数的运算测试实例.txt

由中科院数学机械化研究所开发的 数学机械化平台（Dos平台运行）能进行各类复杂的数学运算。包括数值运算和多项式运算等。 挺不错的

************************
****Groebner basis 
*************************

gb([
x3+x2+x1,
x3*x2+x2*x1+x1*x3,
x3*x2*x1-1]);

gbs([
x3+x2+x1,
x3*x2+x2*x1+x1*x3,
x3*x2*x1-1]);

gb([
x3+x2+x1,
x3*x2+x2*x1+x1*x3,
x3*x2*x1-1],1);

gbs([
x3+x2+x1,
x3*x2+x2*x1+x1*x3,
x3*x2*x1-1],1);

gbs([
x4+x3+x2+x1,
x4*x3+x3*x2+x2*x1+x1*x4,
x4*x3*x2+x3*x2*x1+x2*x1*x4+x1*x3*x4,
x4*x3*x2*x1-1]);

gbs([
x4+x3+x2+x1,
x4*x3+x3*x2+x2*x1+x1*x4,
x4*x3*x2+x3*x2*x1+x2*x1*x4+x1*x3*x4,
x4*x3*x2*x1-1],1);

***************************
**** Geometry package
***************************

** Example 1: natrual languages

geomtocs
geomtopd
ndg
wderive
wprove
geom("Example Centroid. Let ABC be a triangle. D is the midpoint of BC. 
E is the midpoint of AC. G is the intersection of line DA and line EB.  
F is the intersection of lines CG and AB. Show that F is the midpoint of AB.");

geom
geomtocs
geomtopd
ndg
wderive
wprove
("Example Simson. Let D be a point on the circumcircle O of the triangle ABC. 
E is the foot from point D to line AB. F is the foot from point D to line BC. 
G is the foot from point D to line AC. Show that points E, F, and G are collinear.");

geom
geomtocs
geomtopd
ndg
wderive
wprove
("Example Gauss. A1A2A3A4 is a quadrilateral. M1 is the midpoint 
of A1A3.  M2 is the midpoint of A2A4. X is the intersection of the diagonals 
A1A2 and A3A4.  Y is the intersection of the diagonals A1A4 and A2A3. 
M3 is the midpoint of XY.  Show that M1, M2, and M3 are collinear.");


** example 2: constructive form
geom([
geomtopd([
geomtocs([
wprove([
ndg([
wprove([
[[POINT,A,B,C],
 [MIDPOINT,D,B,C],
 [MIDPOINT,E,A,C],
 [INTER,G,[LINE,A,D],[LINE,B,E]],
 [INTER,F,[LINE,A,B],[LINE,C,G]]],
[[MIDPOINT,F,A,B]]]);

geom([
geomtopd([
geomtocs([
wprove([
ndg([
wprove([
[[POINT,A,B,A1,B1],
 [ON,C,[LINE,A,B]],
 [ON,C1,[LINE,A1,B1]],
 [INTER,P,[LINE,A,B1],[LINE,A1,B]],
 [INTER,Q,[LINE,A,C1],[LINE,A1,C]],
 [INTER,R,[LINE,B,C1],[LINE,B1,C]]],
[[coll,P,Q,R]]]);


** Example 3: preducate form

geom([
geomtopd([
geomtocs([
wprove([
ndg([
wprove([
[y_F,x_F,y_G,x_G,y_E,x_E,y_D,x_D,v_C,u_C,v_B,u_B,v_A,u_A],[],
[A,[0,0],B,[0,v_B],C,[u_C,v_C],D,[x_D,y_D],E,[x_E,y_E],G,[x_G,y_G],F,[x_F,y_F]],
[[POINT,A,B,C],
[MIDPOINT,D,B,C],[MIDPOINT,E,A,C],
[coll,G,A,D],[coll,G,B,E],
[coll,F,A,B],[coll,F,C,G]],
[[para,A,D,B,E],[para,A,B,C,G]],
[[MIDPOINT,F,A,B]]]);

** Example 4: Algebraic form
wprove(
[[y_F,x_F,y_G,x_G,y_E,x_E,y_D,x_D,v_C,u_C,v_B,u_B,v_A,u_A],[],
 [2*x_D-u_C,2*y_D-v_C-v_B,2*x_E-u_C,2*y_E-v_C,
  x_D*y_G-y_D*x_G,
  x_E*y_G-y_E*x_G+v_B*x_G-v_B*x_E,
  -v_B*x_F,
  x_G*y_F-u_C*y_F-y_G*x_F+v_C*x_F+u_C*y_G-v_C*x_G],
[x_D*y_E-y_D*x_E-v_B*x_D,-v_B*x_G+v_B*u_C],
[2*x_F,2*y_F-v_B]]);

** example 5: wderive
wderive(
[[x2,x1,k],[a,b,c],
 [b1,[0,0], c1,[a,0], a1, [x1,x2]],
 [[dis, a1, c1, b],
 [dis, a1, b1, c],
 [area, k, a1, b1, c1]],
 [],
 []]);


Example 6: Wronskian

## (x,y,z) is on a plane
depend([x,y,z],[t]);
wronskian([1,x,y,z],t);
> z[3]*y[2]*x[1]-y[3]*z[2]*x[1]-z[3]*y[1]*x[2]+y[3]*z[1]*x[2]+x[3]*z[2]*y[1]-x[3]*z[1]*y[2]

## (x,y) is on a circle
wronskian([1,x,y,x^2+y^2],t);

## (x,y,z) is on a sphere
wronskian([1,x,y,z,x^2+y^2+z^2],t);

## x*y^2 is on a constant
wronskian([1,x*y^2],t);


Example 7. wdprove.

## Kepler's laws imply Newton'w law

depend([a,r,y,x],[t]);
wdprove(
[[a,r,y,x,p,e],[],[],
[r^2-x^2-y^2,
 a^2-diff(x,t,2)^2-diff(y,t,2)^2,
 x* diff(y,t,2) - diff(x,t,2)*y,
 r - p - e * x],
[p],
[diff(a*r^2,t)]]);

depend([a,r,x,y],[t]);
wdprove(
[[a,r,y,x],[],[],
[r^2-x^2-y^2,
 a^2-diff(x,t,2)^2-diff(y,t,2)^2,
 x* diff(y,t,2) - diff(x,t,2)*y,
 diff(a*r^2,t)],
[a],
[wronskian([1,x,y,r],t)]]);

Example 9. prove_curve

##  C is a curve iff t=k'=0.
curve();
wprove_curve([[],[],[],[z,[XY_CIRCLE,C]],[],[t,diff(k,s)]]);
curve();
wprove_curve([[],[],[],[t,diff(k,s)],[],[[FIX_PLANE,C],[FIX_SPHERE,C]]]);



######################################################################3

######################################################################
#特征列和零点分解范例
######################################################################3


   
# cycle problem for n=4;
     
f1:=x1+x2+x3+x4;
f2:=x1*x2+x2*x3+x3*x4+x4*x1;
f3:=x1*x2*x3+x2*x3*x4+x3*x4*x1+x4*x1*x2;
f4:=x1*x2*x3*x4-1;
ps1:=[f1,f2,f3,f4];

ord1:=[x4,x3,x2,x1];
cs1:=wsolve(ps1,ord1);




#ps2 is the cycle problem for n=5.

f1:=x1+x2+x3+x4+x5;
f2:=x1*x2+x2*x3+x3*x4+x4*x5+x5*x1;
f3:=x1*x2*x3+x2*x3*x4+x3*x4*x5+x4*x5*x1+x5*x1*x2;
f4:=x1*x2*x3*x4+x2*x3*x4*x5+x3*x4*x5*x1+x4*x5*x1*x2+x5*x1*x2*x3;
f5:=x1*x2*x3*x4*x5-1;
ps2:=[f1,f2,f3,f4,f5];
ord2:=[x5,x4,x3,x2,x1];


cs2:=wsolve(ps2,ord2);


     



#ps3 is the neural network problem.
p1:=1-c*x+x*y^2+x*z^2;
p2:=1-c*y+y*x^2+y*z^2;
p3:=1-c*z+z*x^2+z*y^2;
ps3:=[p1,p2,p3];
ord3:=[z,y,x,c];
nzero3:=[];

cs3:=wsolve(ps3,ord3,nzero3,"ritt");



#K(a,b)


p1:=2*y^2*(y^2+x^2)+(b^2-3*a^2)*y^2-2*b*y^2*(y+x)+2*a^2*b*(y+x)-a^2*x^2+a^2*(a^2-b^2);
p2:=4*y^3+4*y*(y^2+x^2)-2*b*y^2-4*b*y*(y+x)+2*(b^2-3*a^2)*y+2*a^2*b;
p3:=4*x*y^2-2*b*y^2-2*a^2*x+2*a^2*b;
ps4:=[p1,p2,p3];
ord4:=[y,x];

cs4:=wsolve(ps4,ord4);






#thesis p109;


p1:=x^2*y*z+x*y^2*z+x*y*z^2+x*y*z+x*y+x*z+z*x;
p2:=x^2*y^2*z+x*y^2*z^2+x^2*y*z+x*y*z+y*z+z+x;
p3:=x^2*y^2*z^2+x^2*y^2*z+x^2*y^2*z+x*y^2*z+x*y*z+x*z+z+1;
ps5:=[p1,p2,p3];
ord5:=[z,y,x];

cs5:=wsolve(ps5,ord5,[],"ritt");




#Geodesy K(c1,c2,c3,c4) p102;


p1:=x1+b^2*x1*x4-c1;
p2:=x2+b^2*x2*x4-c2;
p3:=x3+a^2*x3*x4-c3;
p4:=b^2*x1^2+b^2*x2^2+a^2*x3^2-a^2*b^2;
ps6:=[p1,p2,p3,p4];
ord6:=[x4,x3,x2,x1,b,a];
cs6:=wsolve(ps6,ord6);








#gao
p1:=x4-a4+a2;
p2:=x4+x3+x2+x1-a4-a3-a1;
p3:=x3*x4+x1*x4+x2*x3+x1*x3+(-a3-a1)*a4-a1*a3;
p4:=x1*x3*x4-a1*a3*a4;
ps7:=[p1,p2,p3,p4];
ord7:=[x4,x3,x2,x1];

cs7:=wsolve(ps7,ord7);


















######################################################################
#投影算法范例
######################################################################3




psds:=[[2*u1-x2*x5+x3*x4, 2*u2+x5*x1, 2*u3-x1*x3, 2*u4-(x2-x1)*x5+x3*x4-x1*x3], []];
ord:=[x5, x4, x3, u4];
pord:=[x5, x4, x3];
project(psds[1],psds[2],ord,pord);


psds:=[[x5*x4-x3*x6, x6*(x7-x2)-x8*(x5-x2), x4*(x7-x1)-x8*(x3-x1), u1*(x2-x1)+x2, u2*(x7-x3)-x1+x7, u3*x5-x3+x5], []];
ord:=[x8, x7, x6, x5, x2, u3];
pord:=[x8, x7, x6, x5, x2];
project(psds[1],psds[2],ord,pord);


psds:=[[x2*x8-x9*u4, u3*(x4-u1)-x5*(u2-u1), (x7-u3)*(u4-u2)-(x2-u3)*(x6-u2), x7*(x4-x3)-x5*(-x3+x6), x9*(x4-x3)-x5*(x8-x3), r1*(x3-u1)+x3, r2*(x4-u2)-u1+x4, r3*(x6-u4)-u2+x6, r4*x8-u4+x8], []];
ord:=[x9, x8, x7, x6, x5, x4, x3, x2, r4];
pord:=[x9, x8, x7, x6, x5, x4, x3, x2];
project(psds[1],psds[2],ord,pord);


psds:=[[y2*(z1-x1)-w1*(x2-x1), (y3-y2)*(z2-x2)-(w2-y2)*(x3-x2), (y4-y3)*(z3-x3)+y3*(x4-x3), (y5-y4)*(z4-x4)-(w4-y4)*(x5-x4), -y5*(z5-x5)-(w5-y5)*(x1-x5), w4*x2-y2*z4, w5*x3-y3*z5, w1*x4-y4*z1, w2*x5-y5*z2, u1*(z1-x2)-x1+z1, u2*(z2-x3)-x2+z2, u3*(z3-x4)-x3+z3, u4*(z4-x5)-x4+z4, u5*(z5-x1)-x5+z5], [x5-x4]];
ord:=[w5, z5, w4, z4, z3, w2, z2, w1, z1, y5, x5, y4, y3, u5];
pord:=[w5, z5, w4, z4, z3, w2, z2, w1, z1, y5, x5, y4, y3];
project(psds[1],psds[2],ord,pord);


psds:=[[x2^2-x4^2-x3^2, x6^2+x5^2-x4^2-x3^2, (x3-x1)*x6-x4*(x5-x1), x7*x1-x2^2, x4*(x7-x1)-x8*(x3-x1), (x5-x1)*(x7-x3)*u1-(x3-x1)*(x7-x5)], [x5-x3]];
ord:=[x8, x7, x6, x5, x4, u1];
pord:=[x8, x7, x6, x5, x4];
project(psds[1],psds[2],ord,pord);


psds:=[[r^2-x1^2-y1^2, (x2-x1)^2+(y2-y1)^2-n^2-m^2, (x2-x)^2+(y2-y)^2-n^2-m^2, (r-x2)^2+y2^2-(n+2*r)^2-m^2, y*(x1-r)-y1*(x-r)], [x1-x]];
ord:=[y2, x2, y1, x1, x];
pord:=[y2, x2, y1, x1];
project(psds[1],psds[2],ord,pord);


psds:=[[u1^2-x1^2-x2^2, u2^2-(x1-x3)^2-(x2-x4)^2, u2^2-(x3-u1)^2-x4^2, 2*x-x1-x3, 2*y-x2-x4], [x1-u1]];
ord:=[x4, x3, x2, x1, y];
pord:=[x4, x3, x2, x1];
project(psds[1],psds[2],ord,pord);


psds:=[[x1+u2, u1^2+u2^2-(u1-x3)^2-x2^2, 2*x4-u2-x2, 2*x5-x3, -y*(x2-x)+(x3-y)*x, -y*(x4-x)-(x5-y)*(x1-x)], []];
ord:=[x5, x4, x3, x2, x1, y];
pord:=[x5, x4, x3, x2, x1];
project(psds[1],psds[2],ord,pord);


psds:=[[x1+u2, u1^2+u2^2-(u1-x2)^2-x^2, x2*y+(x-x1)*(x-u2)], []];
ord:=[x2, x1, y];
pord:=[x2, x1];
project(psds[1],psds[2],ord,pord);


psds:=[[r^2-x3^2-x4^2, x3^2-(x3-x1)^2, (x4-u1)^2+(x3-x2)^2-x4^2-x3^2, u2^2-x2^2-u1^2, u3^2-(x1-x2)^2-u1^2], [x1]];
ord:=[x4, x3, x2, x1, r];
pord:=[x4, x3, x2, x1];
project(psds[1],psds[2],ord,pord);





######################################################################
############矩阵行列式的一些测试例子：
######################################################################



example:a:=matrix(3,3,[1,2,3,4,5,8,0,0,0]);
	det(a);
	0

example:a:=fmatrix(4,4,[32,42,11,32,121,18,43,6,4,12,17,62,3,42,12,44]);
	det(a);
	1983024

example:a:=matrix(3,3,[2*x,43*y^2,x*y,0,11*(x-2)*y^2,5*y,0,0,3*z^4]);
	det(a);
	66*z^4*y^2*x^2-132*z^4*y^2*x

example:a:=fmatrix(3,3,[x^2,y-3,z^2*x-y^2,12,x^5*y,y*z,3*z,0,y^4*x]);
	det(a);
	x^8*y^5-12*y^5*x+36*y^4*x+3*y^2*z^2-9*y*z^2-3*x^6*y*z^3+3*z*x^5*y^3

example:a:=fmatrix(3,3,[x,y,z,2*x,y,0,0,2*x,y]);
	det(a);
	4*z*x^2-y^2*x

example:a:=fmatrix(5,5,[11,1315,3123,12,43,65,4324,65,4,4324,54654,7657,8768,3543,343,5435,654,654,654,4654,4564,4564,987970,3123,4433]);
	det(a);
	-92410140159906731807

example:a:=fmatrix(4,4,[3*x^3,y^2-3,z^2*x-y^2,12*y,x^5*y,y*z,3*z,5*x^4-5,y^4*x,x2*z,7*y^3-x*z,z^2-3,y*z^3,4*x^2,123*x*y,65*y^3*z*4]);
	det(a);
-369*x^6*y^4+5460*x^3*y^7*z^2-1476*x^6*y^3*x2*z-260*x^5*y^6*x2*z^2+260*x^6*y^4*x2*z^4-1845*x^4*x2*z*y+1845*x^8*x2*z*y-2340*x^3*x2*z^3*y^3+5*y^3*z^4*x2*x^4+5*y*z^6*x2*x-5*y*z^6*x2*x^5-369*x^4*y^2*z^3-885*x^4*y^4*z^3+1107*x^4*y^2*z+260*x^6*y^6*z^2-1820*x^5*y^9*z-369*x^6*y^2*z^2+5460*x^5*y^7*z-657*x^6*y^4*z^2-48*x^8*y^2*z+4*x^7*y^3*z^2+12*x^8*y*z^2-4*x^8*y*z^4-144*y^5*x^3*z-20*y^4*x^4*z^2+20*y^4*x^8*z^2+1476*y^7*x^2*z+260*y^10*x*z^2-260*y^8*x^2*z^4-2340*y^7*x*z^2+780*y^9*x*z^2+y^2*z^8*x-15*y*z^4*x+15*y*z^4*x^5+5*y^3*z^4*x-5*y^3*z^4*x^5+35*y^6*z^3*x^4+36*y^2*z^5*x2-5*y^3*z^4*x2+12*y^3*z^5*x-3*y^2*z^6*x-108*x^5*z+1107*x^6*y^2+420*y^3*x^5-27*y*z^4+9*y*z^6-35*y^6*z^3+9*y^3*z^4-3*y^3*z^6+20*y^6*x^3-20*y^6*x^7-1845*y^5*x^2+1845*y^5*x^6+615*y^7*x^2-615*y^7*x^6+336*x^7*y^5-12*x^7*y^3-60*x^6*z+60*x^10*z-420*x^9*y^3+36*x^5*z^3-84*y^6*z^4+3*y^4*z^4-y^4*z^6+105*y^4*z^3