title(exp):=disp(dpart(exp))$
(title("a classical textbook example of differentiation"))$
x^x^x;
diff(%,x);
(title("differentiation of a nested function"))$
erf(tan(acos(log(x))));
diff(%,x);
title("expanding and factorizing a univariate polynomial")$
(x-1)*(2*x-1)*(3*x-1)^7;
expand(%);
factor(%);
title("factorization of cyclotomic polynomials - which form of an expression is more simple?")$
x^600-1;
factor(%);
title("factorization of a multivariate polynomial")$
(y^3-x^2)^4*(x+y+z)^2;
expand(%);
factor(%);
title(" algebraic simplification example")$
(sqrt(r^2+a^2)+a)*(sqrt(r^2+b^2)+b)/r^2
 -(sqrt(r^2+b^2)+sqrt(r^2+a^2)+b+a)/(sqrt(r^2+b^2)+sqrt(r^2+a^2)-b-a);
ratsimp(%);
title(" algebraic simplification example")$
(z^5-y*z^4+x*z^4-2*y*z^3-6*x*z^3+2*y^2*z^2+4*x*y*z^2-6*x^2*z^2+y^2*z+
6*x*y*z+9*x^2*z-y^3-5*x*y^2-3*x^2*y+9*x^3)/(-z^2+y+3*x)^2;
ratsimp(%);
title("integration of a non-tabulated function")$
1/(x^3+a*x^2+x);
integrate(%,x);
diff(%,x),ratsimp;
title("logarithmic subcase of risch's integration algorithm")$
(log(x)-1)/(log(x)^2-x^2);
integrate(%,x);  
title("taylor series example")$
sin(x+a*x^3);   
taylor(%,x,0,19);
title("taylor series example - puiseux")$
sqrt(log(1+x)+sin(x));
taylor(%,x,0,17); 
title("macsyma to fortran conversion - optimized routine not yet available")$
exp:-gamma^5+delta*gamma^4+2*delta^2*gamma^3-alpha^3*gamma^3-3*alpha^2*gamma^3
-3*alpha*gamma^3-gamma^3-2*delta^3*gamma^2+3*alpha^3*delta*gamma^2
+9*alpha^2*delta*gamma^2+9*alpha*delta*gamma^2+3*delta*gamma^2
+alpha^2*gamma^2-2*alpha*gamma^2+gamma^2-delta^4*gamma-3*alpha^3*delta^2*gamma
-9*alpha^2*delta^2*gamma-9*alpha*delta^2*gamma-3*delta^2*gamma
+2*alpha^2*delta*gamma-4*alpha*delta*gamma+2*delta*gamma+delta^5
+alpha^3*delta^3+3*alpha^2*delta^3+3*alpha*delta^3+delta^3+alpha^2*delta^2
-2*alpha*delta^2+delta^2+alpha^5+alpha^4-2*alpha^3-2*alpha^2+alpha+1;
fortran(%);
factorsum(exp);
fortran(%);
title("bignum arithmetic and arbitrary precision floating point arithmetic")$
6427752177035961102167848369364650410088811975131171341205503;
%^5;
2535301200456458802993406410751;
%th(3)/%;
sqrt(%pi),numer;
fpprec:50;
bfloat(sqrt(%pi));
/* at the moment this runs pretty slowly but would be a nice example of bignums
title("large numbers - the largest known prime number")$
2^(44497)-1; */
title("vandermond's matrix")$
mat1:matrix([1,x,x^2,x^3],[1,y,y^2,y^3],[1,z,z^2,z^3],[1,w,w^2,w^3]);
factor(determinant(mat1));
minor(mat1,4,4);
%^^-1,factor;
title("solving algebraic equations")$
q^2*x^2+p^2*q*x-p*q*x-p^3=0;
solve(%,x);
title("application of cubic formula")$
(breakup:false,eq:4*x^3+a*x+10*b= 0);
h:solve(eq,x)$
first_root:first(h);
second_root:part(h,2);
third_root:last(h);
title("solution of set of simultaneous linear equations")$
[3*a+5*b+7*c+11*d+13*e=17*r,19*a+23*b+29*c+31*d+37*e=41*s,43*a+47*b+53*c+59*d+61*e=67*t,
71*a+73*b+79*c+83*d+89*e=97*x,101*a+103*b+107*c+109*d+113*e=127*y];
solve(%,[a,b,c,d,e]);
title("solution of set of simultaneous non-linear equations")$
[x*y*z = 42,-z+y+x = -2,-3*z+2*y+3*x = -9];
solve(%);
title("finding eigenvalues")$
matrix([0,6,-10,-8],[6,0,8,10],[-10,8,15*a,6],[-8,10,6,15*a]);
solve(charpoly(%,l),l);
title("two dimensional plot")$ 
plotnum:100$
plot(x*sin(x^2),x,0,4,plot(x*sin(x^2),x,0,4));
title("two dimensional parametric plot")$
plotnum:400;
paramplot(s*sin(s),s*cos(s),s,0,80,paramplot(s*sin(s),s*cos(s),s,0,80));
title("three dimensional cartesian plot of a bessel function")$
plotnum:20$
plot3d(j0(sqrt(x^2+y^2)),x,-12,12,y,-12,12,plot3d(j0(sqrt(x^2+y^2)),x,-12,12,y,-12,12));
title("three dimensional polar plot of the same bessel function")$
plot3d(j0(r),th,0,2*%pi,r,0,12,polar,plot3d(j0(r),th,0,2*%pi,r,0,12,polar));
title("three dimensional plot of x*exp(-x^2-y^2)")$
plot3d(x*exp(-x^2-y^2),x,-2,2,y,-2,2,plot3d(x*exp(-x^2-y^2),x,-2,2,y,-2,2));