\documentclass{article} \def\code#1{{\tt #1}} \def\url#1{{\it #1}} \def\sage#1{{\bf #1}} \def\note#1{#1} \usepackage{axiom} \begin{document} \title{A Sage Interface to Axiom} \author{ AUTHORS OF THIS MODULE: - William Stein (2006-10): Initial version based on Maxima interface - Bill Page (2006-10): Axiom version } \abstract{ Interface to Axiom Axiom is a free GPL-compatible (modified BSD license) general purpose computer algebra system whose development started in 1973 at IBM. It contains symbolic manipulation algorithms, as well as implementations of special functions, including elliptic functions and generalized hypergeometric functions. Moreover, Axiom has implementations of many functions relating to the invariant theory of the symmetric group $S_n$. For many links to Axiom documentation see \url{http://wiki.axiom-developer.org}. EXAMPLES: We evaluate a very simple expression in axiom. \begin{verbatim} sage: axiom('3 * 5') 15 Type: PositiveInteger \end{verbatim} We factor $x^5 - y^5$ in Axiom in several different ways. The first way yields a Axiom object. \begin{verbatim} sage: F = axiom.factor('x^5 - y^5') sage: type(F) \end{verbatim} Note that Axiom objects are normally displayed using ASCII art''; to see a normal linear representation of any Axiom object x, use \code{str(x)}. no emph \begin{verbatim} sage: F 4 3 2 2 3 4 - (y - x) (y + x y + x y + x y + x ) \end{verbatim} You can always use \code{x.str()} to obtain the linear representation of an object, even without changing the display2d flag. This can be useful for moving axiom data to other systems. \begin{verbatim} sage: F.str() '-(y - x)*(y^4 + x*y^3 + x^2*y^2 + x^3*y + x^4)' sage: axiom.display2d(False) sage: F -(y - x)*(y^4 + x*y^3 + x^2*y^2 + x^3*y + x^4) \end{verbatim} The \code{axiom.eval} command evaluates an expression in axiom and returns the result as a string. \begin{verbatim} sage: print axiom.eval('factor(x^5 - y^5)') -(y - x)*(y^4 + x*y^3 + x^2*y^2 + x^3*y + x^4) \end{varbatim} We can create the polynomial $f$ as a Axiom polynomial, then call the factor method on it. Notice that the notation \code{f.factor()} is consistent with how the rest of \sage works. \begin{verbatim} sage: f = axiom('x^5 - y^5') sage: f^2 (x^5 - y^5)^2 sage: f.factor() -(y - x)*(y^4 + x*y^3 + x^2*y^2 + x^3*y + x^4) \end{verbatim} Control-C interruption works well with the axiom interface, because of the excellent implementation of axiom. For example, try the following sum but with a much bigger range, and hit control-C. \begin{verbatim} sage: axiom('sum(1/x^2, x, 1, 10)') 1968329/1270080 \end{verbatim} \subsection{Tutorial} We follow the tutorial at \url{http://wiki.axiom-developer.org/AxiomTutorial}. \begin{verbatim} sage: axiom('1/100 + 1/101') 201/10100 sage: a = axiom('(1 + sqrt(2))^5'); a (sqrt(2) + 1)^5 sage: a.expand() 29*sqrt(2) + 41 sage: a = axiom('(1 + sqrt(2))^5') sage: float(a) 82.012193308819747 sage: a.numer() 82.01219330881975 sage: axiom.eval('fpprec : 100') '100' sage: a.bfloat() 8.20121933088197564152489730020812442785204843859314941221237124017312418754011041266612384955016056b1 sage: axiom('100!') 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000 sage: f = axiom('(x + 3*y + x^2*y)^3') sage: f.expand() x^6*y^3 + 9*x^4*y^3 + 27*x^2*y^3 + 27*y^3 + 3*x^5*y^2 + 18*x^3*y^2 + 27*x*y^2 + 3*x^4*y + 9*x^2*y + x^3 sage: f.subst('x=5/z') (5/z + 25*y/z^2 + 3*y)^3 sage: g = f.subst('x=5/z') sage: h = g.ratsimp(); h (27*y^3*z^6 + 135*y^2*z^5 + (675*y^3 + 225*y)*z^4 + (2250*y^2 + 125)*z^3 + (5625*y^3 + 1875*y)*z^2 + 9375*y^2*z + 15625*y^3)/z^6 sage: h.factor() (3*y*z^2 + 5*z + 25*y)^3/z^6 sage: eqn = axiom(['a+b*c=1', 'b-a*c=0', 'a+b=5']) sage: s = eqn.solve('[a,b,c]'); s [[a = (25*sqrt(79)*%i + 25)/(6*sqrt(79)*%i - 34),b = (5*sqrt(79)*%i + 5)/(sqrt(79)*%i + 11), c = (sqrt(79)*%i + 1)/10], [a = (25*sqrt(79)*%i - 25)/(6*sqrt(79)*%i + 34),b = (5*sqrt(79)*%i - 5)/(sqrt(79)*%i - 11), c = - (sqrt(79)*%i - 1)/10]] \end{varbatim} Here is an example of solving an algebraic equation: \begin{verbatim} sage: axiom('x^2+y^2=1').solve('y') [y = - sqrt(1 - x^2),y = sqrt(1 - x^2)] sage: axiom('x^2 + y^2 = (x^2 - y^2)/sqrt(x^2 + y^2)').solve('y') [y = - sqrt(( - y^2 - x^2)*sqrt(y^2 + x^2) + x^2),y = sqrt(( - y^2 - x^2)*sqrt(y^2 + x^2) + x^2)] \end{verbatim} You can even nicely typeset the solution in latex: \begin{verbatim} sage: latex(s) \left[ \left[ a=\frac{25 \sqrt{79} i+25}{6 \sqrt{79} i-34} , b= \frac{5 \sqrt{79} i+5}{\sqrt{79} i+11} , c=\frac{\sqrt{79} i+1}{10} \right] , \left[ a=\frac{25 \sqrt{79} i-25}{6 \sqrt{79} i+34} , b= \frac{5 \sqrt{79} i-5}{\sqrt{79} i-11} , c=-\frac{\sqrt{79} i-1}{10} \right] \right] \end{verbatim} To have the above appear onscreen via \code{xdvi}, type \code{view(s)}. (TODO: For OS X should create pdf output and use preview instead?) \begin{verbatim} sage: e = axiom('sin(u + v) * cos(u)^3'); e cos(u)^3*sin(v + u) sage: f = e.trigexpand(); f cos(u)^3*(cos(u)*sin(v) + sin(u)*cos(v)) sage: f.trigreduce() (sin(v + 4*u) + sin(v - 2*u))/8 + (3*sin(v + 2*u) + 3*sin(v))/8 sage: w = axiom('3 + k*%i') sage: f = w^2 + axiom('%e')^w sage: f.realpart() %e^3*cos(k) - k^2 + 9 sage: f = axiom('x^3 * %e^(k*x) * sin(w*x)'); f x^3*%e^(k*x)*sin(w*x) sage: f.diff('x') k*x^3*%e^(k*x)*sin(w*x) + 3*x^2*%e^(k*x)*sin(w*x) + w*x^3*%e^(k*x)*cos(w*x) sage: f.integrate('x') (((k*w^6 + 3*k^3*w^4 + 3*k^5*w^2 + k^7)*x^3 + (3*w^6 + 3*k^2*w^4 - 3*k^4*w^2 - 3*k^6)*x^2 + ( - 18*k*w^4 - 12*k^3*w^2 + 6*k^5)*x - 6*w^4 + 36*k^2*w^2 - 6*k^4)*%e^(k*x)*sin(w*x) + (( - w^7 - 3*k^2*w^5 - 3*k^4*w^3 - k^6*w)*x^3 + (6*k*w^5 + 12*k^3*w^3 + 6*k^5*w)*x^2 + (6*w^5 - 12*k^2*w^3 - 18*k^4*w)*x - 24*k*w^3 + 24*k^3*w)*%e^(k*x)*cos(w*x))/(w^8 + 4*k^2*w^6 + 6*k^4*w^4 + 4*k^6*w^2 + k^8) sage: f = axiom('1/x^2') sage: f.integrate('x', 1, 'inf') 1 sage: g = axiom('f/sinh(k*x)^4') sage: g.taylor('x', 0, 3) f/(k^4*x^4) - 2*f/(3*k^2*x^2) + 11*f/45 - 62*k^2*f*x^2/945 sage: axiom.taylor('asin(x)','x',0, 10) x + x^3/6 + 3*x^5/40 + 5*x^7/112 + 35*x^9/1152 \end{verbatim} \subsection{Examples involving matrices} We illustrate computing with the matrix whose $i,j$ entry is $i/j$, for $i,j=1,\ldots,4$. \begin{verbatim} sage: f = axiom.eval('f[i,j] := i/j') sage: A = axiom('genmatrix(f,4,4)'); A matrix([1,1/2,1/3,1/4],[2,1,2/3,1/2],[3,3/2,1,3/4],[4,2,4/3,1]) sage: A.determinant() 0 sage: A.echelon() matrix([1,1/2,1/3,1/4],[0,0,0,0],[0,0,0,0],[0,0,0,0]) sage: A.eigenvalues() [[0,4],[3,1]] sage: A.eigenvectors() [[[0,4],[3,1]],[1,0,0, - 4],[0,1,0, - 2],[0,0,1, - 4/3],[1,2,3,4]] \end{verbatim} We can also compute the echelon form in \sage: \begin{verbatim} sage: B = matrix(QQ, A) sage: B.echelon_form() [ 1 1/2 1/3 1/4] [ 0 0 0 0] [ 0 0 0 0] [ 0 0 0 0] sage: B.charpoly().factor() (x - 4) * x^3 \end{verbatim} \subsection{Laplace Transforms} We illustrate Laplace transforms: \begin{verbatim} sage: _ = axiom.eval("f(t) := t*sin(t)") sage: axiom("laplace(f(t),t,s)") 2*s/(s^2 + 1)^2 sage: axiom("laplace(delta(t-3),t,s)") #Dirac delta function %e^-(3*s) sage: _ = axiom.eval("f(t) == exp(t)*sin(t)") sage: axiom("laplace(f(t),t,s)") 1/(s^2 - 2*s + 2) sage: _ = axiom.eval("f(t) := t^5*exp(t)*sin(t)") sage: axiom("laplace(f(t),t,s)") 360*(2*s - 2)/(s^2 - 2*s + 2)^4 - 480*(2*s - 2)^3/(s^2 - 2*s + 2)^5 + 120*(2*s - 2)^5/(s^2 - 2*s + 2)^6 sage: axiom("laplace(f(t),t,s)") 3 5 360 (2 s - 2) 480 (2 s - 2) 120 (2 s - 2) --------------- - --------------- + --------------- 2 4 2 5 2 6 (s - 2 s + 2) (s - 2 s + 2) (s - 2 s + 2) sage: axiom("laplace(diff(x(t),t),t,s)") s*laplace(x(t),t,s) - x(0) sage: axiom("laplace(diff(x(t),t,2),t,s)") -at('diff(x(t),t,1),t = 0) + s^2*laplace(x(t),t,s) - x(0)*s sage.: axiom("laplace(diff(x(t),t,2),t,s)") ! d ! 2 - -- (x(t))! + s laplace(x(t), t, s) - x(0) s dt ! !t = 0 \end{verbatim} Even better, use \code{view(axiom("laplace(diff(x(t),t,2),t,s)"))} to see a typeset version. \subsection{Continued Fractions} A continued fraction $a + 1/(b + 1/(c + \cdots))$ is represented in axiom by the list $[a, b, c, \ldots]$. \begin{verbatim} sage: axiom("cf((1 + sqrt(5))/2)") [1,1,1,1,2] sage: axiom("cf ((1 + sqrt(341))/2)") [9,1,2,1,2,1,17,1,2,1,2,1,17,1,2,1,2,1,17,2] \end{verbatim} \subsection{Special examples} In this section we illustrate calculations that would be awkward to do (as far as I know) in non-symbolic computer algebra systems like MAGMA or GAP. We compute the gcd of $2x^{n+4} - x^{n+2}$ and $4x^{n+1} + 3x^n$ for arbitrary $n$. \begin{verbatim} sage: f = axiom('2*x^(n+4) - x^(n+2)') sage: g = axiom('4*x^(n+1) + 3*x^n') sage: f.gcd(g) x^n \end{verbatim} You can plot 3d graphs (via gnuplot): \begin{verbatim} sage.: axiom('plot3d(x^2-y^2, [x,-2,2], [y,-2,2], [grid,12,12])') [displays a 3 dimensional graph] \end{verbatim} You can formally evaluate sums (note the \code{nusum} command): \begin{verbatim} sage: S = axiom('nusum(exp(1+2*i/n),i,1,n)') sage.: S 2/n + 3 2/n + 1 %e %e ----------------------- - ----------------------- 1/n 1/n 1/n 1/n (%e - 1) (%e + 1) (%e - 1) (%e + 1) \end{verbatim} We formally compute the limit as $n\to\infty$ of $2S/n$ as follows: \begin{verbatim} sage: T = S*axiom('2/n') sage: T.tlimit('n','inf') %e^3 - %e \end{verbatim} \subsection{Miscellaneous} Obtaining digits of $\pi$: \begin{verbatim} sage: axiom.eval('fpprec : 100') '100' sage: axiom(pi).bfloat() 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068b0 \end{verbatim} Defining functions in axiom: \begin{verbatim} sage: axiom.eval('fun(a) == a^2') 'fun[a] := a^2' sage: axiom('fun(10)') 100 \end{verbatim} \subsection{Interactivity} Axiom has a non-interactive mode that is initiated via the command ")read file.input". \subsection{Latex Output} The latex output of Axiom is not perfect. E.g., \begin{verbatim} sage: axiom.eval('tex(sin(u) + sinh(v^2))') '$$\\sinhv^2 + \\sinu$$false' \end{verbatim} Notice the lack of space after the sin macro, which is a latex syntax error. In \sage this is automatically fixed via a substition for trig functions, which may have potentially bad side effects: \begin{verbatim} sage: latex(axiom('sin(u) + sinh(v^2)')) \sinh v^2+\sin u \end{verbatim} It would be nice if somebody would fix this problem. One way would be to improve Axiom by making the fix to Axiom and giving this back to the Axiom people. Here's another example: \begin{verbatim} sage: g = axiom('exp(3*%i*x)/(6*%i) + exp(%i*x)/(2*%i) + c') sage: latex(g) -\frac{i e^{3 i x}}{6}-\frac{i e^{i x}}{2}+c \end{verbatim} \subsection{Long Input} The Axiom interface reads in even very long input (using files) in a robust manner, as long as you are creating a new object. \note{Using \code{axiom.eval} for long input is much less robust, and is not recommended.} \begin{verbatim} sage: t = '"%s"'%10^10000 # ten thousand character string. sage: a = axiom(t) \end{verbatim} } \maketitle <>= r""" Interface to Axiom Axiom is a free GPL-compatible (modified BSD license) general purpose computer algebra system whose development started in 1973 at IBM. It contains symbolic manipulation algorithms, as well as implementations of special functions, including elliptic functions and generalized hypergeometric functions. Moreover, Axiom has implementations of many functions relating to the invariant theory of the symmetric group $S_n$. For many links to Axiom documentation see \url{http://wiki.axiom-developer.org}. AUTHORS OF THIS MODULE: - William Stein (2006-10): Initial version based on Maxima interface - Bill Page (2006-10): Axiom version If the string "error" (case insensitive) occurs in the output of anything from axiom, a RuntimeError exception is raised. EXAMPLES: We evaluate a very simple expression in axiom. sage: axiom('3 * 5') 15 Type: PositiveInteger We factor $x^5 - y^5$ in Axiom in several different ways. The first way yields a Axiom object. sage: F = axiom.factor('x^5 - y^5') sage: type(F) Note that Axiom objects are normally displayed using ASCII art''; to see a normal linear representation of any Axiom object x, use \code{str(x)}. sage: F 4 3 2 2 3 4 - (y - x) (y + x y + x y + x y + x ) You can always use \code{x.str()} to obtain the linear representation of an object, even without changing the display2d flag. This can be useful for moving axiom data to other systems. sage: F.str() '-(y - x)*(y^4 + x*y^3 + x^2*y^2 + x^3*y + x^4)' sage: axiom.display2d(False) sage: F -(y - x)*(y^4 + x*y^3 + x^2*y^2 + x^3*y + x^4) The \code{axiom.eval} command evaluates an expression in axiom and returns the result as a string. sage: print axiom.eval('factor(x^5 - y^5)') -(y - x)*(y^4 + x*y^3 + x^2*y^2 + x^3*y + x^4) We can create the polynomial $f$ as a Axiom polynomial, then call the factor method on it. Notice that the notation \code{f.factor()} is consistent with how the rest of \sage works. sage: f = axiom('x^5 - y^5') sage: f^2 (x^5 - y^5)^2 sage: f.factor() -(y - x)*(y^4 + x*y^3 + x^2*y^2 + x^3*y + x^4) Control-C interruption works well with the axiom interface, because of the excellent implementation of axiom. For example, try the following sum but with a much bigger range, and hit control-C. sage: axiom('sum(1/x^2, x, 1, 10)') 1968329/1270080 \subsection{Tutorial} We follow the tutorial at \url{http://wiki.axiom-developer.org/AxiomTutorial}. sage: axiom('1/100 + 1/101') 201/10100 sage: a = axiom('(1 + sqrt(2))^5'); a (sqrt(2) + 1)^5 sage: a.expand() 29*sqrt(2) + 41 sage: a = axiom('(1 + sqrt(2))^5') sage: float(a) 82.012193308819747 sage: a.numer() 82.01219330881975 sage: axiom.eval('fpprec : 100') '100' sage: a.bfloat() 8.20121933088197564152489730020812442785204843859314941221237124017312418754011041266612384955016056b1 sage: axiom('100!') 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000 sage: f = axiom('(x + 3*y + x^2*y)^3') sage: f.expand() x^6*y^3 + 9*x^4*y^3 + 27*x^2*y^3 + 27*y^3 + 3*x^5*y^2 + 18*x^3*y^2 + 27*x*y^2 + 3*x^4*y + 9*x^2*y + x^3 sage: f.subst('x=5/z') (5/z + 25*y/z^2 + 3*y)^3 sage: g = f.subst('x=5/z') sage: h = g.ratsimp(); h (27*y^3*z^6 + 135*y^2*z^5 + (675*y^3 + 225*y)*z^4 + (2250*y^2 + 125)*z^3 + (5625*y^3 + 1875*y)*z^2 + 9375*y^2*z + 15625*y^3)/z^6 sage: h.factor() (3*y*z^2 + 5*z + 25*y)^3/z^6 sage: eqn = axiom(['a+b*c=1', 'b-a*c=0', 'a+b=5']) sage: s = eqn.solve('[a,b,c]'); s [[a = (25*sqrt(79)*%i + 25)/(6*sqrt(79)*%i - 34),b = (5*sqrt(79)*%i + 5)/(sqrt(79)*%i + 11),c = (sqrt(79)*%i + 1)/10],[a = (25*sqrt(79)*%i - 25)/(6*sqrt(79)*%i + 34),b = (5*sqrt(79)*%i - 5)/(sqrt(79)*%i - 11),c = - (sqrt(79)*%i - 1)/10]] Here is an example of solving an algebraic equation: sage: axiom('x^2+y^2=1').solve('y') [y = - sqrt(1 - x^2),y = sqrt(1 - x^2)] sage: axiom('x^2 + y^2 = (x^2 - y^2)/sqrt(x^2 + y^2)').solve('y') [y = - sqrt(( - y^2 - x^2)*sqrt(y^2 + x^2) + x^2),y = sqrt(( - y^2 - x^2)*sqrt(y^2 + x^2) + x^2)] You can even nicely typeset the solution in latex: sage: latex(s) \left[ \left[ a=\frac{25 \sqrt{79} i+25}{6 \sqrt{79} i-34} , b= \frac{5 \sqrt{79} i+5}{\sqrt{79} i+11} , c=\frac{\sqrt{79} i+1}{10} \right] , \left[ a=\frac{25 \sqrt{79} i-25}{6 \sqrt{79} i+34} , b= \frac{5 \sqrt{79} i-5}{\sqrt{79} i-11} , c=-\frac{\sqrt{79} i-1}{10} \right] \right] To have the above appear onscreen via \code{xdvi}, type \code{view(s)}. (TODO: For OS X should create pdf output and use preview instead?) sage: e = axiom('sin(u + v) * cos(u)^3'); e cos(u)^3*sin(v + u) sage: f = e.trigexpand(); f cos(u)^3*(cos(u)*sin(v) + sin(u)*cos(v)) sage: f.trigreduce() (sin(v + 4*u) + sin(v - 2*u))/8 + (3*sin(v + 2*u) + 3*sin(v))/8 sage: w = axiom('3 + k*%i') sage: f = w^2 + axiom('%e')^w sage: f.realpart() %e^3*cos(k) - k^2 + 9 sage: f = axiom('x^3 * %e^(k*x) * sin(w*x)'); f x^3*%e^(k*x)*sin(w*x) sage: f.diff('x') k*x^3*%e^(k*x)*sin(w*x) + 3*x^2*%e^(k*x)*sin(w*x) + w*x^3*%e^(k*x)*cos(w*x) sage: f.integrate('x') (((k*w^6 + 3*k^3*w^4 + 3*k^5*w^2 + k^7)*x^3 + (3*w^6 + 3*k^2*w^4 - 3*k^4*w^2 - 3*k^6)*x^2 + ( - 18*k*w^4 - 12*k^3*w^2 + 6*k^5)*x - 6*w^4 + 36*k^2*w^2 - 6*k^4)*%e^(k*x)*sin(w*x) + (( - w^7 - 3*k^2*w^5 - 3*k^4*w^3 - k^6*w)*x^3 + (6*k*w^5 + 12*k^3*w^3 + 6*k^5*w)*x^2 + (6*w^5 - 12*k^2*w^3 - 18*k^4*w)*x - 24*k*w^3 + 24*k^3*w)*%e^(k*x)*cos(w*x))/(w^8 + 4*k^2*w^6 + 6*k^4*w^4 + 4*k^6*w^2 + k^8) sage: f = axiom('1/x^2') sage: f.integrate('x', 1, 'inf') 1 sage: g = axiom('f/sinh(k*x)^4') sage: g.taylor('x', 0, 3) f/(k^4*x^4) - 2*f/(3*k^2*x^2) + 11*f/45 - 62*k^2*f*x^2/945 sage: axiom.taylor('asin(x)','x',0, 10) x + x^3/6 + 3*x^5/40 + 5*x^7/112 + 35*x^9/1152 \subsection{Examples involving matrices} We illustrate computing with the matrix whose $i,j$ entry is $i/j$, for $i,j=1,\ldots,4$. sage: f = axiom.eval('f[i,j] := i/j') sage: A = axiom('genmatrix(f,4,4)'); A matrix([1,1/2,1/3,1/4],[2,1,2/3,1/2],[3,3/2,1,3/4],[4,2,4/3,1]) sage: A.determinant() 0 sage: A.echelon() matrix([1,1/2,1/3,1/4],[0,0,0,0],[0,0,0,0],[0,0,0,0]) sage: A.eigenvalues() [[0,4],[3,1]] sage: A.eigenvectors() [[[0,4],[3,1]],[1,0,0, - 4],[0,1,0, - 2],[0,0,1, - 4/3],[1,2,3,4]] We can also compute the echelon form in \sage: sage: B = matrix(QQ, A) sage: B.echelon_form() [ 1 1/2 1/3 1/4] [ 0 0 0 0] [ 0 0 0 0] [ 0 0 0 0] sage: B.charpoly().factor() (x - 4) * x^3 \subsection{Laplace Transforms} We illustrate Laplace transforms: sage: _ = axiom.eval("f(t) := t*sin(t)") sage: axiom("laplace(f(t),t,s)") 2*s/(s^2 + 1)^2 sage: axiom("laplace(delta(t-3),t,s)") #Dirac delta function %e^-(3*s) sage: _ = axiom.eval("f(t) == exp(t)*sin(t)") sage: axiom("laplace(f(t),t,s)") 1/(s^2 - 2*s + 2) sage: _ = axiom.eval("f(t) := t^5*exp(t)*sin(t)") sage: axiom("laplace(f(t),t,s)") 360*(2*s - 2)/(s^2 - 2*s + 2)^4 - 480*(2*s - 2)^3/(s^2 - 2*s + 2)^5 + 120*(2*s - 2)^5/(s^2 - 2*s + 2)^6 sage: axiom("laplace(f(t),t,s)") 3 5 360 (2 s - 2) 480 (2 s - 2) 120 (2 s - 2) --------------- - --------------- + --------------- 2 4 2 5 2 6 (s - 2 s + 2) (s - 2 s + 2) (s - 2 s + 2) sage: axiom("laplace(diff(x(t),t),t,s)") s*laplace(x(t),t,s) - x(0) sage: axiom("laplace(diff(x(t),t,2),t,s)") -at('diff(x(t),t,1),t = 0) + s^2*laplace(x(t),t,s) - x(0)*s sage.: axiom("laplace(diff(x(t),t,2),t,s)") ! d ! 2 - -- (x(t))! + s laplace(x(t), t, s) - x(0) s dt ! !t = 0 Even better, use \code{view(axiom("laplace(diff(x(t),t,2),t,s)"))} to see a typeset version. \subsection{Continued Fractions} A continued fraction $a + 1/(b + 1/(c + \cdots))$ is represented in axiom by the list $[a, b, c, \ldots]$. sage: axiom("cf((1 + sqrt(5))/2)") [1,1,1,1,2] sage: axiom("cf ((1 + sqrt(341))/2)") [9,1,2,1,2,1,17,1,2,1,2,1,17,1,2,1,2,1,17,2] \subsection{Special examples} In this section we illustrate calculations that would be awkward to do (as far as I know) in non-symbolic computer algebra systems like MAGMA or GAP. We compute the gcd of $2x^{n+4} - x^{n+2}$ and $4x^{n+1} + 3x^n$ for arbitrary $n$. sage: f = axiom('2*x^(n+4) - x^(n+2)') sage: g = axiom('4*x^(n+1) + 3*x^n') sage: f.gcd(g) x^n You can plot 3d graphs (via gnuplot): sage.: axiom('plot3d(x^2-y^2, [x,-2,2], [y,-2,2], [grid,12,12])') [displays a 3 dimensional graph] You can formally evaluate sums (note the \code{nusum} command): sage: S = axiom('nusum(exp(1+2*i/n),i,1,n)') sage.: S 2/n + 3 2/n + 1 %e %e ----------------------- - ----------------------- 1/n 1/n 1/n 1/n (%e - 1) (%e + 1) (%e - 1) (%e + 1) We formally compute the limit as $n\to\infty$ of $2S/n$ as follows: sage: T = S*axiom('2/n') sage: T.tlimit('n','inf') %e^3 - %e \subsection{Miscellaneous} Obtaining digits of $\pi$: sage: axiom.eval('fpprec : 100') '100' sage: axiom(pi).bfloat() 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068b0 Defining functions in axiom: sage: axiom.eval('fun(a) == a^2') 'fun[a] := a^2' sage: axiom('fun(10)') 100 \subsection{Interactivity} Axiom has a non-interactive mode that is initiated via the command ")read file.input". \subsection{Latex Output} The latex output of Axiom is not perfect. E.g., sage: axiom.eval('tex(sin(u) + sinh(v^2))') '$$\\sinhv^2 + \\sinu$$false' Notice the lack of space after the sin macro, which is a latex syntax error. In \sage this is automatically fixed via a substition for trig functions, which may have potentially bad side effects: sage: latex(axiom('sin(u) + sinh(v^2)')) \sinh v^2+\sin u It would be nice if somebody would fix this problem. One way would be to improve Axiom by making the fix to Axiom and giving this back to the Axiom people. Here's another example: sage: g = axiom('exp(3*%i*x)/(6*%i) + exp(%i*x)/(2*%i) + c') sage: latex(g) -\frac{i e^{3 i x}}{6}-\frac{i e^{i x}}{2}+c \subsection{Long Input} The Axiom interface reads in even very long input (using files) in a robust manner, as long as you are creating a new object. \note{Using \code{axiom.eval} for long input is much less robust, and is not recommended.} sage: t = '"%s"'%10^10000 # ten thousand character string. sage: a = axiom(t) """ #***************************************************************************** # Copyright (C) 2005 William Stein # # Distributed under the terms of the GNU General Public License (GPL) # # This code is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU # General Public License for more details. # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ #***************************************************************************** import os, re from expect import Expect, ExpectElement, FunctionElement, ExpectFunction, tmp from sage.misc.misc import verbose from sage.misc.multireplace import multiple_replace cnt = 0 seq = 0 from sage.misc.all import pager, verbose, DOT_SAGE, SAGE_ROOT COMMANDS_CACHE = '%s/axiom_commandlist_cache.sobj'%DOT_SAGE import sage.server.support # The Axiom commands ")what thing det" ")show Matrix" and ")display # op det" commands, gives a list of all identifiers that begin in # a certain way. This could maybe be useful somehow... (?) Also # axiom has a lot a lot of ways for getting documentation from the # system -- this could also be useful. class Axiom(Expect): """ Interface to the Axiom interpreter. """ def __call__(self, x): import sage.rings.all if sage.rings.all.is_Infinity(x): return Expect.__call__(self, 'inf') else: return Expect.__call__(self, x) def __init__(self, script_subdirectory=None, logfile=None, server=None): """ Create an instance of the Axiom interpreter. """ eval_using_file_cutoff = 200 self.__eval_using_file_cutoff = eval_using_file_cutoff Expect.__init__(self, name = 'axiom', prompt = '$$[0-9]+$$ -> ', command = "axiom -nox", maxread = 1, # CRUCIAL to use less buffering for Axiom! script_subdirectory = script_subdirectory, restart_on_ctrlc = False, verbose_start = False, init_code = [], logfile = logfile, eval_using_file_cutoff=eval_using_file_cutoff) def __getattr__(self, attrname): if attrname[:1] == "_": raise AttributeError return AxiomExpectFunction(self, attrname) def _start(self): # For some reason sending a single input line at startup avoids # lots of weird timing issues when doing doctests. Expect._start(self) #out = self._eval_line(')set output algebra off', reformat=False) #out = self._eval_line(')set output tex on', reformat="False") out = self._eval_line(')set message autoload off', reformat=False) self._expect.expect(self._prompt) #out = self._expect.before #print "out 0 = '%s'"%out #self(1) def _eval_line_using_file(self, line, tmp): F = open(tmp, 'w') F.write(line) F.close() if self._expect is None: self._start() # For some reason this trivial comp # keeps certain random freezes from occuring. Do not remove this. # The space before the \n is also important. self._expect.sendline(')read "%s"\n'%tmp) self._expect.expect(self._prompt) return '' def __reduce__(self): return reduce_load_Axiom, tuple([]) def _quit_string(self): return ')lisp (quit)' def _eval_line(self, line, reformat=True, allow_use_file=False, wait_for_prompt=True): if not wait_for_prompt: return Expect._eval_line(self, line) line = line.rstrip().rstrip(';') if line == '': return '' global seq seq += 1 #start = SAGE_START + str(seq) #end = SAGE_END + str(seq) #line = '%s;\n%s; %s;'%(start, line, end) if self._expect is None: self._start() if allow_use_file and self.__eval_using_file_cutoff and \ len(line) > self.__eval_using_file_cutoff: return self._eval_line_using_file(line, tmp) try: E = self._expect # debug # print "in = '%s'"%line E.sendline(line) self._expect.expect(self._prompt) out = self._expect.before # debug # print "out = '%s'"%out except KeyboardInterrupt: self._keyboard_interrupt() if 'Incorrect syntax:' in out: raise RuntimeError, out if not reformat: return out if 'error' in out: return out #out = out.lstrip() i = out.find('\n') out = out[i+1:] outs = out.split("\n") i = 0 outline = '' for line in outs: line = line.rstrip() # print "'%s'"%line if line[:4] == ' (': i = line.find('(') i += line[i:].find(')') if line[i+1:] == "": i = 0 outs = outs[1:] break; out = "\n".join(line[i+1:] for line in outs[1:]) return out ########################################### # Interactive help ########################################### def help(self, s): if sage.server.support.EMBEDDED_MODE: os.system('asq "describe(%s); "< /dev/null'%s) else: os.system('asq "describe(%s);"'%s) def example(self, s): if sage.server.support.EMBEDDED_MODE: os.system('axiom -ht -nogr "example(%s);" < /dev/null'%s) else: os.system('axiom -ht "example(%s);"'%s) describe = help def demo(self, s): if sage.server.support.EMBEDDED_MODE: os.system('axiom -ht "demo(%s);" < /dev/null'%s) else: os.system('axiom -ht "demo(%s);"'%s) def completions(self, s): """ Return all commands that complete the command starting with the string s. This is like typing s[tab] in the maple interpreter. """ s = self.eval('apropos(%s)'%s).replace('\\ - ','-') return [x for x in s[1:-1].split(',') if x[0] != '?'] def _commands(self): """ Return list of all commands defined in Axiom. """ try: return self.__commands except AttributeError: self.__commands = sum([self.completions(chr(97+n)) for n in range(26)], []) return self.__commands def trait_names(self, verbose=True, use_disk_cache=True): try: return self.__trait_names except AttributeError: import sage.misc.persist if use_disk_cache: try: self.__trait_names = sage.misc.persist.load(COMMANDS_CACHE) return self.__trait_names except IOError: pass if verbose: print "\nBuilding Axiom command completion list (this takes" print "a few seconds only the first time you do it)." print "To force rebuild later, delete %s."%COMMANDS_CACHE v = self._commands() self.__trait_names = v sage.misc.persist.save(v, COMMANDS_CACHE) return v def _object_class(self): return AxiomElement def _true_symbol(self): return 'true' def _false_symbol(self): return 'false' def function(self, args, defn, repr=None, latex=None): """ Return the Axiom function with given arguments and definition. INPUT: args -- a string with variable names separated by commas defn -- a string (or Axiom expression) that defines a function of the arguments in Axiom. repr -- an optional string; if given, this is how the function will print. EXAMPLES: sage: f = axiom.function('x', 'sin(x)') sage: f(3.2) -0.058374143427580086 sage: f = axiom.function('x,y', 'sin(x)+cos(y)') sage: f(2,3.5) sin(2) - 0.9364566872907963 sage: f sin(x)+cos(y) sage: g = f.integrate('z'); g (cos(y) + sin(x))*z sage: g(1,2,3) 3*(cos(2) + sin(1)) The function definition can be an axiom object: sage: an_expr = axiom('sin(x)*gamma(x)') sage: t = axiom.function('x', an_expr) sage: t gamma(x)*sin(x) sage: t(2) sin(2) sage: float(t(2)) 0.90929742682568171 sage: loads(t.dumps()) gamma(x)*sin(x) """ name = self._next_var_name() defn = str(defn) args = str(args) axiom.eval('%s(%s) := %s'%(name, args, defn)) if repr is None: repr = defn f = AxiomFunction(self, name, repr, args, latex) return f def set(self, var, value): """ Set the variable var to the given value. """ cmd = '%s := %s'%(var, value) out = self._eval_line(cmd, reformat=False) #out = self._eval_line(cmd, reformat=False, allow_use_file=True) if out.find("error") != -1: raise TypeError, "Error executing code in Axiom\nCODE:\n\t%s\nAxiom ERROR:\n\t%s"%(cmd, out) def get(self, var): """ Get the string value of the variable var. """ s = self._eval_line('%s'%var) return s #def clear(self, var): # """ # Clear the variable named var. # """ # if self._expect is None: # return # self._expect.sendline('kill(%s);'%var) # self._expect.expect(self._prompt) def console(self): axiom_console() def plot2d(self, *args): r""" Plot a 2d graph using Axiom / Viewman. axiom.plot2d(f, '[var, min, max]', options) INPUT: f -- a string representing a function (such as f="sin(x)") [var, xmin, xmax] options -- an optional string representing plot2d options in gnuplot format EXAMPLES: sage.: axiom.plot2d('sin(x)','[x,-5,5]') sage.: opts = '[gnuplot_term, ps], [gnuplot_out_file, "sin-plot.eps"]' sage.: axiom.plot2d('sin(x)','[x,-5,5]',opts) The eps file is saved in the current directory. """ self('plot2d(%s)'%(','.join([str(x) for x in args]))) def plot2d_parametric(self, r, var, trange, nticks=50, options=None): r""" Plots r = [x(t), y(t)] for t = tmin...tmax using gnuplot with options INPUT: r -- a string representing a function (such as r="[x(t),y(t)]") var -- a string representing the variable (such as var = "t") trange -- [tmin, tmax] are numbers with tmin1$, using the qunit axiom command. EXAMPLE: sage: u = axiom.unit_quadratic_integer(101) sage: u.parent() Number Field in a with defining polynomial x^2 - 101 sage: u a + 10 sage: u = axiom.unit_quadratic_integer(13) sage: u 5*a + 18 sage: u.parent() Number Field in a with defining polynomial x^2 - 13 """ from sage.rings.all import QuadraticField, Integer # Take square-free part so sqrt(n) doesn't get simplified further by axiom # (The original version of this function would yield wrong answers if # n is not squarefree.) n = Integer(n).square_free_part() if n < 1: raise ValueError, "n (=%s) must be >= 1"%n s = str(self('qunit(%s)'%n)).lower() r = re.compile('sqrt$$.*$$') s = r.sub('a', s) a = QuadraticField(n, 'a').gen() return eval(s) def plot_list(self, ptsx, ptsy, options=None): r""" Plots a curve determined by a sequence of points. INPUT: ptsx -- [x1,...,xn], where the xi and yi are real, ptsy -- [y1,...,yn] options -- a string representing axiom plot2d options. The points are (x1,y1), (x2,y2), etc. EXAMPLES: sage.: zeta_ptsx = [ (pari(1/2 + i*I/10).zeta().real()).precision(1) for i in range (70,150)] sage.: zeta_ptsy = [ (pari(1/2 + i*I/10).zeta().imag()).precision(1) for i in range (70,150)] sage.: axiom.plot_list(zeta_ptsx, zeta_ptsy) sage.: opts='[gnuplot_preamble, "set nokey"], [gnuplot_term, ps], [gnuplot_out_file, "zeta.eps"]' sage.: axiom.plot_list(zeta_ptsx, zeta_ptsy, opts) """ cmd = 'plot2d([discrete,%s, %s]'%(ptsx, ptsy) if options is None: cmd += ')' else: cmd += ', %s)'%options self(cmd) def plot_multilist(self, pts_list, options=None): r""" Plots a list of list of points pts_list=[pts1,pts2,...,ptsn], where each ptsi is of the form [[x1,y1],...,[xn,yn]] x's must be integers and y's reals options is a string representing axiom plot2d options. EXAMPLES: sage.: xx = [ i/10.0 for i in range (-10,10)] sage.: yy = [ i/10.0 for i in range (-10,10)] sage.: x0 = [ 0 for i in range (-10,10)] sage.: y0 = [ 0 for i in range (-10,10)] sage.: zeta_ptsx1 = [ (pari(1/2+i*I/10).zeta().real()).precision(1) for i in range (10)] sage.: zeta_ptsy1 = [ (pari(1/2+i*I/10).zeta().imag()).precision(1) for i in range (10)] sage.: axiom.plot_multilist([[zeta_ptsx1,zeta_ptsy1],[xx,y0],[x0,yy]]) sage.: zeta_ptsx1 = [ (pari(1/2+i*I/10).zeta().real()).precision(1) for i in range (10,150)] sage.: zeta_ptsy1 = [ (pari(1/2+i*I/10).zeta().imag()).precision(1) for i in range (10,150)] sage.: axiom.plot_multilist([[zeta_ptsx1,zeta_ptsy1],[xx,y0],[x0,yy]]) sage.: opts='[gnuplot_preamble, "set nokey"]' sage.: axiom.plot_multilist([[zeta_ptsx1,zeta_ptsy1],[xx,y0],[x0,yy]],opts) """ n = len(pts_list) cmd = '[' for i in range(n): if i < n-1: cmd = cmd+'[discrete,'+str(pts_list[i][0])+','+str(pts_list[i][1])+'],' if i==n-1: cmd = cmd+'[discrete,'+str(pts_list[i][0])+','+str(pts_list[i][1])+']]' # debug # print cmd if options is None: self('plot2d('+cmd+')') else: self('plot2d('+cmd+','+options+')') class AxiomElement(ExpectElement): def __call__(self, x): self._check_valid() P = self.parent() return P('%s[%s]'%(self.name(), x)) def _cmp_(self, other): """ EXAMPLES: sage: a = axiom(1); b = axiom(2) sage: a == b False sage: a < b True sage: a > b False sage: b < a False sage: b > a True We can also compare more complicated object such as functions: sage: f = axiom('sin(x)'); g = axiom('cos(x)') sage: -f == g.diff('x') True """ # thanks to David Joyner for telling me about using "is". P = self.parent() if P.eval("is (%s < %s)"%(self.name(), other.name())) == P._true_symbol(): return -1 elif P.eval("is (%s > %s)"%(self.name(), other.name())) == P._true_symbol(): return 1 elif P.eval("is (%s = %s)"%(self.name(), other.name())) == P._true_symbol(): return 0 else: return -1 # everything is supposed to be comparable in Python, so we define # the comparison thus when no comparable in interfaced system. def numer(self): P = self.parent() return P('numeric(%s)'%self._name) def real(self): return self.realpart() def imag(self): return self.imagpart() def str(self): self._check_valid() P = self.parent() return P.get('%s::InputForm'%self._name) def __repr__(self): self._check_valid() P = self.parent() return P.get(self._name) def diff(self, var='x', n=1): """ Return the n-th derivative of self. INPUT: var -- variable (default: 'x') n -- integer (default: 1) OUTPUT: n-th derivative of self with respect to the variable var EXAMPLES: sage: f = axiom('x^2') sage: f.diff() 2*x sage: f.diff('x') 2*x sage: f.diff('x', 2) 2 sage: axiom('sin(x^2)').diff('x',4) 16*x^4*sin(x^2) - 12*sin(x^2) - 48*x^2*cos(x^2) sage: f = axiom('x^2 + 17*y^2') sage: f.diff('x') 2*x sage: f.diff('y') 34*y """ return ExpectElement.__getattr__(self, 'differentiate')(var, n) derivative = diff def nintegral(self, var='x', a=0, b=1, desired_relative_error='1e-8', maximum_num_subintervals=200): r""" Return a numerical approximation to the integral of self from a to b. INPUT: var -- variable to integrate with respect to a -- lower endpoint of integration b -- upper endpoint of integration desired_relative_error -- (default: '1e-8') the desired relative error maximum_num_subintervals -- (default: 200) axiom number of subintervals OUTPUT: -- approximation to the integral -- estimated absolute error of the approximation -- the number of integrand evaluations -- an error code: 0 -- no problems were encountered 1 -- too many subintervals were done 2 -- excessive roundoff error 3 -- extremely bad integrand behavior 4 -- failed to converge 5 -- integral is probably divergent or slowly convergent 6 -- the input is invalid EXAMPLES: sage: axiom('exp(-sqrt(x))').nintegral('x',0,1) (0.5284822353142306, 4.1633141378838445E-11, 231, 0) Note that GP also does numerical integration, and can do so to very high precision very quickly: sage: gp('intnum(x=0,1,exp(-sqrt(x)))') 0.5284822353142307136179049194 # 32-bit 0.52848223531423071361790491935415653021 # 64-bit sage: _ = gp.set_precision(80) sage: gp('intnum(x=0,1,exp(-sqrt(x)))') 0.52848223531423071361790491935415653021675547587292866196865279321015401702040079 """ from sage.rings.all import Integer v = self.quad_qags(var, a, b, desired_relative_error, maximum_num_subintervals) return v[0], v[1], Integer(v[2]), Integer(v[3]) def integral(self, var='x', min=None, max=None): r""" Return the integral of self with respect to the variable x. INPUT: var -- variable min -- default: None max -- default: None Returns the definite integral if xmin is not None, otherwise returns an indefinite integral. EXAMPLES: sage: axiom('x^2+1').integral() x^3/3 + x sage: axiom('x^2+ 1 + y^2').integral('y') y^3/3 + x^2*y + y sage: axiom('x / (x^2+1)').integral() log(x^2 + 1)/2 sage: axiom('1/(x^2+1)').integral() atan(x) sage.: axiom('1/(x^2+1)').integral('x', 0, infinity) %pi/2 sage: axiom('x/(x^2+1)').integral('x', -1, 1) 0 sage: f = axiom('exp(x^2)').integral('x',0,1); f -sqrt(%pi)*%i*erf(%i)/2 sage: f.numer() # I wonder how to get a real number (~1.463)?? -0.8862269254527579*%i*erf(%i) """ I = ExpectElement.__getattr__(self, 'integrate') if min is None: return I(var) else: if max is None: raise ValueError, "neither or both of min/max must be specified." return I(var, min, max) integrate = integral def __float__(self): return float(str(self.numer())) def __len__(self): """ Return the length of a list. EXAMPLES: sage: v = axiom('[x^i for i in 0..5]') sage: len(v) 6 """ self._check_valid() return int(self.parent().eval('#(%s)'%self.name())) def __getattr__(self, attrname): if attrname[:1] == "_": raise AttributeError return AxiomFunctionElement(self, attrname) def __getitem__(self, n): r""" Return the n-th element of this list. \note{Lists are 0-based when accessed via the \sage interface, not 1-based as they are in the Axiom interpreter.} EXAMPLES: sage: v = axiom('[i*x^i for i in 0..5]'); v [0,x,2*x^2,3*x^3,4*x^4,5*x^5] sage: v[3] 3*x^3 sage: v[0] 0 sage: v[10] Traceback (most recent call last): ... IndexError: n = (10) must be between 0 and 5 """ n = int(n) if n < 0 or n >= len(self): raise IndexError, "n = (%s) must be between %s and %s"%(n, 0, len(self)-1) return ExpectElement.__getitem__(self, n+1) def subst(self, val): P = self.parent() return P('subst(%s, %s)'%(self.name(), val)) def comma(self, args): self._check_valid() P = self.parent() return P('%s, %s'%(self.name(), args)) def _latex_(self): self._check_valid() P = self.parent() s = axiom._eval_line('outputAsTex(%s)'%self.name(), reformat=False) if not '$$' in s: raise RuntimeError, "Error texing axiom object." i = s.find('$$') j = s.rfind('$$') s = s[i+2:j] s = multiple_replace({'\r\n':' ', '\\%':'', '\\arcsin ':'\\sin^{-1} ', '\\arccos ':'\\cos^{-1} ', '\\arctan ':'\\tan^{-1} '}, s) return s def trait_names(self): return self.parent().trait_names() def _matrix_(self, R): r""" If self is a Axiom matrix, return the corresponding \sage matrix over the \sage ring$R$. This may or may not work depending in how complicated the entries of self are! It only works if the entries of self can be coerced as strings to produce meaningful elements of$R\$. EXAMPLES: sage: _ = axiom.eval("f[i,j] := i/j") sage: A = axiom('genmatrix(f,4,4)'); A matrix([1,1/2,1/3,1/4],[2,1,2/3,1/2],[3,3/2,1,3/4],[4,2,4/3,1]) sage: A._matrix_(QQ) [ 1 1/2 1/3 1/4] [ 2 1 2/3 1/2] [ 3 3/2 1 3/4] [ 4 2 4/3 1] You can also use the \code{matrix} command (which is defined in \code{sage.misc.functional}): sage: matrix(QQ, A) [ 1 1/2 1/3 1/4] [ 2 1 2/3 1/2] [ 3 3/2 1 3/4] [ 4 2 4/3 1] """ from sage.matrix.all import MatrixSpace self._check_valid() P = self.parent() nrows = int(P.eval('length(%s)'%self.name())) if nrows == 0: return MatrixSpace(R, 0, 0)(0) ncols = int(P.eval('length(%s[1])'%self.name())) M = MatrixSpace(R, nrows, ncols) s = self.str().replace('matrix','').replace(',',"','").\ replace("]','[","','").replace('([',"['").replace('])',"']") s = eval(s) return M([R(x) for x in s]) def partial_fraction_decomposition(self, var='x'): """ Return the partial fraction decomposition of self with respect to the variable var. EXAMPLES: sage: f = axiom('1/((1+x)*(x-1))') sage: f.partial_fraction_decomposition('x') 1 1 --------- - --------- 2 (x - 1) 2 (x + 1) """ return self.partfrac(var) class AxiomFunctionElement(FunctionElement): def _sage_doc_(self): return self._obj.parent().help(self._name) class AxiomExpectFunction(ExpectFunction): def _sage_doc_(self): M = self._parent return M.help(self._name) class AxiomFunction(AxiomElement): def __init__(self, parent, name, defn, args, latex): AxiomElement.__init__(self, parent, name, is_name=True) self.__defn = defn self.__args = args self.__latex = latex def __call__(self, *x): self._check_valid() P = self.parent() if len(x) == 1: x = '(%s)'%x return P('%s%s'%(self.name(), x)) def __repr__(self): return self.__defn def _latex_(self): if self.__latex is None: return '\\mbox{\\rm %s}'%self.__defn else: return self.__latex def integrate(self, var): return self.integral(var) def integral(self, var): self._check_valid() P = self.parent() f = P('integrate(%s(%s), %s)'%(self.name(), self.__args, var)) if var in self.__args.split(','): args = self.__args else: args = self.__args + ',' + var return P.function(args, str(f)) def is_AxiomElement(x): return isinstance(x, AxiomElement) # An instance axiom = Axiom(script_subdirectory=None) def reduce_load_Axiom(): return axiom import os def axiom_console(): os.system('axiom') def __doctest_cleanup(): import sage.interfaces.quit sage.interfaces.quit.expect_quitall() @ \end{document}