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v53

     New features that have been added to Maple V for Release 3

     ----------------------------------------------------------

Development of Improved User Interfaces

=======================================

Most of the improvements and work done for this release has been to make the

user interfaces consistent across the main platforms, namely under Windows

on the PC, the Macintosh, and under X Windows on Unix platforms.  Many small

improvements and fixes have been made.  The new interfaces provide more tools

and support for getting output from parts of worksheets into PostScript files

for preparing LaTeX documents and books.

Graphics

========

Most of the improvements in the rendering software have been to fix problems

on the various platforms.  New plotting facilities include.

1: Plotting discontinuities

---------------------------

A facility for identifying discontinuities in an algebraic expression.

For example

> plot( tan(x), x = -10..10, y=-4..4, discont = true );

> plot( -1.5 + Heaviside(x) + Heaviside( 2 - x ), x=-5..5, discont = true );

2: Statistical plotting routines

--------------------------------

New statistical plotting routines.

> with(stats[statplots]);

  [boxplot, histogram, notchedbox, quantile, quantile2,

    scatter1d, scatter2d, symmetry, xscale, xshift, xyexchange]

See section below on stats package for examples.

3: New options

--------------

See ?plot,options and ?plot3d,options and ?plot3d,structure for details.

- contours=n specifies the number of contour lines to be drawn

- thickness=n where n is 0,1,2, or 3 for changing line thickness

- symbol=s specifies how a point is to be displayed where s is one of BOX,

  CROSS, CIRCLE, POINT, or DIAMOND

- font=f for changing the font of a TEXT object where f is a list of the

  form [family, style, size], family is one of TIMES, COURIER or

  HELVETICA}, style is one of BOLD, BOLDOBLIQUE, ITALIC,

  or BOLDITALIC and var is the point size.

New and Enhanced Mathematical Facilities

========================================

1: Definite Integration

-----------------------

Work is proceeding on generalizing certain classes of integrals which yield

solutions in terms of special functions.  This solution of the following

integral is computed in terms of the Beta function and its derivative with

respect to its first argument.

> f:=(x^(-1/3))*ln(x)*((1-x)^(1/2)):

> Int(f,x=0..1) = simplify(int(f,x=0..1));

  1

  /              1/2

 |  ln(x) (1 - x)

 |  ---------------- dx = 3/98

 |         1/3

/         x

0

                      1/2

  (- 42 gamma + 7 Pi 3    - 63 ln(3) - 288 - 42 Psi(1/6)) GAMMA(2/3) GAMMA(5/6)

  -----------------------------------------------------------------------------

                                        1/2

                                      Pi

The solution of this integral involves the Beta and hypergeometric functions.

> assume(a>0): assume(b<0):

> f:=x^(a-1)*(1-Kt*x)^(-b):

> Int(f,x=0..1) = int(f,x=0..1);

              1

              /

             |   (a - 1)           (- b)         Beta(a, 1) a

             |  x         (1 - Kt x)       dx = --------------------

             |                                                    b

            /                                   (a - b) (1 - Kt)

            0

                        Beta(a, 1) a

                 - -----------------------

                                        b

                   Kt (a - b) (1 - Kt)

                   Beta(a, 1) a hypergeom([a - 1, b], [a], Kt)

                 + ------------------------------------------------

                                     Kt (a - b)

Is the answer correct?  We include one numerical check by comparing the formula

obtained with the result from numerical integration at some particular values.

> subs(a=1/2,Kt=1/3,b=-6,");

   1

   /            6

  |  (1 - 1/3 x)

  |  ------------ dx =

  |       1/2

 /       x

 0

        128

     - ---- Beta(1/2, 1) + 3/13 Beta(1/2, 1) hypergeom([-6, -1/2], [1/2], 1/3)

       9477

> simplify(");

                                 1

                                 /        6

                                |  (x - 3)       845440

                         1/729  |  -------- dx = ------

                                |     1/2        729729

                               /     x

                               0

> evalf(");

                           1.158567084 = 1.158567084

> a := 'a': b := 'b':

2: Indefinite Integration

-------------------------

Maple now knows how to integrate some Bessel Functions, for example

> int(x^9*BesselY(3,2*x^2),x);

                                8               2

                             2 x  BesselY(4, 4 x )

> assume(a>0):

> int( x^(2*a)*BesselJ(a,x)*BesselJ(a-1,x),x);

      (2 a + 1)

 1/4 x

     (BesselJ(a, x) BesselJ(a - 1, x) + BesselJ(a + 1, x) BesselJ(a, x))/a

> a := 'a':

Maple now supports Hermite reduction for integrals of the form

           /

          |              p(x)

          | f( u x + v) ----- dx

          |              q(x)

         /

where f is either exp, ln, sin or cos and p(x) and q(x) are polynomials.

Examples

> f := exp(-z)/z^2/(z^2 + 6*z + 6)^2:

> Int(f,z) = int(f,z);

     /                                                     2

    |       exp(- z)                  exp(- z) (2 + 5 z + z )

    |  ------------------ dz = - 1/12 ----------------------- + 1/12 Ei(1, z)

    |   2   2           2                   2

   /   z  (z  + 6 z + 6)                  (z  + 6 z + 6) z

> f := exp(-z)/z/(z+1)/(z^3 + 2):

> Int(f,z) = int(f,z);

          /

         |       exp(- z)

         |  ------------------ dz = - 1/2 Ei(1, z) + exp(1) Ei(1, z + 1)

         |              3

        /   z (z + 1) (z  + 2)

               / -----                       2

               |           (2 + 2 _R1 + _R1 ) exp(_R1) Ei(1, z + _R1)|

             - |   )     1/6 ------------------------------------------|

               |  /                                2                   |

               | -----                          _R1                    |

               \R1 = %1                                               /

                                         3

 %1 :=                          RootOf(_Z  - 2)

> f := ln(x)/x/(x^2+1)^2:

> Int(f,x) = int(f,x);

       /

      |     ln(x)                  2

      |  ----------- dx = 1/2 ln(x)  + 1/2 Pi arctan(x) + 1/2 dilog(- I x)

      |      2     2

     /   x (x  + 1)

                                                         2

                                     2            ln(x) x

          + 1/2 dilog(I x) + 1/4 ln(x  + 1) - 1/2 --------

                                                    2

                                                   x  + 1

> f := sin(z)/(z^3-z-1):

> Int(f,z) = int(f,z);

         /                  -----

        |    sin(z)               Si(z - _R1) cos(_R1) + Ci(z - _R1) sin(_R1)

        |  ---------- dz =    )     -------------------------------------------

        |   3                /                            2

       /   z  - z - 1       -----                    3 _R1  - 1

                           _R1 = %1

                                      3

 %1 :=                       RootOf(_Z  - _Z - 1)

3: signum(0) = ?

----------------

The definition of signum has changed so that signum(0) is now undefined by

default.  The value of signum(<expr which evaluates to 0>) will return a

result which could be 0 or any complex number of magnitude 1.  Why not simply

define signum(0) to be 0?  Or 1 as it was previously?  Because any specific

definition will not permit certain desirable simplifications from taking

place.  For example, the transformation

   signum(abs(x)) ==> 1

would be wrong at x=0 if signum(0)=0.  Similarly, for real x and y, the

transformation

  signum(x y) ==> signum(x) signum(y)

would be wrong for x<0 and y=0 if signum(0)=1.  Yet both simplifications

are desired in many contexts e.g. definite integration.  Thus making the

value of signum(0) undefined appears to be the most useful definition.

The value of signum(0) can however be fixed, either by assigning to the

environment variable \Envsignum0 or by using a 3-argument form of signum:

  signum(0,<expr>,s).

This will evaluate  signum(expr)  under the assumption that signum(0)=s .

The leading argument "0" specifies that the 0'th derivative of signum is being

computed.  The csgn function (complex sign) been modified in the same way.

Examples

> assume(y,real);

> signum(abs(x)), signum(-2*x^2), signum(-2*y^2); # using signum(0)=undefined

                                            2

                              1, - signum(x) , -1

> _Envsignum0 := 0; # using signum(0)=0

> signum(abs(x)), signum(-x^2), signum(-2*y^2);

                                             2              2

                  signum(abs(x)), - signum(x) , - signum(y)

> _Envsignum0 := 1; # using the old behavior

> signum(abs(x)), signum(-x^2), signum(-2*y^2);

                                      2              2

                         1, signum(- x ), signum(- y )

> y := 'y': # clear y

4: Laplace transforms

---------------------

The laplace functions have been extended to handle convolutions.  E.g. given

> f := laplace(F(t),t,s):

> g := laplace(G(t),t,s):

> h := laplace(H(t),t,s):

Direct convolutions

> invlaplace(f*g,s,t);

                             t

                             /

                            |

                            |  G(_U1) F(t - _U1) d_U1

                            |

                           /

                           0

> laplace(",t,s); # check answer

                    laplace(F(t), t, s) laplace(G(t), t, s)

> invlaplace(f*g*h,s,t);

                 t  _U1

                 /   /

                |   |

                |   |   H(_U3) G(_U1 - _U3) d_U3 F(t - _U1) d_U1

                |   |

               /   /

               0   0

> invlaplace(exp(s)*f,s,t); # case where one of the products has a transform

                           Heaviside(t + 1) F(t + 1)

> laplace(",t,s);

                            laplace(F(t + 1), t, s)

> invlaplace(exp(-s)*f,s,t);

                           Heaviside(t - 1) F(t - 1)

> invlaplace(s*f,s,t); # derivative case for convolution

                          /  d

                          |---- F(t)| + F(0) Dirac(t)

                          dt      /

> laplace(",t,s);

                             s laplace(F(t), t, s)

> invlaplace(1/(s - 1)*f,s,t); # integration case of convolution

                             t

                             /

                            |

                            |  exp(_U) F(t - _U) d_U

                            |

                           /

                           0

> laplace(",t,s);

                              laplace(F(t), t, s)

                              -------------------

                                     s - 1

5: RootOf

---------

The value RootOf(a(x)=0,x) specifies a root of a(x).  This is used to

represent algebraic numbers and functions in Maple that cannot be expressed

in terms of radicals.  RootOf(a(x)=0,x) represents any of the roots.

The extension in the new version (for algebraic numbers)

   RootOf(a(x)=x,x,z)

specifies the complex root of a(x) which is near the numerical approximation z.

For example (further examples are given below for minimize and solve)

> RootOf(x^4-2,x,+1.2);

                                       4

                              RootOf(_Z  - 2, 1.2)

> evalf(");

                                  1.189207115

An example of a series expansion

> series(RootOf(x^4+k*x-16, x, 2.001*I), k);

              2                                          2                2

     RootOf(_Z  + 4, 2.001 I) + 1/16 k + 1/2048 RootOf(_Z  + 4, 2.001 I) k

                               2                4              5      6

          + 7/8388608 RootOf(_Z  + 4, 2.001 I) k  - 1/8388608 k  + O(k )

7: The minimize command

-----------------------

The definition of minimize now supports the option infinite which means

minimize will minimize over the closed real interval [infinity,infinity].

The default is to minimize over (-infinity,infinity).  For example

> f := x^3-5*x^2+4;

                                     3      2

                               f := x  - 5 x  + 4

> minimize(f,x);

                                       392

                                     - ---

                                        27

> minimize(f,x,infinite);

                                   - infinity

For polynomials, if the minimum cannot be expressed in terms of radicals, it

is now expressed in the RootOf notation where the root is identified as the

second argument.  For example

> g := x^4-x+4;

                                      4

                                g := x  - x + 4

> minimize(g,x);

                                     3

                    - 3/4 RootOf(4 _Z  - 1, .6299605249) + 4

> evalf(");

                                  3.527529606

Also, minimize supports minimization over ranges, for example

> minimize( -2*y^2/((y-x^2)^4+1)+x^2+y^2, {x,y}, {x=-1..1,y=-1..1} );

                    10         6         8         2         4

            - 1/4 %1   - 5/2 %1  + 5/4 %1  - 1/2 %1  + 5/2 %1  - 1/2

 %1 := RootOf(

             2        4        6        8        10        12       14     16

   - 4 + 8 _Z  + 16 _Z  - 56 _Z  + 72 _Z  - 56 _Z   + 28 _Z   - 8 _Z   + _Z  ,

   -.6891910084)

> evalf(");

                                  -.3837893698

6: Solving algebraic equations

------------------------------

A new method for solving large systems which have algebraic functions

solutions.  A better method for finding the solutions during back substitution

not rationalizing the algebraic functions.  An example of such a system

> e1 := H1^2 + (2*H2*H3/3 + H3^2*x1/3) *(x3 - 3/2*x1*x2 + x1^3/2):

> e2 := 2*H1*H2 - H2*H3*(x1^2 - x2) + H3^2*(-x1/2*(x1^2 - x2) +

>     (x3 - 3/2*x1*x2 + x1^3/2)/3):

> e3 := 2*H1*H3 + H2^2 + 2*H2*H3*x1 + H3^2*(x1^2 + x2)/2:

> eqns := {0 = a1 + a2*H1 + a3*e1, 1 = a2*H2 + a3*e2, 0 = a2*H3 + a3*e3}:

where we want to solve for H1,H2,H3 where the solutions are algebraic

functions of the parameters a1,a2,a3,x1,x2,x3.  The solutions involve the

roots of an irreducible polynomial over degree 8.

Solving with radicals improved in general and also selects correct roots

> solve( sqrt(x+2)-sqrt(2*x-1)=1 );

                                          1/2

                                   6 - 2 7

> solve( {x^2-y+2, x*y=sqrt(75), x-sqrt(3)+y-5}, {x,y} );

                                     1/2

                               {x = 3   , y = 5}

Added power in solving systems of trigonometric equations

> e1 := x - (l1*cos(t1) + l2*cos(t1+t2)):

> e2 := y - (l1*sin(t1) + l2*sin(t1+t2)):

> solve({e1,e2}, {t1,t2});

{t1 = 2 arctan(%1),

                      3        2              2        2         2

 t2 = - 2 arctan((%1 x  + %1 l1  x + 2 %1 l1 x  - %1 l2  x + %1 y  x

              2       3    2       2       2      /      2          2         3

      + 2 %1 y  l1 - y  - x  y + l2  y - l1  y)  /  (%1 x  y - %1 l2  y + %1 y

                                                /

               2                               2       2       2      3

      - %1 y l1  + 2 %1 y l1 l2 + 2 l1 x l2 + y  x - l2  x - l1  x + x ))

 }

                 2              2    2    2    2                2    2     2

 %1 := RootOf((l1  + 2 l1 x - l2  + y  + x ) _Z  - 4 _Z y l1 + x  + y  - l2

                  2

     - 2 l1 x + l1 )

Equations involving RootOfs are now handled more generally.  For example,

here is an algebraic function in a.

> r := RootOf(x^2+a*x+1,x):

> solve( r^2+1/r=a^2-1/a, a);

                                4       3                2

                     RootOf(2 _Z  + 2 _Z  + 1 - 2 _Z + _Z )

More generality in the solutions of inequations

> solve( x^3+x > 5-2/x );

                                             4     2

           {0 < x, x < RootOf(- 5 _Z + 2 + _Z  + _Z , .4482633973)},

                                      4     2

               {RootOf(- 5 _Z + 2 + _Z  + _Z , 1.292690487) < x}

8: Complex evaluation of the Riemann Zeta function

---------------------------------------------------

Complex numerical evaluation of the Riemann Zeta function and its derivatives

has been implemented.  Note, Zeta(n,z) is the nth derivative of Zeta(z) and

Zeta(n,z,q) is the nth derivative of the generalized Zeta function Zeta(0,z,q).

For Zeta(0,z,q), for Re(z)<0 the index q is limited to the range 0<q le 1, and

for Zeta(n,z,q), is only available for Re(z)>0.  Examples

> Zeta(3.5+4.5*I);

                          .9241880340 + .01456451995 I

 # The 1st non-trivial 0 (approx):

> Zeta(1/2+14.13472514*I);

                                   -9                 -8

                     .2163160213*10   - .1358779595*10   I

 # Derivatives:

> Zeta(7, -3.5+1.9*I);

                         .01231658152 + .01218445174 I

 # 2 argument Zeta (aka, Generalized zeta function, aka Hurwitz zeta function):

> Zeta(0,3.5+4.5*I,.3);

                          43.96655549 - 51.81043075 I

 # Derivatives:

> Zeta(5,3.5+2.7*I, .7);

                        - .0002070976852 - .03085254975 I

9: Polynomial factorization over finite fields

----------------------------------------------

Maple V Release 3 can factor univariate polynomials over GF(p^k).

Maple represents elements of GF(p^k) as polynomials in alpha where alpha is a

RootOf an irreducible polynomial.  In the following example, elements of

GF(3^3) are represented as polynomials over GF(2) modulo the x^2+2*x+2.

This polynomial representation does not place any restriction on the size of

the field, and it is quite feasible to compute large fields e.g. GF(2^{128}).

Here is a univariate factorization

> alias( alpha=RootOf(x^2+2*x+2) ):

> f := 2*alpha*x^3+(2+2*alpha)*x^2+(alpha+2)*x+1;

                           3                  2

             f := 2 alpha x  + (2 + 2 alpha) x  + (alpha + 2) x + 1

> Factor(f) mod 3;

             2 alpha (x + alpha + 2) (x + 1 + 2 alpha) (x + alpha)

Release 3 can now factor multivariate polynomials over finite fields.

> f := x^3+alpha*x^2*y+alpha^2*x*y^2+alpha^3*y^3;

                       3          2          2    2        3  3

                 f := x  + alpha x  y + alpha  x y  + alpha  y

> Factor(f) mod 3;

               (y + (alpha + 2) x) (y + alpha x) (y + 2 alpha x)

In prior releases we have been using the Cantor-Zassenhaus algorithm for

factorization of univariate polynomials over GF(p) (p prime) and Berlekamps

algorithm for GF(p^k) for k>1.  For GF(p) for large modulus p, Berlekamps

algorithm performs better in general.  We timed the two methods on the

following problem taken from an article by von zur Gathen from the SIGSAM

Bulletin (Vol 26, No. 2, April 1992.), namely to factor the polynomial

x^n + x + 1 mod p for p=3982441812995697363688351114001.

We found that Berlekamps algorithm was approx. 2.6 times faster than

the Cantor-Zassenhaus algorithm as implemented in Maple.  More details

comparing the two algorithms on von zur Gathen's tests can be found in

a follow up article by Michael Monagan in the SIGSAM Bulletin

(Vol 27, No. 2, April 1993).  Note both algorithms are implemented for

univariate factorization over GF(p^k) as well as GF(p).

10: Manipulation of sums, products, integrals and limits

--------------------------------------------------------

We have provided more tools for manipulating unevaluated sums, integrals,

products and limits.  The expand and simplify commands will apply

the following transformations

   expand( F( a * f(x), ... ) ) ==> normal(a) * F(expand(f(x)), ...)

   simplify( F( a * f(x), ... ) ) ==> simplify(a) * F(simplify(f(x)), ...))

where F is any of sum, int, limit, product, Sum, Int, Limit, Product.

For example

> f := Int( sqrt(a)/Pi*exp(-u*x^2)*x^2, x ):

> f = expand(f);

                                                   /      2

                                             1/2  |      x

                                            a     |  --------- dx

                 /  1/2          2   2            |         2

                |  a    exp(- u x ) x            /   exp(u x )

                |  ------------------- dx = ---------------------

                |           Pi                        Pi

               /

> g := sum( a[i]*i/2^n, i=1..n ):

> g = simplify(g);

                        n                   /  n

                      -----                 |-----       |

                          a[i] i    (- n) |          |

                        )   ------ = 2      |  )   a[i] i|

                       /       n            | /          |

                      -----   2             |-----       |

                      i = 1                 i = 1       /

Differential Equations

======================

1: DESol - Solutions of Differential Equations

----------------------------------------------

We have introduced a new structure DESol for representing the solutions

of differential equations (DEs).  It has the following structure

   DESol( differential equation(s), variable(s), initial condition(s) )

The DEs can be described either functionally using the D operator, or as an

expression using the diff command.  For example, the following

describes the solutions to the equation y'' + y = 0.

> DESol( D(D(y)) = -y, y );

                                       (2)

                        de1 := DESol({D   (y) + y}, {y})

> DESol( diff(y(x),x,x) + y(x), y(x), {y(0)=0, D(y)(0)=1} );

                                /   2

                                |  d       |

                  de2 := DESol({|----- y(x)| + y(x)}, {y(x)})

                                |   2      |

                                dx       /

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 in Maple which represents a solution to an

algebraic equation.  Presently Maple knows how to differentiate, integrate,

generate a series expansion, and numerically evaluate a DESol.  Examples

> de := DESol( D(y)-a*y=1, y );

                       de := DESol({D(y) - a y - 1}, {y})

> D(de);

                       a DESol({D(y) - a y - 1}, {y}) + 1

> de(x);

                          /  d

                   DESol({|---- y(x)| - a y(x) - 1}, {y(x)})

                          dx      /

> int( 1/de(x), x );

                     /  d

           ln(DESol({|---- y(x)| - a y(x) - 1}, {y(x)}))

                     dx      /

                              /  d

                - ln(a DESol({|---- y(x)| - a y(x) - 1}, {y(x)}) + 1)

                              dx      /

Series of DESol (simple ones) are now handled.  By simple we mean that a

Taylor series can be computed using the derivatives of the DE.  This, together

with the use of initial conditions, gives nice results.  For example

> series(de(x),x=0);

                                                 2        2               3

     y(0) + (1 + a y(0)) x + 1/2 a (1 + a y(0)) x  + 1/6 a  (1 + a y(0)) x

                  3               4          4               5      6

          + 1/24 a  (1 + a y(0)) x  + 1/120 a  (1 + a y(0)) x  + O(x )

> de := DESol( diff(x(t),t,t) = g/l*sin(x(t)), x(t), {x(0)=0, D(x)(0)=v0} ):

> series(de,t=0);

                              /            3          2

                    g v0  3   |        g v0          g  v0|  5      6

         v0 t + 1/6 ---- t  + |- 1/120 ----- + 1/120 -----| t  + O(t )

                      l       |          l              2 |

                                                     l  /

2: The use of RootOfs

---------------------

For ODEs where the characteristic polynomial cannot be factored, Maple now

expresses the solution in terms of the roots of the characteristic polynomial

using a RootOf.  Example

> ode := diff(y(x),x5 ) + 2*diff(y(x),x) + 2*y(x) = 0;

                       /   5

                       |  d       |     /  d

                ode := |----- y(x)| + 2 |---- y(x)| + 2 y(x) = 0

                       |   5      |     dx      /

                       dx       /

> dsolve(ode,y(x));

                                -----

                        y(x) =    )    _C1[_R] exp(_R x)

                                 /

                                -----

                               _R = %1

                                                5

 %1 :=                      RootOf(2 + 2 _Z + _Z )

> ode :=   diff(y(x),x6) + 4*diff(y(x),x4) +

>        4*diff(y(x),x3) + 4*diff(y(x),x2) + 8*diff(y(x),x):

> dsolve(ode,y(x));

                                                   / -----

                        1/2               1/2      |                       |

  y(x) = _C2 + _C3 cos(2    x) + _C4 sin(2    x) + |   )    _C1[_R] exp(_R x)|

                                                   |  /                      |

                                                   | -----                   |

                                                   \R = %1                  /

                                     3

 %1 :=                      RootOf(_Z  + 2 _Z + 4)

3: Bessel's Equation

--------------------

Maple now handles Bessel's equation shifted from the origin.

> ode := diff(y(x),x,x) + (4*k+1)/(x+2)*

> diff(y(x),x) + (4*(x+2)^6 + 3/(x+2)^2)*y(x);

                                   /  d

          /   2        (4 k + 1) |---- y(x)|

          |  d       |             dx      /   /         6       3

   ode := |----- y(x)| + --------------------- + |4 (x + 2)  + --------| y(x)

          |   2      |           x + 2           |                    2|

          dx       /                                       (x + 2) /

> dsolve(ode,y(x));

bytes used=2005608, alloc=1310480, time=6.73

                 2           (- k)                 2     1/2             4

    y(x) = _C1 (x  + 4 x + 4)      BesselJ(1/4 (4 k  - 3)   , 1/2 (x + 2) )

                 2           (- k)                 2     1/2             4

         + _C2 (x  + 4 x + 4)      BesselY(1/4 (4 k  - 3)   , 1/2 (x + 2) )

4: Exponential solutions of ODEs

--------------------------------

We have implemented the exponential linear ODE algorithm of Manuel Bronstein

(Reference ``Linear Ordinary Differential Equations: Breaking Through

the Order 2 Barrier'', Manuel Bronstein, Proceedings of ISSAC '92, ACM Press.)

This algorithm finds solutions whose logarithmic derivative is in the

coefficient field, i.e. y'(x)/y(x) is a rational function.  This includes

solutions which themselves lie in the coefficient field.  For example

> ode := (-2*x^2+x+n^2)*y(x)+(4*x^2-2*x-n^2)*diff(y(x),x)+

>        (-3*x^2+x)*diff(y(x),x2)+x^2*diff(y(x),x3);

                      2        2             2          2  /  d

         ode := (- 2 x  + x + n ) y(x) + (4 x  - 2 x - n ) |---- y(x)|

                                                           dx      /

                             /   2           /   3

                      2      |  d       |    2 |  d       |

              + (- 3 x  + x) |----- y(x)| + x  |----- y(x)|

                             |   2      |      |   3      |

                             dx       /      dx       /

> dsolve(ode,y(x));

y(x) =

                           /                                 /

                          |                                 |

 _C1 exp(x) + _C2 exp(x)  |  BesselJ(n, x) dx + _C3 exp(x)  |  BesselY(n, x) dx

                          |                                 |

                         /                                 /

> subs(n=1,ode);

                    2                     2            /  d

              (- 2 x  + x + 1) y(x) + (4 x  - 2 x - 1) |---- y(x)|

                                                       dx      /

                                  /   2           /   3

                           2      |  d       |    2 |  d       |

                   + (- 3 x  + x) |----- y(x)| + x  |----- y(x)|

                                  |   2      |      |   3      |

                                  dx       /      dx       /

> dsolve(",y(x));

    y(x) = _C1 exp(x) + _C2 exp(x) BesselJ(0, x) + _C3 exp(x) BesselY(0, x)

> ode := diff(y(x),x2) + 3/x*diff(y(x),x) + (x^2-143)/x^2*y(x)-1/x;

                                     d

                  /   2          ---- y(x)     2

                  |  d       |      dx         (x  - 143) y(x)

           ode := |----- y(x)| + 3 --------- + --------------- - 1/x

                  |   2      |         x               2

                  dx       /                        x

> dsolve(ode,y(x));

                                      2              4            6          8

  y(x) = (980995276800 + 22295347200 x  + 278691840 x  + 2580480 x  + 20160 x

              10    12    /  13   _C1 BesselJ(12, x)   _C2 Y(12, x)

       + 144 x   + x  )  /  x   + ------------------ + ------------

                        /                  x                 x

5: Solutions in terms of DESol

------------------------------

If the solution cannot be computed in closed form, we can now return it in

terms of the DESol data structure.

> ode := diff(y(x),x,x) + diff(y(x),x)*x^2 + y(x)*x;

                        /   2

                        |  d       |   /  d       2

                 ode := |----- y(x)| + |---- y(x)| x  + y(x) x

                        |   2      |   dx      /

                        dx       /

> dsolve(ode,y(x));

                     /   2

                     |  d        |    2 /  d

       y(x) = DESol({|----- _Y(x)| + x  |---- _Y(x)| + x _Y(x)}, {_Y(x)})

                     |   2       |      dx       /

                     dx        /

This example shows a case where a partial solution is obtained using Manuel

Bronstein's algorithm.

> ode := diff(y(x),x3) + 4*diff(y(x),x,x) + 4*diff(y(x),x) - 12*y(x)/x;

                /   3          /   2

                |  d       |     |  d       |     /  d           y(x)

         ode := |----- y(x)| + 4 |----- y(x)| + 4 |---- y(x)| - 12 ----

                |   3      |     |   2      |     dx      /        x

                dx       /     dx       /

> dsolve(ode,y(x));

                                                                 /

                             2      3                2      3   |

      y(x) = _C1 (27 x + 24 x  + 4 x ) + (27 x + 24 x  + 4 x )  |  DESol(

                                                                |

                                                               /

                                 /   2

                       2      3  |  d        |

          {(27 x + 24 x  + 4 x ) |----- _Y(x)|

                                 |   2       |

                                 dx        /

                               2       3       /  d

               + (252 x + 132 x  + 16 x  + 81) |---- _Y(x)|

                                               dx       /

                               2       3

               + (564 x + 192 x  + 16 x  + 360) _Y(x)},    {_Y(x)}) dx

6: Solution Basis

-----------------

An alternative output form is available in terms of a basis.  For example

> ode := 2*diff(y(x),x,x) + 3*diff(y(x),x) + y(x) - exp(-3*x);

                    /   2

                    |  d       |     /  d

           ode := 2 |----- y(x)| + 3 |---- y(x)| + y(x) - exp(- 3 x)

                    |   2      |     dx      /

                    dx       /

> dsolve(ode,y(x),output=basis);

                  [[exp(- x), exp(- 1/2 x)], 1/10 exp(- 3 x)]

7: Numerical Solutions

----------------------

The output form now returns a list for every point (rather than a set).

> p := dsolve( {diff(y(x),x) = -y(x),y(0)=1},y(x),type=numeric);

p := proc(rkf45_x) ... end

> p(1);

                       [x = 1, y(x) = .3678794379225918]

Alternatively, the output can be a list of procedures

> p := dsolve({diff(x(t),t)=y(t),diff(y(t),t)=x(t)+y(t),x(0)=2,y(0)=1},

>    {x(t),y(t)}, type=numeric, output=listprocedure);

   p := [t = proc(t) ... end, x(t) = proc(t) ... end, y(t) = proc(t) ... end]

Now pick off the desired procedures and make a parametric plot.

> X := subs(p,x(t)): Y := subs(p,y(t)):

> plot([X,Y,0..3]);

A new numerical algorithm called dverk78 of W. Enright has been implemented

which guarantees a preset accuracy for the answer.

8: Series Solutions

-------------------

The series option to dsolve now accepts non-zero initial conditions, e.g.

> Order := 3:

> dsolve(diff(y(x),x2) - y(x) = 1, y(x),series);

                                                         2      3

             y(x) = y(0) + D(y)(0) x + (1/2 + 1/2 y(0)) x  + O(x )

> dsolve({diff(y(x),x2) - y(x) = 1,y(1)=y(1)},y(x),series);

                                                            2              3

y(x) = y(1) + D(y)(1) (- 1 + x) + (1/2 + 1/2 y(1)) (- 1 + x)  + O((- 1 + x) )

Sqrt and Radical Simplifications

================================

1: Automatic radical transformations

------------------------------------

The following transformations that Maple previously made automatically

   (x*y)^(n/d) ==> x^(n/d)*y^(n/d)  e.g. (x*y)^(1/2) ==> x^(1/2)*y^(1/2)

   (x^r)^(n/d) ==> x^(r*n/d) e.g. (x^2)^(1/2) ==> x, (1/x)^(1/2) ==> 1/x^(1/2)

which are incorrect for negative x,y, are no longer made automatically by the

internal Maple simplifier.  The only transformations made automatically are

   (i) reduction of integer powers for integers n,a,b

      n^(a/b) ==> n^(q+r/b) ==> (n^q)*n^(r/b)  e.g. 2^(4/3) ==> 2*2^(1/3)

   (ii) rational power simplification for integers n,d, and rational r

      (n/d)^r ==> n^r/d^r  e.g. (2/3)^(1/2) ==> 2^(1/2)/3^(1/2)

   (iii) combining radical powers for rational r, integers n,d

      (x^r)^(n/d) ==> x^(r*n/d) iff -1<r<1  e.g. (x^(1/2))^(3/2) ==> x^(3/4)

2: Radical transformations made by sqrt, combine, simplify and expand

---------------------------------------------------------------------

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 provably correct transformations are

made as determined by the signum, csgn, and Re and Im functions.  For example,

because of the following

> signum(Pi-3);

                                       1

i.e. Maple has proven this to be positive - it used interval arithmetic - then

the following simplifications can be made.

> f := ((Pi-3)^2*x^2*y)^(1/2);

                                          2  2   1/2

                            f := ((Pi - 3)  x  y)

> simplify(f);

                                          2   1/2

                               (Pi - 3) (x  y)

Because Maple does not know anything about x, it cannot simplify this to

(Pi - 3) x sqrt(y).  Another example,

> f := 2^(1/3)*(x+1)^(1/3)*(y+1)^(1/3);

                              1/3        1/3        1/3

                        f := 2    (x + 1)    (y + 1)

> combine(f,radical);

                                   1/3          1/3

                            (y + 1)    (2 x + 2)

Now that 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, i.e. effectively

assuming that the sign of the expressions is positive.  Hence

> sqrt( (Pi-3)^2*x^2, symbolic );

                                   (Pi - 3) x

> sqrt( (3-Pi)^2*x^2, symbolic );

                                   (Pi - 3) x

> sqrt( -a^3*b^2*c, symbolic );

                                            1/2

                                 a b (- c a)

This is useful when you are solving an equation and you just need one

root, the simplest one, and you will construct the other roots from this.

For example

> solve(x^2+a^3*b^2*c,x);

                                   1/2               1/2

                        a b (- c a)   , - a b (- c a)

Alternatively, the user can use the assume facility to tell Maple that

expressions such as a,b,c here are real, real and positive, real and

negative etc.  For example

> assume(x>0);

> sqrt(x^2*y);

                                        1/2

                                    x y

> assume(x,real);

> sqrt(x^6);

                                    2    2 1/2

                                  x  (x )

> simplify(");

                                   3

                                 x  signum(x)

3: The simplify command

------------------------------

The code for simplifying expressions containing square roots and general roots

has been improved.  Each (square) root is first simplified.  The principal

transformation made when a>0 is

   (a^n*b)^(m/d) = (a^(q+r/d)*b)^(m/d) ==> a^(m*q) * (a^(r/d)*b)^(m/d)

For example

> f := (a^8*b)^(1/3);

                                        8   1/3

                                 f := (a  b)

> assume(a>0);

> simplify(f,radical);

                                   2    2   1/3

                                 a  (a  b)

Then dependent integer roots are located in the expression.

For example in the expression

> s := 6^(1/2)-2^(1/2)*3^(1/2)+1;

                                 1/2    1/2  1/2

                           s := 6    - 3    2    + 1

It is determined that sqrt(6) can be written as sqrt(2) sqrt(3) hence

the expression is simplified to

> simplify(s,radical);

                                       1

Finally, the expression is rationalized.  For example, writing the expression

> s := (2*x+2)^(3/2)-2*(2*x+2)^(1/2);

                                     3/2              1/2

                       s := (2 x + 2)    - 2 (2 x + 2)

as (2 x + 2) S - 2 S, then simplifying this as a rational expression, we obtain

> simplify(s,radical);

                                           1/2

                                2 (2 x + 2)    x

4: The combine command

----------------------

Radicals can be combined together with

   combine(expr,radical);

   combine(expr,radical,symbolic);

The principal transformation made is

   x^(m/d) * y^(n/d) ==> (x^m*y^n)^(1/d) iff signum(x)>0 or signum(y)>0

For example

> e := 2^(1/2)*3^(1/2)*(x+1)^(3/2)*y^(3/2);

                               1/2  1/2        3/2  3/2

                         e := 2    3    (x + 1)    y

> combine(e,radical);

                                    3/2          1/2

                           (x + 1) y    (6 x + 6)

> combine(e,radical,symbolic);

                                                  1/2

                           (x + 1) y (6 x y + 6 y)

5: Denesting of radicals and the radnormal command

--------------------------------------------------

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.  For example

> sqrt(3+4*I), sqrt(4-sqrt(12)), sqrt(1/2+sqrt(3)/4);

                              1/2           1/2        1/2

                      2 + I, 3    - 1, 1/4 6    + 1/4 2

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.  For example

> f := sqrt( 2*(3-sqrt(2)-sqrt(3)+sqrt(6)) );

                                  1/2      1/2      1/2 1/2

                     f := (6 - 2 2    - 2 3    + 2 6   )

> readlib(radnormal)(f);

                                  1/2        1/2  1/2

                           - 1 + 3    + 1/3 3    6

6: The rationalize command

--------------------------

The command rationalize will rationalize an expression containing

radicals, i.e. given a rational expression of the form n/d, the result is a

new expression n'/d' where no radicals appear in the new denominator d'.

For example

> f := sqrt(2)/(1+sqrt(3));

                                         1/2

                                        2

                                 f := --------

                                           1/2

                                      1 + 3

> rationalize(f);

                                  1/2         1/2

                             1/2 2    (- 1 + 3   )

> f := 1/(x^(1/3)+y^(3/2));

                                          1

                                f := -----------

                                      1/3    3/2

                                     x    + y

> rationalize(f);

                         3    3/2  1/3    2/3          9/2

                     (- y  + y    x    - x   ) (- x + y   )

                     --------------------------------------

                                       9    2

                                    - y  + x

7: The simplify command

-----------------------

The command  simplify(expr,assume=t)  will simplify an expression assuming

all variables are of type t where t is typically real, positive.  For example

> f := r^2 * sin (t) / sqrt(x^2+r^2-2*x*r*cos(t));

> g := int( int( f, t=0..Pi ), r=0..1 );

                        2 3/2             2 1/2  2             2 1/2

    g := 1/6 (2 ((1 + x) )    - 3 ((1 + x) )    x  - 3 ((1 + x) )    x

                       2 3/2               2 1/2  2               2 1/2

         - 2 ((- 1 + x) )    + 3 ((- 1 + x) )    x  - 3 ((- 1 + x) )    x)/x

> simplify(g,assume=real);

                                                                   3

     - 1/6 (- 2 signum(1 + x) - 3 signum(1 + x) x + signum(1 + x) x

                                                                       3

          - 2 signum(- 1 + x) + 3 signum(- 1 + x) x - signum(- 1 + x) x )/x

New and Enhanced System Facilities

==================================

1: Global statement

-------------------

Procedures may now include an optional global statement which follows the

local statement in a procedure.  For example, the procedure

proc(n) local y; global A;

    y := 1;

    for i to n do y := x*y; A[i] := y od;

end;

has one parameter n, one local variable y, and one global variable A.  The

variables i and x are not defined, and in the older versions of Maple, would be assumed

to be global variables.  In this case, one can guess that x is meant to be

global and i should be local.  In the new version of Maple, in order to reduce

the likelihood of errors, if a variable is not defined, then Maple will declare

it to be local if

   (i)  it appears on the left hand side of an assignment statement

   (ii) it is used as an index of a for statement or seq statement

otherwise Maple will declare it global.  Applying these rules to the above

procedure, the undeclared variable i will be declared to be local, and the

undeclared variable x will be assumed global.  Hence the following action

is taken on the above procedure

Warning, i is assumed to be a local variable

proc(n) local y,i; global A;

    y := 1; for i to n do  y := x*y; A[i] := y od

end

The warning serves as a useful reminder to the user to declare variables.

2: Name protection

------------------

A protection facility has been added to prevent the user from accidentally

assigning to Maple system variables.  For example, the names ``list'' and

``lhs'' are now protected.  Attempting to assign to them will result in

> lhs := x^2-1;

Error, attempting to assign to lhs which is protected

> list := [1,2,3];

Error, attempting to assign to list which is protected

A name can be protected using the protect function.  For example

> height := proc(p,x) local t; max(seq(abs(t),t=[coeffs(p,x)])) end:

> protect(height);

> height := 1;

Error, attempting to assign to height which is protected

Names can be unprotected either by unassigning them, or by using the unprotect

function.  For example

> list := 'list':

> list := proc() convert([args],'list') end:

> list(1,2,3);

                                   [1, 2, 3]

> unprotect(height);

> height := 1;

                                  height := 1

3: march the Maple library archiver

------------------------------------------

Maple 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 Maple library consists of two files, maple.lib and

maple.ind.  The maple.lib file contains the Maple code, a set of .m files,

and the maple.ind file is a index into the file maple.lib for fast access.

This facility yields faster access and is more economical in file space.

It also provides a convenient way to distribute a library of Maple codes.

The command

   march -c archiveDir tableSize

creates a Maple archive in the directory <archiveDir> where <tableSize>

is an estimate of the number of files to be put in the archive.  The commands

   march -a archiveDir fileName indexName

   march -u archiveDir fileName indexName

adds, respectively updates, the archive in the directory <archiveDir>

with the file <fileName> to be called <indexName> in the archive.

There are further options for packing, extracting, removing .m files etc.

4: makehelp

-----------

This utility routine 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

on-line help command ? .  The resulting TEXT object can be saved into a file

to be included as on-help within a package.  See ?makehelp for details

5: The fortran command

----------------------

The fortran function now accepts an optional argument

mode = <modtype> where the mode type must be one of single

(the default), double, complex or generic which

specifies how function names are to be translated.  For example

> fortran( ln(x)+sin(x) );

      t0 = alog(x)+sin(x)

> fortran( ln(x)+sin(x), mode=generic );

      t0 = log(x)+sin(x)

The handling of arrays has been improved (also in the C function)

to output subscripts in lexicographical order and if an array entry is

not assigned, the string undefined is output.  For example

> A := array(symmetric,1..2,1..2,[(1,1)=ln(x),(2,1)=sin(x)]):

> fortran(A);

      A(1,1) = alog(x)

      A(1,2) = sin(x)

      A(2,1) = sin(x)

      A(2,2) = undefined

6: The piecewise function

-------------------------

The piecewise() function allows one to define a function with different forms

on different intervals.  One can also assert a global smoothness at the joints.

At present, diff/max and diff/min produce answers in terms of piecewise.

For example, the quadratic B-spline is given by the following piecewise

quadratic polynomial.  The first argument 1 states that the polynomial is C1

continuous

> b2 := piecewise( 1,

>                     x<0, 0,

>                     x<1, x^2/2,

>                     x<2, -3/2+x*(3-x),

>                     x<3, 9/2+x*(x-3),

>                     0 # otherwise

>                ):

> diff(b2,x);

     piecewise(0, x < 0, 0, x < 1, x, x < 2, - 2 x + 3, x < 3, 2 x - 3, 0)

7: help

=======

The following commands have been added to display parts of on-line help files.

- The command info(foo) will display the first line of the help file

  i.e. the FUNCTION line.

- The command usage(foo) or ??foo will display the CALLING SEQUENCE

  section.

- The command example(foo) or ???foo will display the EXAMPLES section.

- The command related(foo) will display the SEE ALSO section.

Note the old example function has been replaced by this functionality.

8: String and text processing capabilities

------------------------------------------

Maple strings now have no limitation on their length.  Previously the

length limitation was 499 characters.  The SearchText and searchtext

functions (case sensitive and insensitive respectively) can be used to

search a string for a pattern.  The functionality is

   searchtext(t,s)

   searchtext(t,s,a..b)

meaning search for the substring t in the string is (optionally starting

at position a in the string s ending at position b).

> s := `maple and Maple and MAPLE`:

> SearchText(`Maple`,s);

                                       11

> searchtext(`Maple`,s);

                                       1

9: The interface command

------------------------

The version of Maple is available from

> interface(version);

              Maple V, Release 3, Thu Aug 19 15:16:50 MET DST 1993

The number of bits in the machine word is available via

> interface(wordsize);

                                       32

Library Packages

================

The package mechanism has been extended to support subpackages.  This works

in an obvious way and is illustrated by the new stats package below.

1: numtheory - number theory package

------------------------------------

The function factorEQ(m,d) factors the integer m in the Euclidean domain

Z[sqrt(d)].  For example

> numtheory[factorEQ](38477343,11);

                     1/2              1/2             1/2             1/2

     (3) (125 + 34 11   ) (125 - 34 11   ) (85 + 16 11   ) (85 - 16 11   )

The isprime function now uses one strong pseudo primality test and

one Lucas test (not known to fail).  numtheory[safeprime] has also been

updated in the same way.

We have implemented a new iterative algorithm for numtheory[jacobi]

that yields a modest improvement of 15% in running time.

2: linalg - linear and vector algebra package

---------------------------------------------

The functions curl, diverge, grad, and laplacian have been extended to work

in spherical, cylindrical, and in orthogonally curvilinear coordinate systems

in general (default cartesian).

> with(linalg,curl,grad):

> g := [r,sin(theta),z]: v := [r,theta,z]:

> curl(g,v,coords=cylindrical);

                                      sin(theta)

                              [ 0, 0, ---------- ]

                                           r

> g := r^2*sin(theta)*cos(phi): v := [r,theta,phi]:

> grad(g,v,coords=spherical);

        [ 2 r sin(theta) cos(phi), r cos(theta) cos(phi), - r sin(phi) ]

The functions ihermite, ismith, hermite and smith have been modified to

optionally return the multiplier matrices.  That is

  H := ihermite(A,'U');   and   H := hermite(A,x,'U');

compute the Hermite normal form matrix H in the Euclidean domains Z and

F[x] respectively and assigns U the multiplier matrix such that H = U A.

(Hermite normal form is row reduction over a Euclidean ring).  Similarly

  S := ismith(A,'U','V');  and  S := smith(A,x,'U','V');

compute the Smith normal form S in the Euclidean domains Z and F[x] and

assigns U and V the multiplier matrices such that S = U S V.

(Smith normal form is row and column reduction over a Euclidean ring).

> with(linalg,ismith,ihermite,inverse,hilbert):

> A := inverse(hilbert(3));

                                 [  9    -36   30  ]

                                 [                 ]

                            A := [ -36   192  -180 ]

                                 [                 ]

                                 [  30  -180   180 ]

> H := ihermite(A,'U'):

> print(H,U);

                         [ 3   0  30 ]  [ 13   9   7 ]

                         [           ]  [            ]

                         [ 0  12   0 ], [  6   4   3 ]

                         [           ]  [            ]

                         [ 0   0  60 ]  [ 20  15  12 ]

> evalm( H - U &* A );

                                  [ 0  0  0 ]

                                  [         ]

                                  [ 0  0  0 ]

                                  [         ]

                                  [ 0  0  0 ]

> S := ismith(A,'U','V'):

> print(S,U,V);

                [ 3   0   0 ]  [  -3   0   1 ]  [ -5  -6  -30 ]

                [           ]  [             ]  [             ]

                [ 0  12   0 ], [  18   4  -1 ], [  1   1   0  ]

                [           ]  [             ]  [             ]

                [ 0   0  60 ]  [ -10  -5  -3 ]  [  1   1   1  ]

> evalm( S - U &* A &* V );

                                  [ 0  0  0 ]

                                  [         ]

                                  [ 0  0  0 ]

                                  [         ]

                                  [ 0  0  0 ]

Note the new algorithm that is used for Hermite normal forms is, unlike the

old algorithm, a polynomial time algorithm, and is in practice much faster

for larger matrices.

4: GaussInt -- Gaussian integer package

---------------------------------------

The routines GIhermite and GIsmith similarly have been added to compute the

Hermite and Smith normal forms over the Euclidean domain Z[i]

The routines GIchrem implements the Chinese Remainder Algorithm for Z[i]

> GaussInt[GIchrem]([5+13*I,15-9*I],[3+4*I,7-11*I]);

                                  - 17 + 17 I

The routine GInodiv computes the number of non-associated divisors

> GaussInt[GInodiv](2+3*I);

                                       2

The Statistics Package stats

----------------------------

The stats package has been completely redesigned. The data structure for the

arguments to the functions has changed.  This implies that programs using the

previous version of the stats package will need to be updated.

See ?stats[updates] on how to do this.

New types of data are now supported in addition to numbers.

- Missing data is represented by the keyword `missing'.

- Data classes are represented as ranges a..b where the bound a is

  inclusive and the bound b is exclusive.

- Weighted data is represented by the function Weight(data, weight).

  For example, the following are all valid data:

  3, 3..4, missing, Weight(4,6), Weight(missing,7), Weight(3..4,9).

The stats package is subdivided into the following subpackages.

- describe: for descriptive statistics,

- transform: for data transformations,

- random: for generating numbers with a particular distribution,

- statevalf: for numerical evaluations of statistical functions,

- fit: for fitting data to a curve and

- statplots: for statistical graphics.

and the function importdata} which supplements Maple's other input facilities.

Issuing with(stats); defines abbreviations for all the subpackages but not the

subfunctions belonging to them.  Then issuing with(describe); defines

abbreviations for the functions in the describe subpackage.

> with(stats);

      [describe, fit, importdata, random, statevalf, statplots, transform]

> with(describe):

Now the following commands all refer to the same function on the data [1,5]

> mean([1,5]), describe[mean]([1,5]), stats[describe,mean]([1,5]);

                                    3, 3, 3

Certain functions require (and others allow) extra parameters. For example,

describe[quartile[1]] is a function that gives the first quartile of the

data. This allows one to do

> data:=[1,5,Weight(6,3), 9, 10, 15];

                    data := [1, 5, Weight(6, 3), 9, 10, 15]

> describe[quartile[1]](data);

                                       5

Functions like variance allow a parameter to specify if one is computing

the sample variance or the population variance. (The parameter used

the number to be subtracted from the item count, so variance[0] is

the population variance and variance[1] is the sample variance).

1: The descriptive statistics subspackage stats[describe]

---------------------------------------------------------

Contains the following functions

  coefficientofvariation, count, countmissing, covariance, decile,

  geometricmean, harmonicmean, kurtosis, linearcorrelation, mean,

  meandeviation, median, mode, moment, percentile, quadraticmean,

  quantile, quartile, range, skewness, standarddeviation, variance.

2: The transform subpackage stats[transform]

--------------------------------------------

Contains

  apply, classmark, cumulativefrequency, deletemissing, divideby,

  frequency, moving, multiapply, remove, scaleweight, split,

  standardscore, statsort, statvalue, tally, tallyinto.

The function apply is used to apply a transformation on each observation,

whereas moving is to apply a function across a sliding window over the data

(and some can be used to smooth data using, for example, a weighted moving

mean.) The function split is used to split a list of data into multiple

lists, which can be required to be of equal weight. Data are counted with

tally and regrouped into a specified pattern (exceptions can be reported if

desired) with tallyinto.

3: Statistical distributions

----------------------------

The distributions are functions indexed by their parameters.  For example,

poisson[2] is a Poisson distribution with mean lambda = 2.  The following

distributions are supported:

  beta, binomiald (binomial), cauchy, chisquare, discreteuniform, empirical,

  exponential, fratio (variance ratio, Fisher F), gamma, hypergeometric

  laplaced (Laplace), logistic, lognormal, negativebinomial, normald (normal),

  poisson, studentst (T distribution), uniform, weibull.

Given a distribution, random numbers with that distribution can be obtained

using the stats[random] subpackage. One can specify the quantity of

numbers to be generated at once, or that a generator is to be returned.

> stats[random,normald[0,1]](5);

       1.175839568, -.5633641309, .2353939952, -1.442550291, -1.079196234

> T:=stats[random,studentst[3]]('generator'):

> seq(T(),i=1..5);

      -1.749151242, -.5845810002, -2.441129943, -.05425587235, 1.632365981

Numerical evaluations of statistical functions are computed by the

stats[statevalf] subpackage. For example,

> statevalf[pdf, normald](0.5);

                                  .3520653266

gives the height of the standard normal curve at the value 0.5.  Similarly

> statevalf[cdf, chisquare[2]](0.6);

                                  .2591817793

gives the area under the curve of the  Chi square with 2 degrees of freedom

lying  to the left of 0.6.  The inverse of this last function is obtained by

using icdf -- inverse cumulative distribution function

> statevalf[icdf, chisquare[2]](");

                                  .6000000000

Similar facilities are available for discrete distributions.

4: The regression subpackage stats[fit]

---------------------------------------

Provides for regressions. At the present, weighted least square fits

of data to linear models is provided.

5: The plotting subpackage stats[statplots]

--------------------------------------------

Contains various statistical plots and plotting utilities.  The result of

these functions can also be manipulated via the plots[display] function.

Example

> with(statplots);

  [boxplot, histogram, notchedbox, quantile, quantile2,

    scatter1d, scatter2d, symmetry, xscale, xshift, xyexchange]

> Xdata := [4.535, 4.029, 5.407, 1.605, 5.757, 3.527, 7.890, 8.159, 6.092,

>          13.442, 2.845, 5.172, 3.277, 8.810, 3.657, 7.226, 3.851, 2.162,

>           2.668, 4.692]:

> Ydata:=  [7.454, 4.476, 2.873, 5.476, 9.975,-1.476, 1.033, 1.140, 4.813,

>            .450, -.788, 9.389, 4.811,-3.107, 4.407, 5.534, 1.691, -.789,

>          1.684, 1.605]:

> p1 := scatter2d(Xdata,Ydata):   p1;  # scatter plot

> p2 := boxplot[15](Ydata):       p2;  # boxplot in right margin

> p3 := notchedbox[12](Xdata):    p3;  # notched box at top

> plots[display]({ p1, p2,

>                  xyexchange(p3)}, view =[0..17,-4..14], axes=FRAME );

Share Library

=============

The Maple share library is a library of Maple routines (the Maple src code

included), Maple worksheets, and accompanying documentation files which have

been contributed by users of Maple to the Maple user community.

The version of the share library that comes with Maple V Release 3

contains about 25 Maple applications packages, 50 Maple routines, 30 Maple

worksheets and 25 additional PostScript, LaTeX and TeX articles documenting

the routines and packages.

In this version of the share library we have divided the share library

up into the following subject areas to make it easier for the user to find

what s/he is looking for and provided a contents listing for each area

with the given ? command

  Algebra                              ?share,algebra

  Analysis (Calculus)                  ?share,analysis  or  ?share,calculus

  Automatic Differentiation            ?share,autodiff

  Combinatorics                        ?share,combinat

  Engineering                          ?share,engineer

  Graphics and Geometry                ?share,graphics  or  ?share,geometry

  Linear Algebra                       ?share,linalg

  Number Theory                        ?share,numtheory

  Numerics                             ?share,numerics

  Programming                          ?share,programming

  Science                              ?share,science

  Statistics                           ?share,statistics

  System Tools                         ?share,system

For each Maple routine and package in the share library we have simplified

the loading mechanism to make it easier to access the share library.

For example, to access the gfun package in the calculus directory in the

share library, one does

> with(share);  # locate the share library

> readshare(gfun,calculus);  # load the gfun package

> ?gfun  # on-line help

In addition to Maple code, the share library also contains other files.

The files ending with the suffix

- .tex      -- are LaTeX or TeX src files

- .dvi      -- are device independent file (output from latex or tex)

- .ps       -- are PostScript files

- .ms       -- are Maple worksheets

- .in       -- are (plain text) Maple input files containing examples

Files with no suffix are Maple src code files.  The original Maple src code

is there and available to the user should s/he wish to study it or modify it.

Users who would like to contribute Maple code or worksheets to the share

library are encouraged to do so.  Please contact

Dr. Michael Monagan

monagan@inf.ethz.ch

Here is a brief summary of some of the new routines/packages in the

share library.

- perm: a package of routines for computing with permutation groups

- guesss: A routine to guess the next values in a sequence

- invar: a package of routines mainly for computing the invariant ring of

  permutation groups or finite linear groups over Q or Q(alpha)

- macroC: a Maple package for generating C language code

- BesselH: Implementation of Hankel Functions in terms of BesselJ and BesselY

- algcurve:

- pade2: computes a generalized Pade approximation of functions f1,f2,...,fn

- intpak: an experimental interval arithmetic package.

- fft, fht: fast Fourier and Hartley transforms

- surfaces: a set of procedures to calculate basic differential-geometric

  quantities of parametric surfaces in space

- sffge: row reduction of a sparse matrix of polynomials

- genus: the genus of an algebraic curve given by the polynomial f(x,y)

- normform: a package of routines for computing matrix normal forms

  Contains: ismithex, smithex, frobenius, jordan, and ratjordan

- puiseux: a Puiseux series expansion of an algebraic function

- ratlode: the rational solutions to a linear n'th order ODE in y(x) over Q

- coxpoly: the coxeter polynomial of the rooted cycletree

- relpoly: the reliability polynomial of a network

- GB: Buchberger's Groebner bases algorithm over finite fields

- integral\basis: of an algebraic number or function

- FPS: tries to find a formula for the coefficients of a formal power series

- fields: uses Groebner bases methods to answer questions about degrees of

  algebraic field extensions and related questions

- polycon: a package for analysis of polynomial and rational control systems

- fjeforms: a new package for computing with differential forms

- IntSolve: an integral equation solver

- coxeter: procedures for studying roots systems and finite Coxeter groups

- weyl: for manipulating weight vectors and computing multiplicities for

  irreducible representations of semisimple Lie algebras

- ratinterp: rational function interpolation

- elliptic: determines the order of the group of points on a non-singular

  elliptic curve y^2 = x^3+A*x+B over a finite field Z mod p

In addition, new in this version of the share library are a selection of

approximately 30 worksheets showing applications of Maple in various

disciplines, in education and research.

Miscellaneous

=============

- Convert/binomial converts expressions involving factorials and GAMMAs

  into binomials, for example

> f := (n+1)!/k!/(n+1-k)!*(2*n)!/n!^2:

> f = convert(f,binomial);

             (n + 1)! (2 n)!

          --------------------- = binomial(n + 1, k) binomial(2 n, n)

                              2

          k! (n + 1 - k)! (n!)

- The combine/ln function takes an optional argument t to specify which type

  (default type rational) of logs you want to combine, e.g.

> f := (a*ln(x)+3*ln(x)-ln(1-x)+ln(1+x)/2:

> combine(f,ln);

                                        3        1/2

                                       x  (1 + x)

                          a ln(x) + ln(-------------)

                                           1 - x

> combine(f,ln,integer);

                                                      3

                                                     x

                      a ln(x) + 1/2 ln(1 + x) + ln(-----)

                                                   1 - x

> combine(f,ln,anything);

                                  a  3        1/2

                                 x  x  (1 + x)

                              ln(----------------)

                                       1 - x

- numerical comparisons against infinity and -infinity are now allowed

> evalb( 5 < infinity );

                                      true

- new function: numboccur(a,b) count the number of occurrences of b in a

> numboccur( expand(sin(20*x)), x );

                                       20

- The system function lprint has changed a little so that it's output can

  be read back into Maple.  The lprint form of series has changed to output

  a series command.

Changes to Maple V Release 3

============================

- The new global statement will affect many users Maple procedures.  Maple

  will declare variables to be local automatically.  The rules are described

  in the section ``New and Enhanced System Facilities''.  The new release comes

  with a tool to insert global declarations needed to update old code.

- The local variable declaration in -> operators is no longer accepted.

  I.e. the operator  x -> local i; x;  will now yield a syntax error.

- The synonym linalg[range] for linalg[colspace] has been removed.

- The keywords linear and group in the define package have been renamed

  Linear and Group respectively because linear is a Maple type and group

  is a Maple package.

- The example function has been redesigned.  It now displays the EXAMPLE

  section from the help file.

- The stats package has been rewritten.  See the section ``The Statistics

  Package stats above.  Details for upgrading are given in ?stats,updates

- The following sqrt and radical simplifications are no longer applied

  automatically.

> sqrt(x^2), (x^3)^(1/3);

                                      x, x

> sqrt(x*y), (x*y)^(1/3);

                               1/2  1/2   1/3  1/3

                              x    y   , x    y

  See the section ``Sqrt and Radical Simplifications'' for details.  Similarly

  the simplify command no longer makes the above transformations unless

  it can prove that they are correct i.e. x is positive.

- The definition of signum(0) has changed.  See the section ``New and Enhanced

  Mathematical Facilities'' for details.

- The output from lprint of series has changed.  See the

  ``Miscellaneous'' section for details..

- All builtin and readlib defined function names, type names, and system

  constants are now protected.  E.g. exp, gcd, list, true respectively.

  See the section ``New and Enhanced System Facilities'' for details.

- The calling sequence of grobner[gsolve] has changed from

  grobner[gsolve](<polys>,<notzero>,<vars>) to be

  grobner[gsolve](<polys>,<vars>,<notzero>) (where the last two arguments

  are optional) inorder to be consistent with other solvers in Maple.

- The variables gamma and R in the cartan package have been renamed

  `cartan/gamma` and `cartan/R` respectively.