Komputeralgebrai algoritmusokJ\303\241rai AntalEzek a programok csak szeml\303\251ltet\303\251sre szolg\303\241lnak.1. T\303\266rt\303\251net2. Algebrai alapok3. Norm\303\241l form\303\241k, reprezent\303\241ci\303\2634. Aritmetika5. K\303\255nai marad\303\251kol\303\241srestart;E 5.1. P\303\251lda.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnE 5.2. P\303\251lda.a:=-30*x^3*y+90*x^2*y^2+15*x^2-60*x*y+45*y^2;collect(a,[x,y],`distributed`);collect(a,x);collect(a,y);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnE 5.3. P\303\251lda.3/1;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnE 5.4. P\303\251lda.[i$i=-8..8]; map(x->x mod 6,%);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnE 5.5. P\303\251lda.subs(x=5,a); subs(y=2,a);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnE 5.6. P\303\251lda.a:=3*x^2*y^2-x^2*y+5*x^2+x*y^2-3*x*y; b:=2*x*y+7*x+y^2-2;a mod 5; b mod 5;a mod 7; b mod 7;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnE 5.7. P\303\251lda.a:=7*x+5; b:=2*x-3; c:=expand(a*b);subs(x=0,a) mod 5; subs(x=0,b) mod 5; subs(x=0,c) mod 5;subs(x=1,a) mod 5; subs(x=1,b) mod 5; subs(x=1,c) mod 5;subs(x=2,a) mod 5; subs(x=2,b) mod 5; subs(x=2,c) mod 5;subs(x=0,a) mod 7; subs(x=0,b) mod 7; subs(x=0,c) mod 7;subs(x=1,a) mod 7; subs(x=1,b) mod 7; subs(x=1,c) mod 7;subs(x=2,a) mod 7; subs(x=2,b) mod 7; subs(x=2,c) mod 7;c mod 7; c mod 5;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnE 5.8. P\303\251lda.m*i$i=-infinity..infinity;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnE 5.9. P\303\251lda.p:=5*x+2; p*d;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnE 5.10. P\303\251lda.p1:=x; p2:=y;p1*a1+p2*a2;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnE 5.11. P\303\251lda.[i$i=-8..8]; map(x->x mod 6,%);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnE 5.12. P\303\251lda.p:=x^2+1;a:=x^2+8*x+4; rem(a,p,x); b:=2*x^2+8*x+5; rem(b,p,x);p:=x-2; rem(a,p,x); subs(x=2,a); rem(b,p,x); subs(x=2,b);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnE 5.13. P\303\251lda.a:=-30*x^3*y+90*x^2*y^2+15*x^2-60*x*y+45*y^2; a mod 7; subs(y=3,a);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnE 5.14. P\303\251lda.m0:=3; m1:=5; m:=m0*m1; 11=2+3*3; -4=-1+(-1)*3;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnA 5.1. Algoritmus. IntegerCRA:=proc(M,U) local G,N,n,i,j,t;
n:=nops(M)-1;
G:=[0$i=1..n];
N:=[0$i=0..n];
for j to n do
t:=M[1] mod M[j+1];
for i to j-1 do
t:=t*M[i+1] mod M[j+1];
od;
G[j]:=1/t mod M[j+1];
od;
N[1]:=U[1];
for j to n do
t:=N[j];
for i from j-2 to 0 by -1 do
t:=t*M[i+1]+N[i+1] mod M[j+1];
od;
N[j+1]:=(U[j+1]-t)*G[j] mod M[j+1];
od;
t:=N[n+1];
for j from n-1 to 0 by -1 do
t:=t*M[j+1]+N[j+1];
od; t;
end;
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnE 5.15. P\303\251lda.`mod`:=mods; debug(IntegerCRA); IntegerCRA([99,97,95],[49,-21,-30]);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnA 5.2. Algoritmus. NewtonInterp:=proc(a,u,x,p) local i,j,t,n,G,N;
n:=nops(a)-1;
G:=[0$i=1..n];
N:=[0$i=0..n];
for j to n do
t:=a[j+1]-a[1] mod p;
for i to j-1 do
t:=t*(a[j+1]-a[i+1]) mod p;
od;
G[j]:=1/t mod p;
od;
N[1]:=u[1];
for j to n do
t:=N[j];
for i from j-2 to 0 by -1 do
t:=t*(a[j+1]-a[i+1])+N[i+1] mod p;
od;
N[j+1]:=(u[j+1]-t)*G[j] mod p;
od;
t:=N[n+1];
for j from n-1 to 0 by -1 do
t:=t*(x-a[j+1])+N[j+1] mod p;
od; t;
end;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnE 5.16. P\303\251lda.u0:=NewtonInterp([0,1],[-21,-30],y,97);u1:=NewtonInterp([0,1],[20,17],y,97);u2:=NewtonInterp([0,1],[-36,-31],y,97);u:=NewtonInterp([0,1,2],[u0,u1,u2],x,97); expand(u);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnE 5.17. P\303\251lda.a:=7*x+5; b:=2*x-3; c:=expand(a*b);c5:=expand(NewtonInterp([0,1,2],[0,-2,-1],x,5)) mod 5;c7:=expand(NewtonInterp([0,1,2],[-1,2,-2],x,7)) mod 7;c3:=expand(NewtonInterp([0,1,-1],[0,0,1],x,3)) mod 3;expand(IntegerCRA([5,7,3],[-x^2-x,3*x-1,-x^2+x])) mod 105;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn6. Newton-iter\303\241ci\303\263, Hensel-felemel\303\251s7. Legnagyobb k\303\266z\303\266s oszt\303\2638. Faktoriz\303\241l\303\241s9. Egyenletrendszerek10. Gr\303\266bner-b\303\241zisok11. Racion\303\241lis t\303\266rtf\303\274ggv\303\251nyek integr\303\241l\303\241sa12. A Risch-algoritmus.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn