/*  Sample usage of difference equation package  -*-Macsyma-*- */
IF GET('DIFFER,'VERSION)=FALSE THEN LOAD(DIFFER)$

a:matrix([1,-1],[-5,5]);
eigenvalues(a);
eigenvector(a,0);
eigenvector(a,6);
normalize:true$
eigenvector(a,0);

/*  First order equations  */

f[n+1]=f[n];
difference(%,f[n]);

g[0]:1$
g[n+1]=2*g[n];
difference(%,g[n]);

h[n+1]=3*h[n]+2;
difference(%,h[n]);

/*  Systems of homogeneous first order equations  */

differverbose:true$
eqn1:y[k+1]+z[k]-3*y[k]=0;
eqn2:3*y[k]+z[k+1]-5*z[k]=0;
difference([eqn1,eqn2],[y[k],z[k]]);

/*  Second order equations */

y[k+2]-6*y[k+1]+8*y[k]=0;
difference(%,y[k]);

differverbose:false$

y[0]:3$
y[1]:2$
y[k+2]-6*y[k+1]+8*y[k]=0;
difference(%,y[k]);
part(%,2),k=5;

/*  Fibonacci Numbers:   1, 1, 2, 3, 5, 8, 13, 21, ...  */

f[0]:1$
f[1]:1$
f[n+2]=f[n+1]+f[n];
difference(%,f[n]);
ratsimp(ev(part(%,2),n=7));

/* Cleanup */

remvalue(a);