The Linear Algebra package now has performance improvements for floating point kernel computations.

The following commands have been modified to work in any coordinate system in 2 dimensions - this only worked in 3 dimensions in Release 4.

diverge - Compute the divergence of a vector function

grad - Compute the vector gradient of an expression

laplacian - Compute the Laplacian of an expression

e := x^2 + y^3 - sin( z );

$e≔x^2+y^3-\sin \left(z\right)$

linalg[ 'laplacian' ]( e, [ x, y, z ], coords = spherical );

$\frac{6x^2\sin \left(y\right)+3\cos \left(y\right)y^2+6\sin \left(y\right)y+\frac{\sin \left(z\right)}{\sin \left(y\right)}}{x^2\sin \left(y\right)}$

The Combinatorial structures package has been expanded. New generating function capabilities have been added:

gfeqns - Find the system of generating function equations associated with a grammar

gfseries - Return the series of the generating functions associated with a grammar

gfsolve - Solve the system of generating functions associated with a grammar

See the combstruct example worksheet for more information on using this package.

The DEtools package has been significantly expanded and enhanced in this release. More functionality has been added:

abelsol - Find solutions of a first order Abel ODE

adjoint - Find the adjoint of a differential operator

bernoullisol - Find solutions of a Bernoulli first order ODE

buisym - Build the symmetry generator given a solution of an ODE

canoni - Look for a pair of canonical coordinates or for an integrating factor

clairautsol - Find solutions of a Clairaut first order ODE

constcoeffsols - Find solutions of a linear constant coefficient ODE

DFactor - Factor a linear differential operator

de2diffop - Convert a differential equation to a differential operator

diffop2de - Convert a differential operator to a differential equation

eigenring - Find the endomorphisms of the solution space

equinv - Look for the most general ODE invariant under a given symmetry

eta_k - Evaluates the k prolongation of the infinitesimals of a 1-parameter Lie group

eulersols - Find solutions of an Euler type of linear ODE

exactsol - Find exact solutions of a first order ODE

expsols - Find exponential solutions of a linear ODE

exterior_power - Find the exterior power of a differential operator

formal_sol - Generate formal solutions of a homogeneous linear differential equation

GCRD - Identify the Greatest Common Right Divisor of two differential operators

gen_exp - Find the generalized exponents of a linear homogeneous differential equation

genhomosol - Find solutions of a homogeneous first order ODE

infgen - Evaluates the k extension of the infinitesimal generator of a 1-parameter Lie group

integrate_sols - Integrate the solutions of a differential operator or equation

intfactor - Look for a pair of canonical coordinates or for an integrating factor

kovacicsols - Find solutions of a second order linear ODE

LCLM - Calculate Least Common Left Multiple of differential operators

leftdivision - Perform left-hand division on differential operators

linearsol - Find solutions of a first order linear ODE

line_int - Solve for a line integral, given its partial derivatives

matrixDE - Find solutions of a linear system of ODEs in matrix form

matrix_riccati - Solve a Matrix Riccati differential equation

moser_reduce - Generate the moser-irreducible form of a linear differential system

mult - Multiply differential operators

newton_polygon - Find the Newton polygon of a linear ODE

odeadvisor - Analyzer and solving advisor for first and second order ODEs

odepde - Returns the PDE for the coefficients of the symmetry generator

Poincare - Work with subpackage for visualization of Hamiltonians

hamilton_eqs - Generate Hamilton's equations, given a Hamiltonian

generate_ic - Generate a set of lists of initial conditions for plotting a 2PS/3PS

polysols - Find polynomial solutions of a linear ODE

ratsols - Find rational function solutions of a linear ODE

regular_parts - Find regular parts of a linear ODE

riccati_system - Solve a system of Riccati differential equations or convert it into a matrix form

riccatisol - Find solutions of a first order Riccati ODE

RiemannPsols - Find solutions of a second order linear ODE

rightdivision - Perform right-hand division of differential operators

separablesol - Find solutions of a separable first order ODE

super_reduce - Generate super-irreducible form of a linear differential system

symmetric_product - Compute the symmetric product of differential operators

symmetric_power - Calculate the symmetric power of a differential equation or operator

symgen - Look for a symmetry generator for a given first or second order ODE

symtest - Test a given symmetry

transinv - Look for the set of transformations of variables which leave the ODE invariant

New functions in this package:

autodispersion - Compute self-dispersion of a polynomial

firstlin - Solve first order linear recurrences

riccati - Solve recurrences of Riccati type

divconq - Solve "divide and conquer" recurrences

New functions in this package:

RegularPolygon - Define a regular polygon

RegularStarPolygon - Define a regular star polygon

form - Return a given geometric object if it is supported by the geometry package

form - View detailed information about a geometric object

apothem - Return the apothem of a regular polygon

ExteriorAngle - Return the exterior angle of a regular polygon

inradius - Return the radius of the in-circle of a regular polygon

InteriorAngle - Return the interior angle of a regular polygon

perimeter - Compute the perimeter of a regular polygon

restart;

with( geometry ):

RegularPolygon( P, 5, point( O, 1, 1), 2 );

Warning, a geometry object has been assigned to the protected name O.  Use of protected names for geometry objects is not recommended and may break Maple functionality.

$P$

detail( P );

$\begin{array}{ll}\text{name of the object}& P\\ \text{form of the object}& \mathrm{RegularPolygon2d}\\ \text{the side of the polygon}& 4\sin \left(\frac{\mathrm{\pi }}{5}\right)\\ \text{the center of the polygon}& \left[1\,1\right]\\ \text{the radius of the circum-circle}& 2\\ \text{the radius of the in-circle}& 2\cos \left(\frac{\mathrm{\pi }}{5}\right)\\ \text{the interior angle}& \frac{3\mathrm{\pi }}{5}\\ \text{the exterior angle}& \frac{2\mathrm{\pi }}{5}\\ \text{the perimeter}& 20\sin \left(\frac{\mathrm{\pi }}{5}\right)\\ \text{the area}& 20\sin \left(\frac{\mathrm{\pi }}{5}\right)\cos \left(\frac{\mathrm{\pi }}{5}\right)\\ \text{the vertices of the polygon}& \left[\left[3\,1\right]\,\left[1+2\cos \left(\frac{2\mathrm{\pi }}{5}\right)\,1+2\sin \left(\frac{2\mathrm{\pi }}{5}\right)\right]\,\left[1-2\cos \left(\frac{\mathrm{\pi }}{5}\right)\,1+2\sin \left(\frac{\mathrm{\pi }}{5}\right)\right]\,\left[1-2\cos \left(\frac{\mathrm{\pi }}{5}\right)\,1-2\sin \left(\frac{\mathrm{\pi }}{5}\right)\right]\,\left[1+2\cos \left(\frac{2\mathrm{\pi }}{5}\right)\,1-2\sin \left(\frac{2\mathrm{\pi }}{5}\right)\right]\right]\end{array}$

apothem( P );

$2\cos \left(\frac{\mathrm{\pi }}{5}\right)$

The integral transforms package inttrans has been substantially reworked internally.

The inverse Mellin transform, invmellin, is new in this release.

with( inttrans ):

assume( a, 'complex' );

assume( b, 'complex' );

assume( c > 0 );

U := mellin( u( x ), x, s );

$U≔\mathrm{`?`}$

inttrans[ 'invmellin' ]( U, s, x );

$u\left(x\right)$

The command savetable will save a table to a given file.

See the integral transform example worksheet for more information about this package.

To compute Hermite-Pade approximations, use the command hermite_pade.

numapprox[ 'hermite_pade' ]([sin(x),cos(x)],x=Pi,7);

$\left[-6{\left(x-\mathrm{\pi }\right)}^2+15\,{\left(x-\mathrm{\pi }\right)}^3-15x+15\mathrm{\pi }\right]$

You can compute an integral basis for an algebraic number field using the integral_basis command.

numtheory[ 'integral_basis' ]( {2^(1/3), 3^(1/3)} );

$\left[1\,2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\,2^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\,3^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\,2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{3}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\,-\frac{23^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3}-\frac{2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{3}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3}+\frac{2^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{3}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3}\,3^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\,-\frac{23^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3}+\frac{2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{3}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3}\,-\frac{23^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3}-\frac{2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{3}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3}+\frac{2^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{3}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3}\right]$

The new summand command allows you to locate the summand of an inert sum.

student[ 'summand' ](Sum(1/i!,i=1..n));

$\frac{1}{i!}$

The hfarrays are used internally to improve performance.

The hackware package now has greater robustness.