Sat May 30 16:10:28 PDT 1992
REDUCE 3.4.1, 15-Jul-92 ...
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Time: 0 ms
4: 4: COMMENT Greatest Common Divisor Test Suite;
% The following examples were introduced in Moses, J. and Yun, D.Y.Y.,
% "The EZ GCD Algorithm", Proc. ACM 73 (1973) 159-166, and considered
% further in Hearn, A.C., "Non-modular Computation of Polynomial GCD's
% Using Trial Division", Proc. EUROSAM 79, 227-239, 72, published as
% Lecture Notes on Comp. Science, # 72, Springer-Verlag, Berlin, 1979.
on gcd;
% The following is the best setting for this file.
on ezgcd;
% In systems that have the heugcd code, the following is also a
% possibility, although not all examples complete in a reasonable time.
% load heugcd; on heugcd;
% The final alternative is to use neither ezgcd nor heugcd. In that case,
% most examples take excessive amounts of computer time.
share n;
operator xx;
% Case 1.
for n := 2:5
do write gcd(((for i:=1:n sum xx(i))-1)*((for i:=1:n sum xx(i)) + 2),
((for i:=1:n sum xx(i))+1)
*(-3xx(2)*xx(1)**2+xx(2)**2-1)**2);
1
1
1
1
% Case 2.
let d = (for i:=1:n sum xx(i)**n) + 1;
for n := 2:7 do write gcd(d*((for i:=1:n sum xx(i)**n) - 2),
d*((for i:=1:n sum xx(i)**n) + 2));
2 2
XX(2) + XX(1) + 1
3 3 3
XX(3) + XX(2) + XX(1) + 1
4 4 4 4
XX(4) + XX(3) + XX(2) + XX(1) + 1
5 5 5 5 5
XX(5) + XX(4) + XX(3) + XX(2) + XX(1) + 1
6 6 6 6 6 6
XX(6) + XX(5) + XX(4) + XX(3) + XX(2) + XX(1) + 1
7 7 7 7 7 7 7
XX(7) + XX(6) + XX(5) + XX(4) + XX(3) + XX(2) + XX(1) + 1
for n := 2:7 do write gcd(d*((for i:=1:n sum xx(i)**n) - 2),
d*((for i:=1:n sum xx(i)**(n-1)) + 2));
2 2
XX(2) + XX(1) + 1
3 3 3
XX(3) + XX(2) + XX(1) + 1
4 4 4 4
XX(4) + XX(3) + XX(2) + XX(1) + 1
5 5 5 5 5
XX(5) + XX(4) + XX(3) + XX(2) + XX(1) + 1
6 6 6 6 6 6
XX(6) + XX(5) + XX(4) + XX(3) + XX(2) + XX(1) + 1
7 7 7 7 7 7 7
XX(7) + XX(6) + XX(5) + XX(4) + XX(3) + XX(2) + XX(1) + 1
% Case 3.
let d = xx(2)**2*xx(1)**2 + (for i := 3:n sum xx(i)**2) + 1;
for n := 2:5
do write gcd(d*(xx(2)*xx(1) + (for i:=3:n sum xx(i)) + 2)**2,
d*(xx(1)**2-xx(2)**2 + (for i:=3:n sum xx(i)**2) - 1));
2 2
XX(2) *XX(1) + 1
2 2 2
XX(3) + XX(2) *XX(1) + 1
2 2 2 2
XX(4) + XX(3) + XX(2) *XX(1) + 1
2 2 2 2 2
XX(5) + XX(4) + XX(3) + XX(2) *XX(1) + 1
% Case 4.
let u = xx(1) - xx(2)*xx(3) + 1,
v = xx(1) - xx(2) + 3xx(3);
gcd(u*v**2,v*u**2);
2 2
3*XX(3) *XX(2) - XX(3)*XX(2) + XX(3)*XX(2)*XX(1) - 3*XX(3)*XX(1)
2
- 3*XX(3) + XX(2)*XX(1) + XX(2) - XX(1) - XX(1)
gcd(u*v**3,v*u**3);
2 2
3*XX(3) *XX(2) - XX(3)*XX(2) + XX(3)*XX(2)*XX(1) - 3*XX(3)*XX(1)
2
- 3*XX(3) + XX(2)*XX(1) + XX(2) - XX(1) - XX(1)
gcd(u*v**4,v*u**4);
2 2
3*XX(3) *XX(2) - XX(3)*XX(2) + XX(3)*XX(2)*XX(1) - 3*XX(3)*XX(1)
2
- 3*XX(3) + XX(2)*XX(1) + XX(2) - XX(1) - XX(1)
gcd(u**2*v**4,v**2*u**4);
4 2 3 3 3 2
9*XX(3) *XX(2) - 6*XX(3) *XX(2) + 6*XX(3) *XX(2) *XX(1)
3 3 2 4
- 18*XX(3) *XX(2)*XX(1) - 18*XX(3) *XX(2) + XX(3) *XX(2)
2 3 2 2 2
- 2*XX(3) *XX(2) *XX(1) + XX(3) *XX(2) *XX(1)
2 2 2 2 2 2
+ 12*XX(3) *XX(2) *XX(1) + 12*XX(3) *XX(2) - 12*XX(3) *XX(2)*XX(1)
2 2 2 2
- 12*XX(3) *XX(2)*XX(1) + 9*XX(3) *XX(1) + 18*XX(3) *XX(1)
2 3 3
+ 9*XX(3) - 2*XX(3)*XX(2) *XX(1) - 2*XX(3)*XX(2)
2 2 2
+ 4*XX(3)*XX(2) *XX(1) + 4*XX(3)*XX(2) *XX(1)
3 2
- 2*XX(3)*XX(2)*XX(1) - 8*XX(3)*XX(2)*XX(1) - 12*XX(3)*XX(2)*XX(1)
3 2
- 6*XX(3)*XX(2) + 6*XX(3)*XX(1) + 12*XX(3)*XX(1) + 6*XX(3)*XX(1)
2 2 2 2 3
+ XX(2) *XX(1) + 2*XX(2) *XX(1) + XX(2) - 2*XX(2)*XX(1)
2 4 3 2
- 4*XX(2)*XX(1) - 2*XX(2)*XX(1) + XX(1) + 2*XX(1) + XX(1)
% Case 5.
let d = (for i := 1:n product (xx(i)+1)) - 3;
for n := 2:5 do write gcd(d*for i := 1:n product (xx(i) - 2),
d*for i := 1:n product (xx(i) + 2));
XX(2)*XX(1) + XX(2) + XX(1) - 2
XX(3)*XX(2)*XX(1) + XX(3)*XX(2) + XX(3)*XX(1) + XX(3) + XX(2)*XX(1)
+ XX(2) + XX(1) - 2
XX(4)*XX(3)*XX(2)*XX(1) + XX(4)*XX(3)*XX(2) + XX(4)*XX(3)*XX(1)
+ XX(4)*XX(3) + XX(4)*XX(2)*XX(1) + XX(4)*XX(2) + XX(4)*XX(1)
+ XX(4) + XX(3)*XX(2)*XX(1) + XX(3)*XX(2) + XX(3)*XX(1) + XX(3)
+ XX(2)*XX(1) + XX(2) + XX(1) - 2
XX(5)*XX(4)*XX(3)*XX(2)*XX(1) + XX(5)*XX(4)*XX(3)*XX(2)
+ XX(5)*XX(4)*XX(3)*XX(1) + XX(5)*XX(4)*XX(3)
+ XX(5)*XX(4)*XX(2)*XX(1) + XX(5)*XX(4)*XX(2) + XX(5)*XX(4)*XX(1)
+ XX(5)*XX(4) + XX(5)*XX(3)*XX(2)*XX(1) + XX(5)*XX(3)*XX(2)
+ XX(5)*XX(3)*XX(1) + XX(5)*XX(3) + XX(5)*XX(2)*XX(1) + XX(5)*XX(2)
+ XX(5)*XX(1) + XX(5) + XX(4)*XX(3)*XX(2)*XX(1) + XX(4)*XX(3)*XX(2)
+ XX(4)*XX(3)*XX(1) + XX(4)*XX(3) + XX(4)*XX(2)*XX(1) + XX(4)*XX(2)
+ XX(4)*XX(1) + XX(4) + XX(3)*XX(2)*XX(1) + XX(3)*XX(2)
+ XX(3)*XX(1) + XX(3) + XX(2)*XX(1) + XX(2) + XX(1) - 2
clear d,u,v;
% The following examples were discussed in Char, B.W., Geddes, K.O.,
% Gonnet, G.H., "GCDHEU: Heuristic Polynomial GCD Algorithm Based
% on Integer GCD Computation", Proc. EUROSAM 84, 285-296, published as
% Lecture Notes on Comp. Science, # 174, Springer-Verlag, Berlin, 1984.
% Maple Problem 1.
gcd(34*x**80-91*x**99+70*x**31-25*x**52+20*x**76-86*x**44-17*x**33
-6*x**89-56*x**54-17,
91*x**49+64*x**10-21*x**52-88*x**74-38*x**76-46*x**84-16*x**95
-81*x**72+96*x**25-20);
1
% Maple Problem 2.
g := 34*x**19-91*x+70*x**7-25*x**16+20*x**3-86;
19 16 7 3
G := 34*X - 25*X + 70*X + 20*X - 91*X - 86
gcd(g * (64*x**34-21*x**47-126*x**8-46*x**5-16*x**60-81),
g * (72*x**60-25*x**25-19*x**23-22*x**39-83*x**52+54*x**10+81) );
19 16 7 3
34*X - 25*X + 70*X + 20*X - 91*X - 86
% Maple Problem 3.
gcd(3427088418+8032938293*x-9181159474*x**2-9955210536*x**3
+7049846077*x**4-3120124818*x**5-2517523455*x**6+5255435973*x**7
+2020369281*x**8-7604863368*x**9-8685841867*x**10+4432745169*x**11
-1746773680*x**12-3351440965*x**13-580100705*x**14+8923168914*x**15
-5660404998*x**16 +5441358149*x**17-1741572352*x**18
+9148191435*x**19-4940173788*x**20+6420433154*x**21+980100567*x**22
-2128455689*x**23+5266911072*x**24-8800333073*x**25-7425750422*x**26
-3801290114*x**27-7680051202*x**28-4652194273*x**29-8472655390*x**30
-1656540766*x**31+9577718075*x**32-8137446394*x**33+7232922578*x**34
+9601468396*x**35-2497427781*x**36-2047603127*x**37-1893414455*x**38
-2508354375*x**39-2231932228*x**40,
2503247071-8324774912*x+6797341645*x**2+5418887080*x**3
-6779305784*x**4+8113537696*x**5+2229288956*x**6+2732713505*x**7
+9659962054*x**8-1514449131*x**9+7981583323*x**10+3729868918*x**11
-2849544385*x**12-5246360984*x**13+2570821160*x**14-5533328063*x**15
-274185102*x**16+8312755945*x**17-2941669352*x**18-4320254985*x**19
+9331460166*x**20-2906491973*x**21-7780292310*x**22-4971715970*x**23
-6474871482*x**24-6832431522*x**25-5016229128*x**26-6422216875*x**27
-471583252*x**28+3073673916*x**29+2297139923*x**30+9034797416*x**31
+6247010865*x**32+5965858387*x**33-4612062748*x**34+5837579849*x**35
-2820832810*x**36-7450648226*x**37+2849150856*x**38+2109912954*x**39
+2914906138*x**40);
1
% Maple Problem 4.
g := 34271+80330*x-91812*x**2-99553*x**3+70499*x**4-31201*x**5
-25175*x**6+52555*x**7+20204*x**8-76049*x**9-86859*x**10;
10 9 8 7 6
G := - 86859*X - 76049*X + 20204*X + 52555*X - 25175*X
5 4 3 2
- 31201*X + 70499*X - 99553*X - 91812*X + 80330*X + 34271
gcd(g * (44328-17468*x-33515*x**2-5801*x**3+89232*x**4-56604*x**5
+54414*x**6-17416*x**7+91482*x**8-49402*x**9+64205*x**10
+9801*x**11-21285*x**12+52669*x**13-88004*x**14-74258*x**15
-38013*x**16-76801*x**17-46522*x**18-84727*x**19-16565*x**20
+95778*x**21-81375*x**22+72330*x**23+96015*x**24-24974*x**25
-20476*x**26-18934*x**27-25084*x**28-22319*x**29+25033*x**30),
g * (-83248+67974*x+54189*x**2-67793*x**3+81136*x**4+22293*x**5
+27327*x**6+96600*x**7-15145*x**8+79816*x**9+37299*x**10
-28496*x**11-52464*x**12+25708*x**13-55334*x**14-2742*x**15
+83128*x**16-29417*x**17-43203*x**18+93315*x**19-29065*x**20
-77803*x**21-49717*x**22-64749*x**23-68325*x**24-50163*x**25
-64222*x**26-4716*x**27+30737*x**28+22972*x**29+90348*x**30));
10 9 8 7 6 5
86859*X + 76049*X - 20204*X - 52555*X + 25175*X + 31201*X
4 3 2
- 70499*X + 99553*X + 91812*X - 80330*X - 34271
% Maple Problem 5.
gcd(-8472*x**4*y**10-8137*x**9*y**10-2497*x**4*y**4-2508*x**4*y**6
-8324*x**9*y**8-6779*x**9*y**6+2733*x**10*y**4+7981*x**7*y**3
-5246*x**6*y**2-274*x**10*y**3-4320,
15168*x**3*y-4971*x*y-2283*x*y**5+3074*x**6*y**10+6247*x**8*y**2
+2849*x**6*y**7-2039*x**7-2626*x**2*y**7+9229*x**6*y**5+2404*y**5
+1387*x**4*y**8+5602*x**5*y**2-6212*x**3*y**7-8561);
1
% Maple Problem 6.
g := -19*x**4*y**4+25*y**9+54*x*y**9+22*x**7*y**10-15*x**9*y**7-28;
9 7 7 10 4 4 9 9
G := - 15*X *Y + 22*X *Y - 19*X *Y + 54*X*Y + 25*Y - 28
gcd(g*(91*x**2*y**9+10*x**4*y**8-88*x*y**3-76*x**2-16*x**10*y
+72*x**10*y**4-20),
g*(34*x**9-99*x**9*y**3-25*x**8*y**6-76*y**7-17*x**3*y**5
+89*x**2*y**8-17));
9 7 7 10 4 4 9 9
15*X *Y - 22*X *Y + 19*X *Y - 54*X*Y - 25*Y + 28
% Maple Problem 7.
gcd(6713544209*x**9+8524923038*x**3*y**3*z**7+6010184640*x*z**7
+4126613160*x**3*y**4*z**9+2169797500*x**7*y**4*z**9
+2529913106*x**8*y**5*z**3+7633455535*y*z**3+1159974399*x**2*z**4
+9788859037*y**8*z**9+3751286109*x**3*y**4*z**3,
3884033886*x**6*z**8+7709443539*x*y**9*z**6
+6366356752*x**9*y**4*z**8+6864934459*x**3*y**2*z**6
+2233335968*x**4*y**9*z**3+2839872507*x**9*y**3*z
+2514142015*x*y*z**2+1788891562*x**4*y**6*z**6
+9517398707*x**8*y**7*z**2+7918789924*x**3*y*z**6
+6054956477*x**6*y**3*z**6);
1
% Maple Problem 8.
g := u**3*(x**2-y)*z**2+(u-3*u**2*x)*y*z-u**4*x*y+3;
4 3 2 2 3 2 2
G := - U *X*Y + U *X *Z - U *Y*Z - 3*U *X*Y*Z + U*Y*Z + 3
gcd(g * ((y**2+x)*z**2+u**5*(x*y+x**2)*z-y+5),
g * ((y**2-x)*z**2+u**5*(x*y-x**2)*z+y+9) );
4 3 2 2 3 2 2
U *X*Y - U *X *Z + U *Y*Z + 3*U *X*Y*Z - U*Y*Z - 3
% Maple Problem 9.
g := 34*u**2*y**2*z-25*u**2*v*z**2-18*v*x**2*z**2-18*u**2*x**2*y*z+53
+x**3;
2 2 2 2 2 2 2 2 3
G := - 25*U *V*Z - 18*U *X *Y*Z + 34*U *Y *Z - 18*V*X *Z + X + 53
gcd( g * (-85*u*v**2*y**2*z**2-25*u*v*x*y*z-84*u**2*v**2*y**2*z
+27*u**2*v*x**2*y**2*z-53*u*x*y**2*z+34*x**3),
g * (48*x**3-99*u*x**2*y**2*z-69*x*y*z-75*u*v*x*y*z**2
-43*u**2*v+91*u**2*v**2*y**2*z) );
2 2 2 2 2 2 2 2 3
25*U *V*Z + 18*U *X *Y*Z - 34*U *Y *Z + 18*V*X *Z - X - 53
% Maple Problem 10.
gcd(-9955*v**9*x**3*y**4*z**8+2020*v*y**7*z**4
-3351*v**5*x**10*y**2*z**8-1741*v**10*x**2*y**9*z**6
-2128*v**8*y*z**3-7680*v**2*y**4*z**10-8137*v**9*x**10*y**4*z**4
-1893*v**4*x**4*y**6+6797*v**8*x*y**9*z**6
+2733*v**10*x**4*y**9*z**7-2849*v**2*x**6*y**2*z**5
+8312*v**3*x**3*y**10*z**3-7780*v**2*x*y*z**2
-6422*v**5*x**7*y**6*z**10+6247*v**8*x**2*y**8*z**3
-7450*v**7*x**6*y**7*z**4+3625*x**4*y**2*z**7+9229*v**6*x**5*y**6
-112*v**6*x**4*y**8*z**7-7867*v**5*x**8*y**5*z**2
-6212*v**3*x**7*z**5+8699*v**8*x**2*y**2*z**5
+4442*v**10*x**5*y**4*z+1965*v**10*y**3*z**3-8906*v**6*x*y**4*z**5
+5552*x**10*y**4+3055*v**5*x**3*y**6*z**2+6658*v**7*x**10*z**6
+3721*v**8*x**9*y**4*z**8+9511*v*x**6*y+5437*v**3*x**9*y**9*z**7
-1957*v**6*x**4*y*z**3+9214*v**3*x**9*y**3*z**7
+7273*v**2*x**8*y**4*z**10+1701*x**10*y**7*z**2
+4944*v**5*x**5*y**8*z**8-1935*v**3*x**6*y**10*z**7
+4029*x**6*y**10*z**3+9462*v**6*x**5*y**4*z**8-3633*v**4*x*y**7*z**5
-1876,
-5830*v**7*x**8*y*z**2-1217*v**8*x*y**2*z**5
-1510*v**9*x**3*y**10*z**10+7036*v**6*x**8*y**3*z**3
+1022*v**9*y**3*z**8+3791*v**8*x**3*y**7+6906*v**6*x*y*z**10
+117*v**7*x**2*y**4*z**4+6654*v**6*x**5*y**2*z**3
-7302*v**10*x**8*y**3-5343*v**8*x**5*y**9*z
-2244*v**9*x**3*y**8*z**9-3719*v**5*x**10*y**6*z**8
+2629*x**3*y**2*z**10+8517*x**9*y**6*z**7-9551*v**5*x**6*y**6*z**2
-7750*x**10*y**7*z**4-5035*v**5*x**2*y**5*z-5967*v**9*x**5*y**9*z**5
-8517*v**3*x**2*y**7*z**6-2668*v**10*y**9*z**4+1630*v**5*x**5*y*z**8
+9099*v**7*x**9*y**4*z**3-5358*v**9*x**5*y**6*z**2
+5766*v**5*y**3*z**4-3624*v*x**4*y**10*z**10
+8839*v**6*x**9*y**10*z**4+3378*x**7*y**2*z**5+7582*v**7*x*y**8*z**7
-85*v*x**2*y**9*z**6-9495*v**9*x**10*y**6*z**3+1983*v**9*x**3*y
-4613*v**10*x**4*y**7*z**6+5529*v**10*x*y**6
+5030*v**4*x**5*y**4*z**9-9202*x**6*y**3*z**9
-4988*v**2*x**2*y**10*z**4-8572*v**9*x**7*y**10*z**10
+4080*v**4*x**8*z**8-382*v**9*x**9*y**2*z**2-7326);
1
end;
5: 5:
Time: 14790 ms plus GC time: 731 ms
6: 6:
Quitting
Sat May 30 16:10:44 PDT 1992