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updates

v52

New features that have been added to Maple for version V release 2

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

Development of Improved User Interfaces

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

New user interfaces which include as a minimum 2 and 3-dimensional graphics,

and some command line editing facility have been designed since the release

of Maple V for the X Window system, Sunview, PC MS/DOS, Macintosh, Amiga DOS,

Vax VMS, NeXT, and the PC under Windows. The new user interfaces for Maple V

release 2 will include worksheets, real mathematical formulae output,

and a help browser.

1: Maple Worksheets

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

Beginning with the Macintosh user interface for Maple V, the new user

interfaces will all support the concept of a "worksheet" which integrates text,

Maple input commands, Maple output, and graphics into one document. Worksheets

have been standardized and are being ported to other platforms for release 2,

notably X windows. They provide editable input and text fields, simple

commands for recalculating parts of worksheets etc. A collection of examples

of applications of Maple will be now be distributed in the form of worksheets.

A new save facility is being added that will also save the state of a worksheet

as well as the display.

2: Two-dimensional output

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

Beginning with release 2 of the X Window interface, new interfaces will

display output using mathematical, Greek, and italics fonts for improved

display of mathematical formulae.

3: Help topic browser

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

Beginning with Sunview user interface in Maple V, new user interfaces will

include a help browser for accessing the on-line help information.

4: Unix Maple interface

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

Now supports command line editing using a vi and emacs like editor.

Hitting the interrupt key Ctrl-C twice will no longer kill a Maple session, it

will only interrupt the current calculation. The quit key Ctrl-can be used

to immediately kill the Maple session. Also, running a Maple session in a

shell under the X Window system, interface(plotdevice=x11); will allow you to

access the X plotting facilities.

5: X Window interface

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

The interface has been redesigned to follow Motif style conventions. The help

index is accessible from a menu. The user input fields and the Maple output

fields are completely separated.

6: Improved error messages

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

The new type checking facility results in better error messages, e.g.

> f := proc(n:integer) if n < 0 then 0 else n fi end:

> f(x);

Error, f expects its 1st argument, n, to be of type integer, but received x

Graphics

========

The Maple plotting facility consists of two parts, the rendering of the picture

which is written in C, and the point generation which is done in Maple. The

basic 2 dimensional plotting command is plot(f(x),x=a..b); It creates a

PLOT data structure which is then displayed. The 3 dimensional plotting command

is plot3d(f(x,y),x=a..b,y=c..d); It creates a PLOT3D data structure.

Thus the PLOT and PLOT3D data structures define the what can be displayed.

Details on these data structures can be obtained from ?plot3d,structure

1: Improvements to rendering and the PLOT and PLOT3D data structures

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

- better (more robust but slower) hidden line removal algorithm

- the PLOT3D data structure supports text and polygon primitives

- PLOT and PLOT3D include support for 2 and 3 D frame sequence animation

- the PLOT3D data structure supports a contour plot style option,

which is also accessible interactively from the style menu

- The plotting primitives in the PLOT and PLOT3D data structures have

been made more flexible e.g. each object can have its style and

color assigned individually.

- a wider choice of colors, with more user control of surface coloring

(PLOT3D).

- addition of Lighting models

- We have also made the PLOT and PLOT3D data structures compatible.

2: Device support

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

- encapsulated postscript, color postscript and grayscale shading postscript

output are now supported

- addition of HP Laserjet output

- a wider selection of output devices for PLOT3D are supported in particular

any device that is supported in 2 dimensions including tektronics, i300

imagen laser printer, Unix plot, DEC ln03 laser printer etc.

3: New plotting facilities and improvements (plots package)

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

This is a list of the new plotting tools. All are written in Maple, making

use of the PLOT and PLOT3D data structures. For details see the section

on the plots package in the section on the Library Packages.

- 2d and 3d animation

- 2d and 3d implicit plotting

- 2d and 3d vectorfield and gradient plots

- Density plots

- Plotting of dsolve(..,numeric) procedures in ODEplot

- Plotting of polygons and polyhedra

- Log and log-log plots e.g.

- Phase portraits and direction field plots, plotting of 1-st order and

2-nd order differential equations (in DEtools package)

- Ability to specify color function for objects in plot3d, tubeplot,

vectorfield and gradient plots, etc.

New and Enhanced Mathematical Facilities

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

1: New Mathematical functions known to Maple

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

- The floor and ceiling functions floor(x) and ceil(x)

- The Elliptic integrals where 0<k<1 and let c = sqrt(1-k^2)

LegendreE(x,k) = int( (1-k^2*t^2)/sqrt(1-t^2)/sqrt(1-k^2*t^2), t=0..x )

LegendreEc(k) = int( (1-k^2*t^2)/sqrt(1-t^2)/sqrt(1-k^2*t^2), t=0..1 )

LegendreEc1(k) = int( (1-k^2*t^2)/sqrt(1-t^2)/sqrt(1-c^2*t^2), t=0..1 )

LegendreF(x,k) = int( 1/sqrt(1-t^2)/sqrt(1-k^2*t^2), t=0..x )

LegendreKc(k) = int( 1/sqrt(1-t^2)/sqrt(1-k^2*t^2), t=0..1 )

LegendreKc1(k) = int( 1/sqrt(1-t^2)/sqrt(1-c^2*t^2), t=0..1 )

LegendrePi(x,k,a) = int( 1/(1-a*t^2)/sqrt(1-t^2)/sqrt(1-k^2*t^2), t=0..x )

LegendrePic(k,a) = int( 1/(1-a*t^2)/sqrt(1-t^2)/sqrt(1-k^2*t^2), t=0..1 )

LegendrePic1(k,a) = int( 1/(1-a*t^2)/sqrt(1-t^2)/sqrt(1-c^2*t^2), t=0..1 )

- Routines for complex numbers Re(z), Im(z), argument(z), conjugate(z), csgn(z)

- Routines for computing the magnitude of a number ilog[b](x), ilog10(x)

- Utility routine polar(z) to allow computing with complex numbers in polar

form i.e. polar(a,b) = a cos(b) + I a sin(b)

- The logarithmic integral Li(x) = Ei(ln(x) for x > 1

- The exponential integrals Ei(n,x) = int(exp(t)/t^n,t=1..infinity) for

Re(x)>0; extended by analytic continuation to the entire complex plane

except for 0 when n=1). Note Ei(x) is the Cauchy principle value.

- Ssi(x) the shifted sine integral

- W(n,x), the W function satisfying exp(W(x))*W(x) = x where n, any integer

specifies a branch cut. Note W(x) is the principle branch

- erfc(n,x) = int(erfc(n-1,t),t=x..infinity), for n=1,2,... where

erfc(0,x) = erfc(x). Note erfc(n,z) corresponds to "i^n erfc z" notation

in Abramowitz & Stegun

- The derivatives of signum, csgn, trunc, frac, round, floor, ceil implemented

as signum(1,x), csgn(1,x) etc.

2: Enhanced numerical facilities

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

- automatic complex numerical arithmetic, i.e. +, -, *, /, ^

- extend evalf to the complex domain for all elementary functions and erf(z),

erfc(z), erfc(n,z), GAMMA(z), GAMMA(a,z) (incomplete Gamma function),

Beta(u,v), Psi(z), Psi(n,z), W(z), W(n,z) (all branches), Ei(z),

Ci(z), Si(z), Chi(z), Shi(z), Ssi(z)

- Dawson's integral dawson(x) = exp(-x^2) * int(exp(t^2),t=0..x)

- Elliptic integrals, LegendreE(x,k), LegendreEc(k), LegendreEc1(k),

LegendreF(x,k), LegendreKc(k), LegendreKc1(k), LegendrePi(x,k,a),

LegendrePic(k,a), LegendrePic1(k,a)

- Li(x) = Ei(ln(x) the logarithmic integral defined only for real x > 1

- fsolve has a new algorithm for solving a univariate polynomial over R or C

which should always succeed and, all digits should be accurate to 0.6 ulps.

- Addition of the remez algorithm for minimax rational function approximation

- Addition of an adaptive double exponential algorithm for numerical integration

- Addition of Levin's u transform for the numerical evaluation of infinite

sums, products and limits

- The routines gauselim, gaussjord, rowspace, colspace, eigenvals, eigenvects,

inverse, det, in the linalg package now all handle matrices of floating point

and complex floating point entries

3: Algebraic numbers and algebraic functions

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

The evala facility in Maple V supports polynomial operations over algebraic

number fields. This facility is used in indefinite integration, eigenvectors,

and solving polynomial equations. E.g. Maple can factor the polynomial

3 1/2 2 1/2 1/2

x + (- 28 + 16 2 ) x + (- 288 2 + 384) x - 1792 + 1280 2

over the algebraic number field Q(sqrt(2)) as follows

> a := x^3+(-28+16*sqrt(2))*x^2+(-288*sqrt(2)+384)*x-1792+1280*sqrt(2):

> factor(a,sqrt(2));

1/2 1/2

(x - 8) (x - 8 + 8 2 ) (x - 12 + 8 2 )

Because Maple can do this factorization, it means Maple can compute the

eigenvalues (also the eigenvectors -- see below) of the following matrix, e.g.

> A := matrix( [[5-5*sqrt(2), -2+4*sqrt(2), 5+sqrt(2)],

> [-5+sqrt(2), 10-4*sqrt(2), 3+3*sqrt(2)],

> [-3+3*sqrt(2), -2+4*sqrt(2), 13-7*sqrt(2)]] ):

> eigenvals(A);

1/2 1/2

8, 12 - 8 2 , 8 - 8 2

The generalization of the evala facility to algebraic functions in Maple V

release 2 means that we can replace the sqrt in the above problem by a sqrt of

a formula and not just an integer, e.g. sqrt(a). Many parts of Maple work

better because of this. The examples in the following sections on integration,

polynomial factorization, and symbolic eigenvectors, Norms and Traces, etc.

illustrate other new capabilities in Maple which build on this facility.

4: Polynomial factorization over algebraic number and function fields

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

Maple V can factor multivariate polynomials over the rational numbers Q and

univariate polynomials over algebraic number fields Q(alpha) as in the above

example. Maple V release 2 can factor multivariate polynomials over

algebraic number fields. For example let's factor the polynomial x^6+y^6

first over the rationals Q then over Q(i) i.e. the complex rationals.

> factor(x^6+y^6);

2 2 4 2 2 4

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

> factor(x^6+y^6,I);

2 2 2 2

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

Maple V release 2 can also factor multivariate polynomials over algebraic

function fields e.g. let's factor the following polynomial over Q(sqrt(a)).

> f := x^3 + (16*a^(1/2)-28)*x^2 + (64*a-288*a^(1/2)+256)*x

> -512*a+1280*a^(1/2)-768:

> factor(f,sqrt(a));

1/2 1/2

(x - 12 + 8 a ) (x + 8 a - 8) (x - 8)

Two new factorization routines split and AFactor have been added to the

library. split(a,x) factors a univariate polynomial in x into linear factors

over its splitting field. evala(AFactor(a)) does a complete factorization

of a multivariate polynomial, i.e. over the algebraic closure of Q, e.g.

> evala(AFactor(x^2-2*y^2));

2 2

(x - RootOf(_Z - 2) y) (x + RootOf(_Z - 2) y)

The RootOf value that you see in the output above is Maples notation for

the algebraic numbers which are the solutions to the polynomial equation

_Z^2 - 2 = 0. In this case, the solutions are +- sqrt(2), i.e. they can

be written in terms of exact radicals. The user can convert from (and to)

the RootOf notation to the radical notation using the convert function

> convert(",radical);

1/2 1/2

(x - 2 y) (x + 2 y)

5: Symbolic Eigenvalue and Eigenvector computation.

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

The eigenvector routine linalg[eigenvects] returns the eigenvectors of a matrix

represented exactly in terms of the roots of the characteristic polynomial.

For example, for our matrix A used previously, we obtain the eigenvectors

> A := matrix( [[5-5*sqrt(2), -2+4*sqrt(2), 5+sqrt(2)],

> [-5+sqrt(2), 10-4*sqrt(2), 3+3*sqrt(2)],

> [-3+3*sqrt(2), -2+4*sqrt(2), 13-7*sqrt(2)]] ):

> eigenvects(A);

1/2 1/2 1/2

[8 - 8 2 , 1, {[ -1, - 2 , 1 ]}], [12 - 8 2 , 1, {[ 1, -1, 1 ]}],

[8, 1, {[ 1, 1, 1 ]}]

The output is a sequence of lists [e,m,b] where e is the eigenvalue, m

its multiplicity, and b a basis for the eigenspace for e. In Maple V, the

eigenvects command could handle algebraic numbers, including rational numbers.

Now it can handle algebraic functions, hence matrices of polynomials, e.g.

> A := matrix(4,4,[[a,a,a,0], [b,b,0,c], [b,0,c,c], [0,a,a,a]]);

[ a a a 0 ]

[ ]

[ b b 0 c ]

A := [ ]

[ b 0 c c ]

[ ]

[ 0 a a a ]

> factor(charpoly(A,x));

3 2 2 2 2 2

(x - a) (x - x a - c x - b x + b c x - c a x - b x a + c a + b a + a b c)

> alias(alpha=RootOf(op(2,"),x)): # Denote the roots of the cubic by alpha

> eigenvects(A);

[a, 1, {[ - c/b, 0, 0, 1 ]}],

2 b a - alpha + alpha c + alpha a + c a

[alpha, 1, {[ 1, - ----------------------------------------,

a (b - c)

b a + alpha b - alpha + alpha a + 2 c a

----------------------------------------, 1 ]}]

a (b - c)

Note: the optional argument 'radical' to the eigenvects command can asks Maple

to express roots of polynomials as exact radicals where possible instead of in

terms of the RootOf representation.

6: ODEs with Bessel and Hypergeometric function solutions

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

Many more cases are handled with a better algorithm, which gives results

in terms of Bessel functions rather than hypergeometric functions, e.g.

> 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 (1 + 4 k) |---- y(x)|

| d | dx / / 6 3

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

| 2 | x + 2 | 2|

dx / (x + 2) /

> dsolve(",y(x));

(- 2 k) 2 1/2 4

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

2 1/2 4

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

7: Symbolic Fourier transform

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

The call fourier(f(s),s,w) computes the fourier transform

infinity

| - I t w

F(w) = | f(t) e dt

- infinity

For example,

> fourier(s,s,w);

2 I Pi Dirac(1, w)

> fourier(1/(4 - I*t)^(1/3),t,w);

1/2

3 GAMMA(2/3) exp(- 4 w) Heaviside(w)

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

2/3

8: New limit algorithm

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

The idea behind the new algorithm is the following. First locate the most

varying part of the expression. Treat this indeterminate as a symbol and

compute a series expansion in this symbol alone. If there is no constant

coefficient in the resulting series, you are done, otherwise, recurse on

that constant coefficient. Here are some examples

> f := exp(exp(exp(Psi(Psi(Psi(n))))))/n;

exp(exp(exp(Psi(Psi(Psi(n))))))

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

> limit(f,n=infinity);

> f := n*(GAMMA(n-1/GAMMA(n))-GAMMA(n)+ln(n));

/ 1

f := n |GAMMA(n - --------) - GAMMA(n) + ln(n)|

GAMMA(n) /

> limit( f, n=infinity );

1/2

9: Symbolic simplifications for min, max, trunc, round, ceil, floor, Re, Im

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

These routines now do quite a number of symbolic simplifications.

Note, interval arithmetic is used by the signum function in these examples

to prove that the constant Pi < sqrt(10) and that cos(2) < 0.

> min( min(Pi,sqrt(10)), min(a+1,a-1), a );

min(Pi, a - 1)

> Re( ln(Pi-sqrt(10)) + sqrt(cos(2)) );

1/2

ln(- Pi + 10 )

10: Norms and Traces in algebraic number and function fields

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

Norm(alpha,L,K) and Trace(alpha,L,K) compute the norm (trace) of an algebraic

number or function in L over field K. I.e. the input is over the field L and

the output is over the field K a subfield of L. The fields L and K are

specified by sets of RootOf's which define the field extensions. For example

> alias( s2 = RootOf(x^2-2=0), s3 = RootOf(x^2-3=0) ):

> L := {s2,s3}: # so L = Q(sqrt(2),sqrt(3)) and

> K := {s2}; # K = Q(sqrt(2))

> evala(Trace(x-s2-s3,L,K));

2 x - 2 s2

> evala(Norm(x-s2-s3,L,K));

x - 2 s2 x - 1

> evala(Norm(x-s2-s3)); # K = Q

4 2

x - 10 x + 1

11: Computing with field extensions: Indep, Primfield

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

These are tools for computing with nested field extensions. Indep(L) searches

for relations between between the set of field extensions (RootOfs) L.

Primfield(L,K) computes a primitive element alpha for a set of field

extensions L over the subfield K of L, e.g.

> evala(Primfield({s2,s3}));

3 3

[ [%1 = s3 + s2], [s3 = 11/2 %1 - 1/2 %1 , s2 = - 9/2 %1 + 1/2 %1 ]]

4 2

%1 := RootOf(_Z - 10 _Z + 1)

The result is a list of two elements. The first element defines the primitive

element alpha as a RootOf the minimal polynomial over K. We see that the

minimal polynomial in this example is x^4-10*x^2+1. It equates alpha in terms

of L, i.e. here alpha = sqrt(3)+sqrt(2). The second entry expresses the field

extensions of K in terms of alpha. Here sqrt(2) = alpha^3/2-9*alpha/2, and

sqrt(3) = 11/2*alpha-alpha^3/2.

12: Continued fractions

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

The cfrac routine in the numtheory package now computes continued fraction

expansions for real numbers, polynomials, series, and formulae. Several forms

are available including simple, regular, simregular, etc. The utility routines

nthconver, nthnumer, nthdenom return the nth convergent, numerator, denominator

e.g. here is the continued fraction for Pi, and the first few convergents

> with(numtheory):

> cf := cfrac(Pi,5);

cf := 3 + --------------------------

7 + ----------------------

15 + -----------------

1 + -------------

292 + -------

1 + ...

> seq( nthconver(cf,i), i=0..5 );

333 355 103993 104348

3, 22/7, ---, ---, ------, ------

106 113 33102 33215

> cf := cfrac(exp(x),x,4,simregular);

cf := 1 + -------------------------------

1 - 1/2 -----------------------

1 + 1/6 ---------------

1 - 1/6 -------

1 + ...

> seq(nthconver(cf,i),i=0..3);

1 + 1/2 x 1 + 2/3 x + 1/6 x

1, 1 + x, ---------, ------------------

1 - 1/2 x 1 - 1/3 x

13: Diophantine approximation

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

The functions minkowski and kronecker in the number theory package solve

the linear diophantine approximation problem, namely, given real or p-adic

numbers a[i][j] and epsilon[i], and m equations of the form

|| sum( a[i][j]*x[j], j=1..m ) || <= epsilon[i], for i=1..m

where || z || means, in the real case, the distance between z and the nearest

integer z, for p-adic case || z || means p-adic valuation (see function valuep

in padic package), solve for integers x[j]. I.e. find integers x[j] and y[i]

such that abs( sum(a[i][j]*x[j],j=1..n) - y[i] ) <= epsilon[i] for i=1..m

(in the real case). This homogeneous case is solved by the minkowski function

where the equations are input in the form

abs( sum( a[i][j]*x[j], j=1..m ) - y[i] ) <= epsilon[i], for i=1..m

For example

> eqn1 := abs( E*x1 + 2^(1/2)*x2 - y1 ) <= 10^(-2):

> eqn2 := abs( 3^(1/3)*x1 + Pi*x2 - y2 ) <= 10^(-4):

> minkowski( {eqn1,eqn2}, {x1,x2}, {y1,y2} );

[x1 = 7484], [x2 = -2534], [y1 = 16760], [y2 = 2833]

The non-homogeneous case (more difficult) i.e. for real or p-adic alpha[i],

equations of the form

|| sum( a[i][j]*x[j], j=1..n ) + alpha[i] || <= epsilon[i], for i=1..m

are solved by the kronecker function.

New and Enhanced Integration Facilities

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

1: Dirac and Heaviside

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

Integration now knows about the Heaviside and Dirac functions, e.g.

> int(f(x)*Dirac(x),x=-infinity..infinity);

f(0)

> int(f(x)*Heaviside(x),x=-3..4);

| f(x) dx

2: Elliptic integrals

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

Integration now recognizes Elliptic integrals and reduces them to a normal

form in terms of Legendre functions

> f := sqrt( (1-x^2) * (1/25 + x^2/4) );

2 1/2 2 1/2

f := 1/10 (1 - x ) (4 + 25 x )

> int(1/f,x=1/2..1);

10 1/2 1/2 1/2

---- 29 LegendreF(1/2 3 , 5/29 29 )

> evalf(");

2.327355206

3: Algebraic functions

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

Integration of algebraic functions is now using the Risch-Trager algorithm.

This includes a full implementation of the algebraic part including integrands

which involve parameters and algebraic numbers, e.g.

> alias(alpha=RootOf(y^3-x^2-a,y); # i.e. alpha = (x^2+a)^(1/3)

> f := (4*alpha^2*x^3+(5*x^4+3*x^2*a)*alpha-3*x^2-3*a)/(x^2+a)/x^2;

2 3 4 2 2

4 alpha x + (5 x + 3 x a) alpha - 3 x - 3 a

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

2 2

(x + a) x

> int(f,x);

x - 1 2

- 3 ----- + 3 x alpha + 3 alpha

and an implementation of the transcendental part for algebraic functions

which do not contain parameters, e.g.

> alias(alpha=RootOf(y^2-x^3-1,y)); # i.e. beta = sqrt(x^3+1)

> f := (3*x^4-6*x^3+5*x^2-2*x+2)/(x^5-2*x^4-x+2*x^3+1)*beta

> +(3*x^4+5*x^2-5*x^3-2*x)/(x^5-2*x^4-x+2*x^3+1);

4 3 2 4 2 3

(3 x - 6 x + 5 x - 2 x + 2) beta 3 x + 5 x - 5 x - 2 x

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

5 4 3 5 4 3

x - 2 x - x + 2 x + 1 x - 2 x - x + 2 x + 1

> int(f,x);

2 3

x + x + 1 - 2 x beta

2 beta - ln(----------------------)

3 2 2

(x - x + 1)

4: Integration of the W function

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

Indefinite integration now uses an inverse function transformation to

transform integrals involving the W function (where note W(x) is the function

defined to satisfy exp(W(x))*W(x)=x ) of linear arguments to elementary

functions, hence the following integrals are computed

> int(x/W(x),x);

2 2

x x

1/2 ---- + 1/4 -----

W(x) 2

W(x)

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

---- - Ei(1, - W(x))

W(x)

5: The Exponential integral, error function, and related integrals

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

Indefinite integration of the forms g(x)*sin(f(x)) and g(x)*cos(f(x)) where

g(x) is a rational function in x and f(x) is a quadratic polynomial in x are

expressed in terms of the Fresnel integrals FresnelC(x) and FresnelS(x),

and the cosine and sine integrals Ci(x) and Si(x). Note, Maple V can do

these examples, but the answer involves complex error functions and

Exponential integrals of complex arguments

> int( 5*x^4*sin(2*x^2+1), x );

3 2 15 2

- 5/4 x cos(2 x + 1) + ---- x sin(2 x + 1)

15 1/2 / x x

- ---- Pi |cos(1) FresnelS(2 -----) + sin(1) FresnelC(2 -----)|

32 | 1/2 1/2 |

Pi Pi /

> int( sin(u*x^2-v)/x, x );

2 2

1/2 cos(v) Si(u x ) - 1/2 sin(v) Ci(u x )

6: New classes of definite integrals

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

Several new classes of infinite indefinite integrals are now handled.

These are computed by evaluating derivatives of special functions

and exploiting symmetry. For example, for p>0 and q>0, the class

infinity

| (a - 1) s q

| x exp(- p x - ----) dx

| s

/ x

in general can be expressed in terms of the Bessel K function e.g.

> assume(p>0);

> assume(q>0);

> int( x^2*exp(-p*x^3-q/x^3), x=0..infinity );

1/2 1/2 1/2

q BesselK(1, 2 p q )

2/3 -------------------------------

1/2

Note, the appearance of p and q in this answer indicates that assumptions

about p and q have been made.

Also, several classes involving Bessel functions e.g. for u>0,v>0, the class

infinity

| s w s/2

| exp(- u t ) t BesselJ(v, c t ) dt

in general results in 1 F 1, the confluent hypergeometric function, e.g.

> int( exp(-t)*t^2*BesselJ(1,sqrt(t)), t=0..infinity );

15 1/2

---- Pi hypergeom([7/2], [2], -1/4)

Also classes involving the error functions, e.g. for u>0

> assume(u>0);

> Int( exp(-u*x^2)*erf(b*x), x=0..infinity ) =

> int( exp(-u*x^2)*erf(b*x), x=0..infinity );

infinity arctan(-----)

/ 1/2

| 2 u

| exp(- u x ) erf(b x) dx = -------------

| 1/2 1/2

/ Pi u

7: Handling singularities in an indefinite integral

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

Definite integration now uses a new algorithm to resolve integrals which have

discontinuities (removable or non-removable) in the result from indefinite

integration. For example,

> int( 1/(5+3*cos(x)), x=0..13*Pi/2 );

1/2 arctan(1/2) + 3/2 Pi

This result was obtained by first computing the indefinite integral

> int( 1/(5+3*cos(x)), x );

1/2 arctan(1/2 tan(1/2 x))

then by taking limits given the following information about the discontinuities

> readlib(discont)(",x);

{2 Pi _Z1 + Pi}

A second example showing that Maple can prove that this integral diverges,

instead of returning unevaluated -- meaning I don't know.

> int(1/x^2,x=-1..1);

infinity

Also, the addition of the CauchyPrincipalValue option to int handles this one

> int( 1/x, x=-1..1, CauchyPrincipalValue);

> int( exp(-t)/t, t=-2..infinity, CauchyPrincipalValue );

- Ei(2)

8: Numerical integration uses exponential quadrature

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

An adaptive, double-exponential routine has been implemented to handle

those integrals for which the default Clenshaw-Curtis quadrature routine

(based on Chebyshev series) does not succeed. This new routine replaces the

use of an 8'th order adaptive Newton-cotes routine, which, being a fixed

order method, does not perform well at high precision.

9: User interface for indefinite integration

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

An interface for indefinite integration has been added to allow users to

teach Maple about how to compute the indefinite integral of an expression f(x)

involving the function g(x) by writing a Maple procedure for `int/g` just as

one would write `diff/g` for differentiating g(x). The call int(g(x),x) will

result in the calling `int/g`(f(_X)) i.e. the whole expression in terms of

the global _X will be passed to `int/g`. The routine should return FAIL if

it cannot compute the integral.

New and Enhanced System Facilities

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

1: Type declarations for parameters

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

Procedures accept type declarations as illustrated by the following example.

The purpose of this facility note is to encourage better type checking.

proc( a:t )

....

end

is equivalent to writing

proc(a)

if nargs > 0 and not type(a,'t') then ERROR(message) fi;

....

end

Note, the type checking is not static, it is done when the procedure is called.

The error message generated automatically is illustrated by this example

> f := proc(n:integer) if n < 0 then 0 else n fi end:

> f(x);

Error, f expects its 1st argument, n, to be of type integer, but received x

2: D extended to computation sequences and programs

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

The D operator can now compute partial derivatives of functions which are

defined as procedures. This is known as automatic differentiation. Consider

> f := proc(x) local t1,t2;

> t1 := x^2;

> t2 := sin(x);

> 3*t1*t2+2*x*t1-x*t2

> end:

> # The following computes the derivative of f wrt x, the first argument

> D[1](f);

proc(x) local t1x,t2x,t1,t2;

t1x := 2*x;

t1 := x^2;

t2x := cos(x);

t2 := sin(x);

3*t1x*t2+3*t1*t2x+2*t1x*x+2*t1-t2x*x-t2

end

The reader can check that this really does compute the derivative of f by

verifying that diff(f(x),x) - D(f)(x) = 0. The advantage of automatic

differentiation is twofold. Firstly, it is much more efficient, in general, to

represent a function as a program instead of as a formula. Secondly, it is

more general as functions can have conditional statements and loops.

For example, given the array(0..n) of the coefficients of a polynomial b,

we can represent the polynomial as a program in Horner form as follows.

> f := proc(x,b,n) local i,s;

> s := 0;

> for i from n by -1 to 0 do s := s*x+b[i] od;

> s

> end:

> f(x,b,4);

(((b[4] x + b[3]) x + b[2]) x + b[1]) x + b[0]

> fx := D[1](f);

fx := proc(x,b,n)

local sx,i,s;

sx := 0;

s := 0;

for i from n by -1 to 0 do sx := sx*x+s; s := s*x+b[i] od;

end

3: Program optimization -- optimize extended to computations sequences

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

The optimize routine has been extended to optimize Maple procedures.

Currently it does common subexpression optimization on computation sequences,

i.e. procedures with assignment statements to local variables only, e.g.

> f := proc(x) local t; t := x^2; 3*t*sin(x)+2*x*t-x*sin(x) end:

> readlib(optimize)(f);

proc(x) local t,t1; t := x^2; t1 := sin(x); 3*t*t1+2*x*t-x*t1 end

4: Multiple libraries -- readlib takes multiple pathnames

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

Maple automatically reads code from the Maple library using the readlib

function. The global variable libname specifies where the Maple library is.

This can now be assigned a sequence of names. E.g. if the user does

> libname := dir1, libname, dir3;

this means that when readlib(f) is executed, Maple will search for f first

under dir1, and if unsuccessful, it will search under the Maple library,

and if unsuccessful under dir3. This means users can have their own

development library and have it override the Maple library.

5: Assume facility

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

One of the deficiencies of Maple and other systems is in handling problems

which contain symbolic parameters where the result depends on the domain or

range of values that the symbolic parameter(s) takes. E.g. consider

> Int( exp(-a*t)*ln(t), t=0..infinity );

infinity

| exp(- a t) ln(t) dt

The answer to this integral depends on the value of the parameter a. If a is

real and positive, the answer is finite. If a is real and non-positive, the

answer is infinite. How can the user tell Maple that a is real and positive?

The solution adopted in Maple V release 2 is to state the assumption about

a, i.e. that a is real and positive as follows

> assume(a>0);

The result of this assumption is that the variable a has been assigned a new

value which prints as a which Maple knows is real (implicitly) and positive.

> a;

> about(a);

Originally a, renamed a:

is assumed to be: RealRange(Open(0),infinity)

The assume facility is presently being integrated into Maple, in particular,

Maple can now compute the definite integral above because Maple can determine

that a is positive because signum(a) returns 1.

> signum(a);

> int( exp(-a*t)*ln(t), t=0..infinity );

ln(a) gamma

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

a a

Note: Maple V users should stop using signum(a) := 1; because this is a

side effect and does not work in conjunction with option remember.

The assume facility includes a data base of known properties, and a deduction

facility for answering questions like

> is(a+1>0);

true

A deficiency of the assume facility is that the user needs to know what to

assume in order to get answers out of facilities like definite integration.

This is a topic of current research and future versions of Maple will provide

different tools for this.

6: Automatic complex numeric arithmetic

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

Complex arithmetic in Maple V had to be done with the evalc function.

Now, complex numeric arithmetic is automatic.

The evalc function is used only for symbolic arithmetic. Example

This includes complex rational arithmetic

> (2+I/3)^2;

35/9 + 4/3 I

And complex floating point arithmetic

> sin( (2.0+I/3)^2 );

- 1.378776230 - 1.294704958 I

The evalc function is now used only for exact symbolic expansions e.g.

> evalc(exp(2+Pi/3*I));

1/2

1/2 exp(2) + 1/2 I exp(2) 3

> evalc( sqrt(a+I*b) );

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

(1/2 (a + b ) + 1/2 a) + I csgn(b - I a) (1/2 (a + b ) - 1/2 a)

> assume(a>0);

> assume(b>0);

> evalc(sqrt(a+b*I));

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

(1/2 (a + b ) + 1/2 a) + I (1/2 (a + b ) - 1/2 a)

7: Arrow operators now accept if statements

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

The arrow operators, e.g. x -> x/(x^2-1); have been extended to handle

if statements so that they can be used to define piecewise functions.

E.g. x -> if x < 0 then 0 elif x < 1 then x else 0 fi;

8: I/O facilities and changes

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

- printf: output a given expression with format (as in the "C"

printf, with the added option of %a for algebraic), and also

includes the C escapes: b for backspace, t for tab, n for newline, e.g.

> printf(`n = %dtvar = %stresult = %an`, 10, y, 3*y^2+1);

n = 10 var = y result = 3*y^2+1

- sscanf: decode a string according to a format ("inverse" of

printf, identical to the "C" sscanf function).

- input (tty and read) and output (lprint and save) of numbers in scientific

E/e notation e.g. 1.2e+3 versus Float(12,2) or 1.2*10^3

- readline: read a line of an arbitrary file and return it as a string

- parse: parse a string into a Maple expression

- readdata: reads a file of numerical data arranged in columns

- output (lprint and save) of powers using ^ instead of **

9: New types

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

- type(a,arctrig) tests if a is an arc trigonometric function

- type(a,complex(t)) tests if an expression of the form a + b*I where a and b

are of type t. Typical cases of t are complex(integer), i.e. Z[i], the

Gaussian integers, complex(rational), i.e. Q(i), complex(float), i.e. a

complex floating point number, and complex(numeric) which is any of these.

- type(a,nothing) always returns false

- type(f,mathfunc) tests if a function is a "math" function

The definition is that f is either an operator, or evalf/f exists,

i.e. Maple knows how to evaluate it in floating point.

This includes abs, exp, erf, BesselJ, etc. but also int, sum, limit,

and arrow procedures with option operator e.g. x -> sin(x)*cos(x);

- type(a,TEXT) tests if a is a TEXT object (used for help files)

10: What is Maple doing?

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

The userinfo facility, that was introduced in Maple V as a replacement

for assigning printlevel to 2, 3, or 4, has been completed.

For information on what Maple is doing, and what algorithms it is using

for a specific function, e.g. integration, one does infolevel[int] := 4;

For information on all functions one does infolevel[all] := 4;

11: Single library archive

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

On the personal computers, the Macintosh, PC, Amiga and NeXT, the Maple V

library is now being distributed as a single file. This saves a significant

amount of storage, and avoids problems with systems which have difficulty

maintaining large numbers of files. This will be done on other platforms

as well. An override mechanism is also provided so files can be updated.

The library archive file also resolves problems on systems which have limited

filename lengths and case insensitive filenames.

12: errorbreak

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

Maple has added some user control for how Maple responds on encountering errors

when reading in a file. This is provided by the interface option errorbreak.

- With interface(errorbreak=0); the entire input file is read and all errors

that are encountered are displayed. This was the Maple V behavior.

- With interface(errorbreak=1); processing stops after the first syntax error

thus avoiding nonsensical multiple syntax error messages that are often

consequences of the first one. This is the new default behavior.

- With interface(errorbreak=2); processing stops after the first error,

whatever it is, i.e. syntax (parse time) or semantic (execution) errors.

Algorithmic Improvements

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

1: Integration of rational functions

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

The Trager-Rothstein algorithm computes the integral of a rational function

f(x) over a field K as a sum of logs over K(alpha), an algebraic extension

of K of minimal degree. E.g. consider the following rational function

in Q(x) where note the denominator is irreducible over Q.

> f := (6*x^5+6*x^2-4*x+8*x^3-4)/(x^6+2*x^3+1-2*x^2);

5 2 3

6 x + 6 x - 4 x + 8 x - 4

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

6 3 2

x + 2 x + 1 - 2 x

Applying the Trager-Rothstein algorithm yields the following result

-----

(1 + alpha) ) ln(x - alpha x + 1)

-----

(alpha - 2) = 0

I.e. there are two logs corresponding to two roots +- sqrt(2) of the

polynomial alpha^2-2 = 0. Hence, Maple gets the following nice result

> int(f,x);

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

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

The first improvement in release 2 is in handling coefficient fields which are

function fields, and not just number fields. I.e. the input rational function

could involve parameters as well as numbers. This is because of the extension

of evala to handle algebraic functions and not just algebraic numbers.

The Lazard-Trager improvement allows this answer to be obtained directly

without having to compute in the more complicated field K(alpha), i.e. in

this example, it doesn't have to compute over Q(sqrt(2)), only Q.

2: Bivariate polynomial Gcd's over Z

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

We have implemented the dense modular method of Collins for computing Gcds

of polynomials with in 2 variables with integer coefficients. The method

works by computing computing many gcd's of polynomials in 1 variable modulo

different primes and combining these solutions using polynomial interpolation

and the Chinese remainder theorem. Note, in Maple V, we implemented this

method for univariate polynomials. Also, this is a dense method. For sparse

polynomials in many variables it is not a good method. Comparing this with

the default EEZGCD algorithm in Maple V, we obtain the following improvement.

We created three dense bivariate polynomials g,a,b each of degree n with

random n digit coefficients and computed gcd(a*g, b*g).

n 10 15 20

Maple V 65.0 1,877 18,763

Maple V rel 2 10.6 33.7 393.7

3: Integer version of the LLL lattice reduction algorithm

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

The LLL (Lenstra Lenstra Lovasz) lattice reduction algorithm has been sped

up by a factor of 3 or so for the case of an integer matrix.

4: Factorization over algebraic number fields

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

We have implemented Lenstra's algorithm for factorization of polynomials over an

algebraic number field. For some polynomials it runs considerably faster than

the default algorithm. But it depends very much on the polynomial.

5: Faster numerical evaluation of the arctrig functions

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

The running time of the algorithms used for the numerical evaluation of

exp(x) and ln(x) at high precision is O(n^2 log_2 n) for n digits.

This is O(log_2 n) integer multiplications each of which is O(n^2).

The algorithms for the trig functions sin, cos, tan use repeated argument

reduction to obtain an O(n^(5/2)) algorithm. The same idea is now used

by the arctrig functions, e.g.

Digits arcsin(Pi/4) arctan(Pi/4)

Maple V Maple V rel 2 Maple V Maple V rel 2

500 25 3.0 23 2.5

1000 176 11.0 148 11.2

2000 1245 50.4 1038 45.2

6: Optimized the evaluation of list and set subscripts

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

List and set subscripts i.e. s[i] where s is a list or set have been sped up

by a factor of 3 to 5.

7: Optimized the evaluation of numerical arithmetic

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

Evaluation of numerical expressions e.g. n-i+1, 2*n-2*j+2, has been sped up to

avoid the creation of an intermediate data structure. This is a significant

improvement in the cases of small integers, e.g. array subscript calculations.

Consider these two codes for computing the Fibonacci numbers F(n)

> F1 := proc(n) if n < 2 then n else F1(n-1)+F1(n-2) fi end;

> F2 := proc(n) F2(n-1)+F2(n-2) end; F2(0) := 0; F2(1) := 1;

The improvement in the arithmetic that occurs in F(n-1)+F(n-2) yields

Maple V Maple V rel 2

n F1 F2 F1 F2

16 1.05 0.73 0.78 0.47

18 2.73 2.17 2.17 1.32

20 7.30 5.20 5.92 3.55

22 20.17 13.45 15.20 8.53

8: Floating point solutions to polynomials

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

We have rewritten the code for computing floating point approximations to roots

of polynomials. The new code will increase the intermediate precision so that

the roots found are fully accurate, i.e. accurate to 0.6 ulp. We have

also implemented a routine in the kernel in C to use hardware floating point

arithmetic. If the roots found using this routine are accurate then the fsolve

routine will execute faster, as illustrated in the following examples. T(n,x)

is the n'th Chebyshev polynomial of the first kind. It has all real roots.

F(n,x) is the n'th Fibonacci polynomial which has all complex roots except 0.

The jump in times indicates where hardware precision is no longer sufficient.

Maple V Release 2

n T(n,x) F(n,x) T(n,x) F(n,x)

10 1.68 1.86 0.45 0.23

20 8.95 11.43 2.17 1.43

30 FAIL 36.72 6.08 3.47

40 FAIL 34.22 7.57

50 106.2 50.45

9: Character tables for Sn

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

In computing the character table for Sn the symmetric group on n letters we

are now making use of the conjugate partitions and symmetry in the table.

The improvement is better than a factor of 2 for large n. Note the table has

dimension p(n) by p(n) where p(n) is the number of partitions of n so requires

exponential space in n to store.

The data here are for computing combinat[character](n)

Maple V Release 2

8 6.6 3.5

10 26.0 12.6

12 102.0 43.4

14 446.8 160.3

New Library Functions

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

AFactor, AFactors -- absolute factorization over the complex numbers

argument -- complex argument of a complex number

assume -- for making assumptions about variables e.g. assume(a>0);

ceil -- ceiling function

csgn -- complex sign

discont -- compute a set of possible discontinuities of an expression

floor -- floor function

fourier, invfourier -- symbolic fourier transform and inverse

ilog, ilog10 -- IEEE integer logarithm function

Indep -- test algebraic extensions over Q for independence

invfunc -- table of inverse functions

is, isgiven -- for testing properties against the assume database

Norm -- computes the norm of the algebraic number or function over a field K

Primfield -- computes a primitive element over a given algebraic number field

powmod -- computes a(x)^n mod b(x) using binary powering

ratrecon, Ratrecon -- rational function reconstruction (Euclidean algorithm)

readdata -- reads a file of numerical data arranged in columns

spline -- computes a natural cubic spline from a set of data

split -- splits a polynomial into linear factors over its splitting field

sqrfree -- a square-free factorization of a multivariate polynomial

symmdiff -- symmetric difference of sets

Trace -- computes the trace of the algebraic number or function over a field K

unload -- unload a Maple library function

Library Packages

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

The package mechanism has been extended to support an initialization routine.

The initialization routine should be placed either in the package as the

routine package[init], or as the routine `package/init` in the library in

the file `package/init.m`; This routine is executed by with(package);

It is up to the initialization routine to decide what to do if it is called

a second time.

Below is a list of the new packages and modifications to existing packages.

1: approx (Numerical Approximation Package)

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

- infnorm: compute the (weighted) infinity norm of an approximation

- minimax: compute a (weighted) minimax numerical approximation

- remez: Remez algorithm for minimax rational function approximation

The main function in this package is minimax which has the following

functionality: minimax(f,x=a..b,n,w) or minimax(f,x=a..b,[m,n],w); It uses

the Remez algorithm to compute the best minimax polynomial of degree <= n,

or rational function with numerator of degree <= m, and denominator of

degree <= n to the function f on the range [a,b]. The optional 4'th parameter

w specifies a weight function (default 1). For example, here is a [4,4]

rational approximation to erf(x) on [0,3] which I've rounded to 7 decimal

digits. The approximation is accurate to almost 5 digits on [0,3].

> evalf(minimax(erf(x),x=0..3,[4,4]), 7);

.00002407481 + (1.127063 + ( - .2493939 + (.09458902 + .06665265 x) x) x) x

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

1. + ( - .2313770 + (.4509384 + ( - .06758241 + .08086507 x) x) x) x

2: combinat (Combinatorial Functions Package)

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

- conjpart: conjugate partition

3: DEtools (Differential Equation Plots Package)

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

A package for plotting differential equations including systems of ODEs

and some special PDEs, allowing the associated vector field to be plotted

and solution curves for different initial conditions.

- DE1: plot a first order linear ODE for given initial conditions

- DE2: plot a system of two first order linear ODEs

- PDEplot: plot the surface of a first order, quasi-linear PDE of the form

P(x,y,u) * D[1](u)(x,y) + Q(x,y,u) * D[2](u)(x,y) = R(x,y u)

- dfieldplot: plot a field plot for a single or two first order linear ODEs

- phaseportrait: plot a phase plot for a single or two first order linear ODEs

4: Gauss

--------

A package which allows one to create "domains" of computation in a similar way

that the AXIOM system allows. The advantage is that routines for computing

with polynomials, matrices, and series etc. can be parameterized by any

coefficient ring. E.g. given a ring R, the Gauss statement

> P := DenseUnivariatePolynomial(R,x):

creates a dense univariate polynomial domain P which contains all the routines

needed to do arithmetic with polynomials over R. The advantage is that the

the code supplied by the DenseUnivariatePolynomial function works for any

coefficient domain R which is a ring, i.e. has operations +, -, *, etc.

It is expected that many Maple library routines will be recoded in Gauss over

the next few versions. The initial implementation includes basic facilities

for computing with polynomials, matrices and series over number rings,

finite fields, polynomial rings, and matrix rings.

5: GaussInt (Gaussian Integer Package)

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

This is a set of routines for computing with Gaussian (complex) integers, i.e.

numbers of the form a + b I where a and b are integers. Included are routines

for computing the Gcd of two Gaussian integers, the factorization of a Gaussian

integer, and a test for primality, e.g.

> GIfactor(13);

(3 + 2 I) (3 - 2 I)

6: linalg (Linear Algebra Package)

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

- blockmatrix: utility routine to create a block matrix

- gausselim, gaussjord, rowspace, colspace: now handle matrices of

floats, complex floats, and complex rationals

- eigenvals, eigenvects: now handle matrices of floats and complex floats

- entermatrix: utility routine for entering the entries of a matrix

- randvector: utility routine to create a random vector

- ratform: (synonym for frobenius) computes the rational canonical form

- Wronskian: computes the Wronskian matrix

7: numtheory: (Number Theory Package)

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

- cfrac: modified to handle different forms of continued fractions

real numbers, series, polynomials.

- cfracpol: computes continued fractions for the real roots of a polynomial

- kronecker: Diophantine approximation in the inhomogeneous case

- minkowski: Diophantine approximation in the homogeneous case

- nthnumer: returns the denominator of the n'th convergent

- nthdenom: returns the numerator of the n'th convergent

- nthconver: returns the n'th convergent of a simple continued fraction

- thue: compute the resolutions of Thue equations or Thue inequalities

- sq2factor: factorization in the UFD Z[sqrt(2)]

8: padic (Padic Number Package)

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

A package for computing P-adic approximations to real numbers, e.g.

> with(padic):

> evalp(sqrt(2),7);

2 3 4 5 6 7 8 9

3 + 7 + 2 7 + 6 7 + 7 + 2 7 + 7 + 2 7 + 4 7 + O(7 )

> evalp(exp(3),3);

2 3 4 6 8 9

1 + 3 + 3 + 2 3 + 2 3 + 3 + 3 + O(3 )

9: plots (Plots Package)

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

- animate: animate one or more functions in 2 dimensions, e.g.

a) sine - cosine waves

animate({sin(x*t),cos(x*t)},x=-2*Pi..Pi,t=0..2*Pi);

b) a polar coordinate unraveling

animate([sin(x*t),x,x=-4..4],t=1..4,coords=polar,numpoints=100,frames=100);

- animate3d: animate one or more curves or surfaces in 3 dimensions, e.g.

animate3d(cos(t*x)*sin(t*y), x=-Pi..Pi, y=-Pi..Pi,t=1..2);

- conformal: routine is typically 3 times faster

- contourplot: contour plot of a 3 dimensional surface, e.g.

contourplot(sin(x*y),x=-Pi..Pi,y=-Pi..Pi,grid=[30,30]);

- densityplot: density plot, e.g.

densityplot(sin(x*y),x=-Pi..Pi,y=-Pi..Pi,grid=[30,30]);

- fieldplot: field plot of a linear first order ODE

vfield := [x/(x^2+y^2+4)^(1/2),-y/(x^2+y^2+4)^(1/2)]:

fieldplot(vfield,x=-2..2,y=-2..2);

- fieldplot3d: field plot of a linear first order ODE

fieldplot3d([2*x,2*y,1],x=-1..1,y=-1..1,z=-1..1,

grid=[10,10,5],arrows=THICK,style=patch);

- gradplot: plot the gradient vector field of a 2 dimensional function

- gradplot3d: plot the gradient vector field of a 3 dimensional function, e.g.

gradplot3d( (x^2+y^2+z^2+1)^(1/2),x=-2..2,y=-2..2,z=-2..2);

- implicitplot: plot a curve defined implicitly by F(x,y) = 0, e.g.

implicitplot( x^2 + y^2 = 1, x=-1..1,y=-1..1 );

- implicitplot3d: plot a surface defined implicitly by F(x,y,z) = 0, e.g.

implicitplot3d( x^2+y^2+z^2 = 1, x=-1..1, y=-1..1, z=-1..1 );

- logplot: log plot, e.g. logplot(10^(20*exp(-x)),x=1..10);

- loglogplot: log log plot, e.g. loglogplot(10^(20*exp(-x)),x=1..100);

- odeplot: plot the solution of an ODE from dsolve

p := dsolve({D(y)(x) = y(x), y(0)=1}, y(x), numeric):

odeplot(p,[x,y(x)],-1..1);

- polygonplot: draw one or more polygons in 2 dimensions

ngon := n -> [seq([ cos(2*Pi*i/n), sin(2*Pi*i/n) ], i = 1..n)]:

polygonplot(ngon(8));

- polygonplot3d: draw one or more polygons in 3 dimensions

- polyhedraplot: for plotting tetrahedrons, octahedrons, hexahedrons,

dodecahedrons,icosahedrons e.g.

polyhedraplot([0,0,0],polytype=dodecahedron);

- surfdata: plot a surface input as a grid of data points

cosdata := [seq([ seq([i,j,evalf(cos((i+j)/5))], i=-5..5)], j=-5..5)]:

F:= (x,y) -> x^2 + y^2:

surfdata( cosdata, axes=frame, color=F );

- textplot: for plotting labelled points in 2 dimensions, e.g.

textplot([1,2,`one point in 2d`],align={ABOVE,RIGHT});

- textplot3d: for plotting labelled points in 3 dimensions, e.g.

textplot3d({[1,2,3,`one point in 3d`],[3,2,1,`second point in 3d`]});

10: networks: (Graph Networks Package)

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

A package of routines for creating, manipulating and computing properties of

graphs, i.e. graphs with edges and vertices. Primitives exist for computing

properties such as edge connectivity, biconnected components, and edge or

vertex set induced subgraphs. The graphs may have loops or multiple edges,

both directed and undirected, and arbitrary weights can be associated with

both edges and vertices. Graphs can be drawn as a picture (a plot) for

display. For example, here we create the Petersen graph and compute and factor

its characteristic polynomial.

> with(networks):

> factor(charpoly(petersen(),x));

4 5

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

The spanpoly(G,p) function computes a polynomial for the undirected graph G

where p is a probability of failure associated with each edge. When G is

connected, this is the all-terminal reliability polynomial of G. Here is the

all-terminal reliability polynomial for the complete graph on 4 vertices.

> spanpoly(complete(4),p);

3 2 3

- p (- 16 + 33 p - 24 p + 6 p )

11: liesymm: (Lie Symmetries Package)

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

This package is for characterizing the contact symmetries of a system

of partial differential equations. The basic routines makeforms() and annul()

for automatic translation to and from differential forms now maintain and make

use of additional information which allow the full reconstruction of the

original system of PDEs, and handle a broader class of systems of PDEs.

> with(liesymm):

> e1 := Diff(h(t,x),x,x) = Diff(h(t,x),t):

> makeforms(e1,h(t,x),w);

[d(h) - w1 d(t) - w2 d(x), (d(w2) &^ d(t)) - w1 (d(x) &^ d(t))]

> annul(");

/ 2

| d | / d

{|----- h(t, x)| - |---- h(t, x)| = 0}

| 2 | dt /

dx /

In addition, knowledge of the original system of equations has been maintained

and used to carry out some basic simplifications of the system of PDEs

characterizing the contact symmetries. New routines include:

- indepvars, depvars: routines for enquiring or setting the independent and

dependent variables for the current problem. E.g. indepvars(); returns

[t, x] and depvars(); returns [h] in the example. This information is

established through use of commands like makeforms() and determine()

- choose: analogous to isolate() but for wedge products.

- reduce: makes use of knowledge of the dependencies to simplify the resulting

determining equations.

- autosimp: carries out some simple integrations and uses the results

to simplify the given system of PDEs.

- TD, Eta, extgen, prolong: for generating the extended operator

- value, dvalue: the value routine has been renamed to dvalue to avoid conflict

with the main value routine. This routine evaluates inert derivatives

(Diff) and returns its result in the inert form .

Share Library

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

1: Contributions from Maple users

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

The share library is a repository of applications codes, contributed primarily

by Maple users. It is distributed with each version of Maple in both src and

Maple ".m" format. However, our ambitions have expanded. The share library

has been completely reorganized. Beginning with this version, it also includes

- TeX/LaTeX introductory documents on using Maple

- TeX/LaTeX support documentation for applications codes

- updates to library routines, i.e. bug fixes, efficiency improvements etc.

- examples of applications of Maple to problem solving

We have also endeavored to ensure that each contribution comes with on-line

Maple style help files, and Maple style test files to help us ensure that the

code is working correctly.

2: How to access code from the share library from inside Maple

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

Users can access routines in the share library by reading them in directly

from the files using the read command. Also, where appropriate, one can now

access them directly from inside Maple using the with command.

The command with(share); looks to see if the share library exists, i.e. it

is adjacent to the Maple library (it simply tries to read a file from it).

If successful, it assigns the global variable "sharename" to the path where

it found the share library, and then it executes the statement

> libname := libname, sharename;

Now Maple's readlib command and with command will automatically search the

share library after first searching the Maple library. If unsuccessful, the

user can locate the share library and set the libname variable directly.

For example, the fit routine in the share library can be used to "fit" a

formula to some data in the least squares sense.

> with(share): # locate the share library

> readlib(fit): # now finds the fit routine from the share library

> fit([-1.5,0.5,1.5],[0,-2,-1.5], a*t^2+b*t+c, t);

.4999999997 t - .5000000001 t - 1.874999999

Documentation for the code is in the file. After reading in the fit routine,

additional documentation is included in TeX/LaTeX files.

See ?share,contents for a summary of the contents of the share library.

3: Electronic distribution version of the Maple share library

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

The share library will also be distributed electronically, firstly, over the

internet by using anonymous ftp, and secondly, by electronic mail, with an

electronic mail server. The electronic version is be updated periodically,

about once every 3 months, with new contributions, corrections to previous

contributions etc. Unlike the Maple version, it contains a directory of

updates to the Maple library, primarily simple bug fixes.

For directions on how to get code using anonymous ftp, electronic mail,

and how to install it and use it, see ?share for details.

4: Policy of the contributions to the share library

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

The code in the share library has been contributed freely by the authors to be

made freely available to all Maple users. It is not to be sold for profit or

changed without the author's permission. Authors who contribute code to the

share library will be asked to write Maple style on-line help documentation and

a test file. Authors will also be asked to sign a non-exclusive agreement to

let us distribute your code, and to let us change your code to maintain it.

Miscellaneous

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

- the catenation operator . now expands in ``.(a,b,c).(1..3) to yield

a1, a2, a3, b1, b2, b3, c1, c2, c3

- expand of @ and @@ now handled e.g. expand((sin@@)(2)(x) ); ==> sin(sin(x))

- evalf of @ and @@ now handled e.g. evalf((sin@@2)(2)); ==> .7890723436

- anames(t) returns all names of type t

- a[1] := x; is disallowed if a is assigned an object other than

an array or table

- f()[1] is now allowed; it returns unevaluated

- Changes to fortran and C

- both now use appendto instead of writeto to append output to

a file instead of overwriting it

- both now translate subscripts e.g. a[i-2*j] where the 2 here

is understood to be an integer

- both now convert symbolic constants e.g. Pi to floating point

- optional parameter digits = n for specifying the precision to be used

in converting constants to floating point

- C now also breaks up large expressions which otherwise cause

some compilers to break

- Changes to dsolve/numeric

The output of dsolve({equations in x},{functions of x},numeric)

is a procedure f where f(x) returns a tuple of values. E.g

dsolve({...},{y(x),z(x)},numeric); returns f a procedure where

f(1) returns x=1, y(x)=y(1), z(x)=z(1)

The odeplot routine in the plots package can be used to plot the result.

- Mellin speeded up by a factor of 4-30

- ifactor has been sped up for the case of a perfect power

- series(f(x),x=a); now allows +-infinity for a

> series(ln(1-x),x=-infinity);

1 1 1 1 1

- ln(- 1/x) - 1/x - ---- - ---- - ---- - ---- + O(----)

2 3 4 5 6

2 x 3 x 4 x 5 x x

- signum(f) where f is a symbolic constant will use interval arithmetic at

Digits precision to determine the sign of f, e.g. signum(Pi-4*sin(1));

- the for loop clauses "by", "to", "from" and "while" can now

appear in any order

- Changes to automatic simplifications

- the ln(1/x) ==> -ln(x) automatic simplification has been turned off.

ln(a^n) ==> n ln(a) is now done iff signum(a) = 1, i.e. x is provably +ve

- the ln(exp(x)) ==> x automatic simplification has been turned off.

ln(exp(x)) ==> x is now down iff Im(x) returns 0, i.e. x is provable real

- the automatic simplification sqrt(x^2) ==> x (incorrect for -ve x) is being

fixed for the next version (the change is more difficult than the others)

- the arcsin(sin(x)) ==> x arctrig simplifications have been turned off

- inverse functions simplifications have been added e.g. sin@@(-1) ==> arcsin

- trig simplifications for multiples of Pi/8, Pi/10, Pi/12 have been added

> sin(Pi/8), sin(Pi/10), sin(Pi/12);

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

1/2 (2 - 2 ) , 1/4 5 - 1/4, 1/2 (2 - 3 )

- Changes to evalhf

- can now be interrupted

- now accepts break/next in for loops

- now understands round, floor, ceil, Dirac and Heaviside.

- improved error messages

- rounding on input to 1/2 ulp

- rounding on output to less than 1/2 ulp (except that result is

truncated to 4 words later and this may cause up to 2.0 ulps)

- arrays passed as local arguments now work

- a primitive printlevel debugging facility added for tracing assignments

though note this may yield a large output

- now understands true and false

- the old library file name limit of 9 characters has been removed.

The limit is now 55 characters for the path e.g. int/rischalg/intbasis

- fsolve recognizes the option real as well as complex. The default

is to compute the real roots.

- Additional simplifications for Dirac and Heaviside are provided through

the general purpose simplify facility, e.g.

> simplify( Heaviside(x-1)*Heaviside(x-3) );

Heaviside(x - 3)

> simplify( F(x+3)*Dirac(x) );

F(3) Dirac(x)

- Additional simplification routines for the arctrig functions are provided

through the general purpose simplifier, e.g.

> simplify( arctan(x)+arctan(1/x) );

1/2 signum(x) Pi

- A new description field has been added to procedures for stating a short

description of what a procedure computes. The idea is that most of the time,

people use the ? help facility to remind them of what the parameters are,

their order, the options, and what the computes, in particular, the return

type. The print facility provides this in conjunction with the new parameter

type declarations, and the description field, e.g. a library routine

> F := proc(n:integer)

> description `computes the n'th Fibonacci number, an integer`;

> if n < 2 then n else F(n-1)+F(n-2) fi;

> end:

> print(F);

proc(n:integer)

description `computes the n'th Fibonacci number, an integer`;

...

end

Changes in Functionality

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

We have endeavored NOT to change the functionality of existing functions

in the system, language, and library from version V release I to release 2

so that users' code will work, and the new manuals will remain accurate.

In some cases it was necessary to make a change in order to fix a problem or

inconsistency. We have listed the changes that are most likely to affect

users below. Please note however that the .m format has changed, so all

users will need to reload any code from Maple V or Maple 4.3.

1: Complex numerical arithmetic is now automatic, i.e. +, -, *, /, ^, of

complex integers, complex rationals, and complex floats is done automatically.

Also evalf now works over the complex floats. Consequently many functions

now work better, and faster, with complex numbers.

2: Simple partial derivatives use the diff notation not the D notation, i.e.

the output of diff(f(x,y),x,y,x); which used to return D[1,1,2](f)(x,y)

now returns diff(diff(diff(f(x,y),x),x),y) which is displayed as follows

-------- f(x, y)

dy dx

Release 2 still assumes that partial derivatives commute as did Maple V.

This is coded in the routine `diff/diff` and can be changed as necessary.

Note, use convert to convert between the two representations, i.e.

convert(...,D); and convert(...,diff);

3: The function combinat[combine] has been renamed combinat[choose] to not

collide with the combine function and the trace function has been renamed

debug to not collide with linalg[trace], i.e. the trace of a matrix.

4: The meaning of x in plot(..., x=a..b); and plot(..., a..b) has changed to

be consistent with the 3-dimensional case. If you give a range as the 2nd

argument, i.e. plot(f,0..1); and plot3d(g,0..1,0..1); then Maple

understands this to mean: treat f and g as functions or mappings, e.g.

plot( sin, -3..3 ); and plot( binomial, -3..3, -3..3 );

On the other hand, if you give an equation/range as the second argument, i.e.

plot(f(x),x=0..1); or plot3d(g(x,y),x=0..1,y=0..1); then this is

understood to mean plot the expression or formula f(x) and g(x,y), e.g.

plot( sin(x), x=-3..3 ); and plot( binomial(n,k), n=-3..3, k=-3..3 );

Hence you can plot Maple procedures and operator expressions e.g.

plot( x -> if x < 0 then 1 else cos(x) fi, -2..2 );

plot( sin^2+cos@log, 0..3 );

5: The output from dsolve/numeric has changed to include the variables, e.g.

> sys := diff(y(x),x)=z(x),diff(z(x),x)=-y(x): fcns := {y(x), z(x)}:

> p:= dsolve({sys,y(0)=0,z(0)=1},fcns,numeric):

> p(0.5);

{x = .5000000000, y(x) = .4794255401, z(x) = .8775825652}

This can be plotted by by then doing

> plots[ODEplot](p,[[x,y(x)],[x,z(x)]],0..10);

6: Output from lprint and save of powers now uses ^ instead of ** and of

floating point numbers now uses scientific e/E notation instead of Float(m,e)

7: Some automatic simplifications have been turned off e.g. ln(1/x) ==> -ln(x)

A complete list is given in the section Changes to automatic simplifications

below under Miscellaneous. The ones most likely to affect users are those

simplifications which have been turned off -- because they are not correct

for all complex values of parameters.

8: Hitting the interrupt key twice will no longer kill a Maple session, it

will only interrupt the current calculation. The quit key can be used to

kill Maple. On Unix this usually is ^by default.

9: Indefinite integrals which require the exponential integral now return

in terms of Ei(1,x) not Ei(x) which is the Cauchy principle value. E.g.

> int( exp(-x)/x^3, x );

exp(- x) exp(- x)

- 1/2 -------- + 1/2 -------- - 1/2 Ei(1, x)

2 x

10: New semantics of evalf(f,n). The old semantics was first evaluate f,

then simplify the result (at Digits precision) yielding f', then evaluate and

simplify n yielding n' and assign Digits to n'. Now evalf f' at n' digits

precision, simplify the result and return. The new semantics is, first

evaluate n to n' and assign Digits to n', then evaluate f and simplify the

result (at precision Digits = n') yielding f', then evalf f' at

Digits = n' precision. This example illustrates what is going on here

x := 7.0;

evalf( 2/x*Pi, 20 );

In Maple V, 2/x*Pi evaluates to 2/7.0*Pi, and simplifying 2/7.0 is done

at Digits = 10 (the default) precision yielding .2857142857*Pi. Then

evalfing this at 20 digits we get .89759790129493458131 . In Maple V

release 2 all the arithmetic is being done at 20 digits precision

and we get .89759790102565521098 .

11: The fsolve command, when given a range, now solves on the closed interval

not the open interval. E.g. for the polynomial 1-x^2 with roots at +- 1,

fsolve(1-x^2,x,-1..1) now returns -1 and +1.

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