaded: ------------------------------------------------------------------------ gcdex extended Euclidean algorithm for polynomials isolve solve for integer solutions mgcdex extended Euclidean algorithm for polynomials modulo p msolve solve for solutions in the integers modulo p resultant resultant of two polynomials testeq random-polynomial-time equivalence tester New functions (use "help(f);" in a maple session for information about f): ------------------------------------------------------------------------- chebyshev compute the Chebyshev series of a function compoly find a non-trivial composition of a polynomial chrem Garner's Chinese remainder algorithm define define characteristics of an operator difforms differential forms package dsolve differential equation solver edit expression editor (for valid Maple expressions) eqn generation of EQN output for typesetting equations eval one-level or full evaluation of an expression evalf/int numerical integration for a definite integral fixdiv compute the fixed divisor of a polynomial fsolve floating-point equation solver grobner package for Groebner basis computations igcdex extended Euclidean algorithm for integers interp polynomial interpolation iroot the nth root of an integer laplace Laplace transforms and inverse Laplace transforms linalg[bezout] create Bezout matrix from polynomials a(x) and b(x) [genmatrix] generate coefficient matrix from a linear system [ismith] determine the Smith normal form for an integer matrix [smith] determine the Smith normal form for a polynomial matrix maxnorm the infinity norm of a polynomial (max abs of coeffs) mfactor polynomial factorization over integers modulo p minterp polynomial interpolation modulo p mres modular resultant in Zp[x] mtaylor multivariate Taylor series optimize common subexpression optimization powseries formal power series package priqueue priority queue facility proot the nth root of a polynomial realroot isolating intervals for real roots of a polynomial residue compute the algebraic residue of a function round round a number to an integer rsolve recurrence equation solver signum sign of a real or complex number simplex package for linear optimization, including functions such as feasible, maximize, minimize, pivot simplify general purpose simplifier sinterp sparse polynomial interpolation sort general purpose sorting routine (uses shellsort) symmpoly test for symmetric polynomial translate evaluate a polynomial p(x) at x = x+t write primitive facility for file output Modified functions (use "help(f);" in a maple session for information): ---------------------------------------------------------------------- convert/confrac new case: convert ratpoly to confrac convert/ratpoly new case: Chebyshev-Pade approximation degree,ldegree new functionality allows for the total degree, both lexicographic and homogeneous ordering eulermac using a better algorithm evalf maps onto parts of a general expression; when applied to an unevaluated int, performs numerical integration gc now prints the first three status values has has(x,y) or has(x,z) becomes has(x,[y,z]) intbound deleted (new name is maxnorm) length now computes the length or size of any object limit improved functionality minimize solve a system of linear inequalities, using the simplex algorithm; improved algorithm save now allows subscripted names and unassigned names seq functionality seq(f(i),i=n..n+3) no longer allowed; the range values must be constants; the syntax now uses the$ operator:  f(n+i) $i=0..3 solve several syntax and functionality changes; new routine solve/float` for linear systems with floating-point coefficients; substantial improvements all around Psi1, Psi2, ... Psi.n becomes Psi(n,x) tdegree deleted (see degree) type/linear type(expr,linear,...) is defined to be type(expr,polynom,...) and degree(expr,...) = 1 type/quadratic type(expr,quadratic,...) is defined to be type(expr,polynom,...) and degree(expr,...) = 2 type/polynom type(expr,polynom,integer), type(expr,polynom,rational) and type(expr,polynom,numeric) mean that expr is a polynomial over the domain of integers, rationals, or reals, respectively; type(expr,polynom,a) is now valid as is type(expr,polynom,f(x)); type(expr,polynom,integer[x,y,z]) becomes type(expr,polynom,[x,y,z],integer) type/ratpoly is available with argument sequences as in polynom Internal efficiency improvements: -------------------------------- coeff avoid unnecessary copying for products; avoid unnecessary fragmentation for sums diff special code for polynomials and series; avoid unnecessary fragmentation for sums divide trial division is now performed over the integers rather than over the rationals; remember table is used for successful polynomial divisions; to divide out a polynomial by its icontent, do: divide(p,icontent(p),'p'); not p := p/icontent(p); expand special code for recognizing monomials; includes a better algorithm for large univariate polynomials for-loops,$     improved for small ranges frontend         now includes a check for the common case where no "freezing" is required, thus avoiding allocation of linear space which is non-garbage-collectible;  the fourth argument option has been deleted genpoly          now encoded internally so as to use linear space; doing one integer division has doubled the speed maxnorm          now encoded internally so as to use constant space; extended functionality for series and products member           now internal for direct access to the elements read             the following maple expressions can now be read from source format in linear space: lists, sets, sequences, tables, sums, products, functions set union,intersect,minus now using variations of merging rather than bubblesort sets             sorting using shellsort rather than bubblesort simpl            sort sums of length > 20 terms using shellsort during the collection of like terms table            sorting for the indexing functions symmetric and antisymmetric has been improved to use shellsort taylor           special code for converting a polynomial into a series; improved code for sums;  improved increment sequence for shellsort (used to sort exponents);  using Horner's rule for evaluation of a dense univariate series with integer coefficients at an integer type/polynom     now does one pass rather than n passes where there are n indeterminates