Many improvements have been made to the CodeGeneration package.  New exports allow translation of Maple code to the Visual Basic® and MATLAB® languages and provide users the ability to extend the current translation facilities.  The translators accept modules as input, in addition to expressions, computation sequences, and procedures. These new features, along with other improvements in efficiency, coverage of the Maple language, and quality of the output, are described as follows.

with(CodeGeneration);

$\left[C\,\mathrm{CSharp}\,\mathrm{Fortran}\,\mathrm{IntermediateCode}\,\mathrm{Java}\,\mathrm{JavaScript}\,\mathrm{Julia}\,\mathrm{LanguageDefinition}\,\mathrm{Matlab}\,\mathrm{Names}\,\mathrm{Perl}\,\mathrm{Python}\,R\,\mathrm{Save}\,\mathrm{Swift}\,\mathrm{Translate}\,\mathrm{VisualBasic}\right]$

The DEtools package contains a new function DEtools[dperiodic_sols], which finds solutions of linear ODEs that have doubly-periodic coefficients. Doubly-periodic functions are typically represented in Weierstrass form (rational expressions in $P$ and $P\'$ ) or Jacobi form (rational expressions of the Jacobi functions, for example, $\mathrm{sn}$, $\mathrm{cn}$, and $\mathrm{dn}$).

alias( P = WeierstrassP(x,g2,g3), Pp = WeierstrassPPrime(x,g2,g3),
       sn = JacobiSN(x,k), cn = JacobiCN(x,k), dn = JacobiDN(x,k) ):

ode := diff(y(x),x$2) - (2*P + B)*y(x):

DEtools[dperiodic_sols](ode, y(x) );

$\left[{\ⅇ}^{-\frac{\sqrt{4B^3-B\mathrm{g2}-\mathrm{g3}}\left(\int \frac{1}{B-P}\ⅆx\right)}{2}}\sqrt{B-P}\,{\ⅇ}^{\frac{\sqrt{4B^3-B\mathrm{g2}-\mathrm{g3}}\left(\int \frac{1}{B-P}\ⅆx\right)}{2}}\sqrt{B-P}\right]$

k := 3:

ode := diff(y(x),x$2) + (k^2*sn*cn/dn)*diff(y(x),x) + 9*dn^2*y(x):

DEtools[dperiodic_sols](ode, y(x) );

$\left[-\frac{4{\mathrm{JacobiSN}\left(x\,3\right)}^3}{3}+\mathrm{JacobiSN}\left(x\,3\right)\,\left(-\frac{4{\mathrm{JacobiSN}\left(x\,3\right)}^3}{3}+\mathrm{JacobiSN}\left(x\,3\right)\right)\left(\frac{18\mathrm{JacobiSN}\left(x\,3\right)\mathrm{JacobiCN}\left(x\,3\right)}{{\mathrm{JacobiDN}\left(x\,3\right)}^2+\frac{23}{4}}+\frac{9\mathrm{JacobiSN}\left(x\,3\right)\mathrm{JacobiCN}\left(x\,3\right)}{{\mathrm{JacobiDN}\left(x\,3\right)}^2-1}\right)\right]$

ode := (Pp + (P^2))*diff(y(x),x$2) + ((P)^3-P*Pp-diff(P,x$2))*diff(y(x),x)
       + ((Pp)^2-(P)^2*Pp-P*diff(P,x$2))*y(x):

DEtools[dperiodic_sols](ode, y(x));

$\left[P\,-{\ⅇ}^{\mathrm{WeierstrassZeta}\left(x\,\mathrm{g2}\,\mathrm{g3}\right)}\right]$

k := 'k':

The DEtools package is extended by means of LinearFunctionalSystems package in searching for polynomial and rational solutions of systems of differential equations.

DEtools[matrixDE] accepts an option of the form solution = polynomial or solution = rational, which specifies to search for polynomial or rational solution. In this case, the function LinearFunctionalSystems[PolynomialSolution] or LinearFunctionalSystems[RationalSolution] is used to find solutions.

M := matrix(6,6,[0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,2,2*x,0,0,1,0,x^2,0,2*x,0,0,1,0,x^2,-2,0,0,0]);

$M≔\left[\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ 0& 0& 1& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 2& 2x& 0& 0& 1& 0\\ x^2& 0& 2x& 0& 0& 1\\ 0& x^2& \mathrm{-2}& 0& 0& 0\end{array}\right]$

DEtools['matrixDE'](M,x,solution=polynomial);

$\left[\left[\begin{array}{cccccc}0& 1& x& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 1& -x& -x^2& 0& 0& 0\\ \mathrm{-1}& x& x^2& 0& 0& 0\\ 0& \mathrm{-1}& -2x& 0& 0& 0\\ -2x& x^2& x^3-2& 0& 0& 0\end{array}\right]\,\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\end{array}\right]\right]$

For DEtools[polysols] and DEtools[ratsols], a system of differential equations can be given either as a list of the equations in the first argument and a list of the variables in the second argument or as a list of lists of lists of coefficients of linear ODEs in the first argument, a list of the right-hand sides of such equations in the second argument, and the independent variable in the third argument. Each equation is represented as the sum of LODEs in one variable, each of them is given as a list of coefficients, so one equation is given as a list of lists of coefficients and a system as a list of lists of lists. The latter sequence is convenient for programming with the routines.

sys := [diff(y1(x), x) - y2(x), diff(y2(x), x) - y3(x) - y4(x), diff(y3(x), x) - y5(x),
        diff(y4(x), x) - 2*y1(x) - 2*x*y2(x) - y5(x), diff(y5(x), x) - x^2*y1(x) - 2*x*y3(x) - y6(x),
        diff(y6(x), x) - x^2*y2(x) + 2*y3(x)];

$\mathrm{sys}≔\left[\frac{\ⅆ}{\ⅆx}\mathrm{y1}\left(x\right)-\mathrm{y2}\left(x\right)\,\frac{\ⅆ}{\ⅆx}\mathrm{y2}\left(x\right)-\mathrm{y3}\left(x\right)-\mathrm{y4}\left(x\right)\,\frac{\ⅆ}{\ⅆx}\mathrm{y3}\left(x\right)-\mathrm{y5}\left(x\right)\,\frac{\ⅆ}{\ⅆx}\mathrm{y4}\left(x\right)-2\mathrm{y1}\left(x\right)-2x\mathrm{y2}\left(x\right)-\mathrm{y5}\left(x\right)\,\frac{\ⅆ}{\ⅆx}\mathrm{y5}\left(x\right)-x^2\mathrm{y1}\left(x\right)-2x\mathrm{y3}\left(x\right)-\mathrm{y6}\left(x\right)\,\frac{\ⅆ}{\ⅆx}\mathrm{y6}\left(x\right)-x^2\mathrm{y2}\left(x\right)+2\mathrm{y3}\left(x\right)\right]$

vars := [y1(x), y2(x), y3(x), y4(x), y5(x), y6(x)];

$\mathrm{vars}≔\left[\mathrm{y1}\left(x\right)\,\mathrm{y2}\left(x\right)\,\mathrm{y3}\left(x\right)\,\mathrm{y4}\left(x\right)\,\mathrm{y5}\left(x\right)\,\mathrm{y6}\left(x\right)\right]$

DEtools['polysols'](sys, vars);

$\left[\left[0\,1\,x\right]\,\left[0\,0\,1\right]\,\left[1\,-x\,-x^2\right]\,\left[\mathrm{-1}\,x\,x^2\right]\,\left[0\,\mathrm{-1}\,-2x\right]\,\left[-2x\,x^2\,x^3-2\right]\right]$

sysp := [[[0,1],[-1]],[[],[0,1],[-1],[-1]],[[],[],[0,1],[],[-1]],
        [[-2],[-2*x],[],[0,1],[-1]],[[-x^2],[],[-2*x],[],[0,1],[-1]],
        [[],[-x^2],[2],[],[],[0,1]]];

$\mathrm{sysp}≔\left[\left[\left[0\,1\right]\,\left[\mathrm{-1}\right]\right]\,\left[\left[\right]\,\left[0\,1\right]\,\left[\mathrm{-1}\right]\,\left[\mathrm{-1}\right]\right]\,\left[\left[\right]\,\left[\right]\,\left[0\,1\right]\,\left[\right]\,\left[\mathrm{-1}\right]\right]\,\left[\left[\mathrm{-2}\right]\,\left[-2x\right]\,\left[\right]\,\left[0\,1\right]\,\left[\mathrm{-1}\right]\right]\,\left[\left[-x^2\right]\,\left[\right]\,\left[-2x\right]\,\left[\right]\,\left[0\,1\right]\,\left[\mathrm{-1}\right]\right]\,\left[\left[\right]\,\left[-x^2\right]\,\left[2\right]\,\left[\right]\,\left[\right]\,\left[0\,1\right]\right]\right]$

DEtools['polysols'](sysp, [0,0,0,0,0,0], x);

$\left[\left[0\,1\,x\right]\,\left[0\,0\,1\right]\,\left[1\,-x\,-x^2\right]\,\left[\mathrm{-1}\,x\,x^2\right]\,\left[0\,\mathrm{-1}\,-2x\right]\,\left[-2x\,x^2\,x^3-2\right]\right]$

The LibraryTools package contains a new routine LibraryTools[Browse]. The LibraryTools[Browse] routine displays a Maplet application interface that allows you to easily manipulate multiple libraries. By clicking buttons, you can create new libraries, navigate, and edit the contents of existing libraries.

LibraryTools[Browse]();

The LinearAlgebra package contains three new related routines: MatrixExponential, MatrixPower, and MatrixFunction. The more general MatrixFunction routine computes F(A) for an analytic function F and a Matrix A.  The two routines MatrixExponential and MatrixPower are provided as convenient means to use this functionality for two common functions. More detail is available on the MatrixFunction help page.

The ListTools package contains a new function ListTools[Occurrences], which counts the number of occurrences of an item in a list. In addition to literal equality, other kinds of equivalences can be specified.

The LREtools[hypergeomsols] function uses an implementation based on the paper: M. van Hoeij, ``Finite Singularities and Hypergeometric Solutions of Linear Recurrence Equations.'' J. Pure Appl. Algebra, 139, p. 109-131 (1999). This algorithm can be substantially faster when the leading or trailing coefficient of the recurrence relation is a polynomial with complicated roots. As an example,

LREtools[hypergeomsols](u(n+2) = 2*(n^4+4*n^3+6*n^2+5*n+3)*(n^4+n+1)*u(n),
u(n), {}, 'output'='basis');

$\left[{\left(-\sqrt{2}\right)}^n\left(\prod _{k=0}^{n-1}\left(k^4+k+1\right)\right)\,{\left(\sqrt{2}\right)}^n\left(\prod _{k=0}^{n-1}\left(k^4+k+1\right)\right)\right]$

The LREtools package is extended by means of LinearFunctionalSystems package in searching for polynomial and rational solutions of systems of recurrence equations.

x := 'x':

sys := [y2(x)*x^2+3*y2(x)*x+2*y2(x)-2*y1(x)*x^2-4*y1(x)*x+y1(x+1)*x^2+y1(x+1)*x,y2(x+1)-y1(x)];

$\mathrm{sys}≔\left[x^2\mathrm{y2}\left(x\right)+3x\mathrm{y2}\left(x\right)+2\mathrm{y2}\left(x\right)-2x^2\mathrm{y1}\left(x\right)-4\mathrm{y1}\left(x\right)x+\mathrm{y1}\left(x+1\right)x^2+\mathrm{y1}\left(x+1\right)x\,\mathrm{y2}\left(x+1\right)-\mathrm{y1}\left(x\right)\right]$

vars := [y1(x), y2(x)];

$\mathrm{vars}≔\left[\mathrm{y1}\left(x\right)\,\mathrm{y2}\left(x\right)\right]$

LREtools['polysols'](sys, vars,{}, 'output'='basis');

$\left[{\mathrm{\_C}}_1+x\left({\mathrm{\_C}}_1+2{\mathrm{\_C}}_2\right)+x\left(x-1\right){\mathrm{\_C}}_2\,x{\mathrm{\_C}}_1+x\left(x-1\right){\mathrm{\_C}}_2\right]$

The LREtools package contains a new function LREtools[IsDesingularizable], which desingularizes linear difference operators with polynomial coefficients.

LREtools[IsDesingularizable]((n-2)*(n^3+2)*E+n,E,n,'leading','output'='operator');

$\mathrm{true},1+\left(6n^3+36n^2+72n+60\right)E^3+\left(n^3-2n^2+6\right)E+\left(4n^3+8n^2+6\right)E^2$

For information on the new LREtools[HypergeometricTerm] subpackage, see New Packages in Maple 9.

The Slode package contains a new function Slode[dAlembertian_formal_sol], which finds formal solutions with d'Alembertian series to a homogeneous linear differential equation with polynomial coefficients.

ode:=(-4-x^2+2*x)*y(x)+(2*x-3*x^3-x^2)*diff(y(x),x)+(x^3-x^4)*diff(y(x),`$`(x,2));

$\mathrm{ode}≔\left(-x^2+2x-4\right)y\left(x\right)+\left(-3x^3-x^2+2x\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\left(-x^4+x^3\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)$

Slode[dAlembertian_formal_sol](ode,y(x));

$x^2\left(-\frac{\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}{x}^{\mathrm{\_n}}\right)}{2}+\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\left(\sum _{\mathrm{\_n1}=0}^{\mathrm{\_n}-1}\frac{{\left(-\frac{1}{2}\right)}^{\mathrm{\_n1}}}{\prod _{\mathrm{\_k}=0}^{\mathrm{\_n1}-1}\frac{\mathrm{\_k}+2}{{\left(\mathrm{\_k}+3\right)}^2}}\right)x^{\mathrm{\_n}}\right)\right){\mathrm{\_C}}_0+\frac{{\ⅇ}^{\frac{2}{x}}\left(\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}{x}^{\mathrm{\_n}}\right)-\frac{1}{3}\right){\mathrm{\_C}}_1}{x}$

ode:=(x-1)^2*diff(y(x),`$`(x,3))-(x-1)*(x-7)*diff(y(x),`$`(x,2))-2*(2*x-5)*diff(y(x),x)-2*y(x);

$\mathrm{ode}≔{\left(x-1\right)}^2\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)-\left(x-1\right)\left(x-7\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-2\left(2x-5\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-2y\left(x\right)$

Slode[dAlembertian_formal_sol](ode,y(x));

${\mathrm{\_C}}_0\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}{x}^{\mathrm{\_n}}\right)+{\mathrm{\_C}}_1\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\mathrm{\_n}{x}^{\mathrm{\_n}}\right)+{\mathrm{\_C}}_2\left(\sum _{\mathrm{\_n}=0}^{\mathrm{\infty }}\left(\sum _{\mathrm{\_n1}=0}^{\mathrm{\_n}-1}\sum _{\mathrm{\_n2}=0}^{\mathrm{\_n1}-1}\prod _{\mathrm{\_k}=0}^{\mathrm{\_n2}-1}\frac{1}{\mathrm{\_k}+3}\right)x^{\mathrm{\_n}}\right)$

The SNAP package contains a new function QRGCD, which computes an approximate right GCD of two univariate polynomials. For more information, see SNAP[QRGCD].

The SolveTools package contains a new function PolynomialSystem, which solves general systems of polynomial equations. For more information, see SolveTools[PolynomialSystem].

SolveTools[PolynomialSystem]({x^2+y^2-3,x*y+5},{x,y});

$\left\{x=\frac{{\mathrm{RootOf}\left({\mathrm{\_Z}}^4-3{\mathrm{\_Z}}^2+25\right)}^3}{5}-\frac{3\mathrm{RootOf}\left({\mathrm{\_Z}}^4-3{\mathrm{\_Z}}^2+25\right)}{5}\,y=\mathrm{RootOf}\left({\mathrm{\_Z}}^4-3{\mathrm{\_Z}}^2+25\right)\right\}$

The SolveTools[CancelInverses] function has an option 'safe' that applies only those transformations that are valid everywhere. For more information, see SolveTools[CancelInverses].