Le jeudi 7 mai 2015 02:52:31 UTC+2, Waldek Hebisch a écrit : > > Souleiman omar hoche wrote: > > > > Hi i want to ask if it is possible to solve the sytem of equations > below > > with fricas: > > > > eq1:= po^2+2*p1*p4+2*p2*p3+qo^2+q1^2+q2^2+q3^2+q4^2-po=0; > > > > eq2:=2*po*p1 +2*p2*p4+p3^2+qo*q4+q1*qo+q2*q1+q3*q2+q4*q3-p1=0; > > > > eq3:= 2*po*p2+p1^2+2*p3*p4+qo*q3+q1*q4+q2*qo+q3*q1+q4*q2-p2=0; > > > > eq4:=2*po*p3+2*p1*p2+p4^2+qo*q2+q1*q3+q2*q4+q3*q0+q4*q1-p3=0; > > ^^ > Do you mean q0 or maybe qo here? > > > > > eq5:=2*po*p4+2*p1*p3+p2^2+qo*q1+q1*q2+q2*q3+q3*q4+q4*qo-p4=0; > > > > eq6:=2*po*qo+p1*q4+p2*q3+p3*q2+p4*q1+q1*p1+q2*p2+q3*p3+q4*p4-qo=0; > > > > eq7:=2*po*q1+p1*qo+p2*q4+p3*q3+p4*q2+qo*p4+q2*p1+q3*p2+q4*p3-q1=0; > > > > eq8:=2*po*q2+p1*q1+p2*qo+p3*q4+p4*q3+qo*p3+q1*p4+q3*p1+q4*p2-q2=0; > > > > eq9:=2*po*q3+p1*q2+p2*q1+p3*qo+p4*q4+qo*p2+q1*p3+q2*p4+q4*p1-q3=0; > > > > eq10:=2*po*q4+p1*q3+p2*q2+p3*q1+p4*qo+qo*p1+q1*p2+q2*p3+q3*p4-q4=0; > > > > I tried to do it with the function *radicalSolve* and *solve*, and i got > > the message: "*Error detected within library code: system does not have > a > > finite number of solutions*". > > > > I want to precise that i first tried to do it with *matlab *and *maple > *and > > i got two different possible solution one with matlab (po=1/10; > p1=1/10; > > p2=1/10; p3=1/10; p4=1/10; qo=1/10; q1=1/10; q2=1/10; q3=1/10; > > q4=1/10) and the other one with maple > > > > (p1 = 0.03344940640, p2 = -0.06720617766, p3 = 0.06720617766, p4 = > > -0.03344940640, po = 0.5000000000, q1 = -0.04300931701, q2 = > -0.3199584031, > > q3 = -0.2454933756, q4 = 0.2674194114, qo = -0.1589583157). > > > > Also another possible solution (po=1/10; p1=1/10; p2=1/10; p3=1/10; > > p4=1/10; qo=-1/10; q1=-1/10; q2=-1/10; q3=-1/10; q4=-1/10). > > > > > > I need all the solution (if it is possible). Do you think that fricas > can > > help for this? If yes, wich command or function do you advice me? > > With q0 in eq4 values above fail to solve the system, so I assume > you want qo. Using: > > leq := [eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8, eq9, eq10]; > lp := [lhs(eq) for eq in leq] > vars := [p1, p2, p3, p4, po, q1, q2, q3, q4, qo] > hd := HDMP(vars, Integer) > lp1 := [eq::hd for eq in lp] > gbf := groebnerFactorize(lp1) > > I get 119 simpler systems. Solutions to your original system > are union of solutions of those 119 systems. Some systems > directy give solutions, for example: > > [10p1 + 1, 10p2 + 1, 10p3 + 1, 10p4 + 1, 10po - 9, 10q1 + 1, 10q2 + 1, > 10q3 + 1, 10q4 + 1, 10qo + 1] > , > > [10p1 - 1, 10p2 - 1, 10p3 - 1, 10p4 - 1, 10po - 1, 10q1 + 1, 10q2 + > 1, > 10q3 + 1, 10q4 + 1, 10qo + 1] > , > [10p1 + 1, 10p2 + 1, 10p3 + 1, 10p4 + 1, 10po - 9, 10q1 - 1, 10q2 - > 1, > 10q3 - 1, 10q4 - 1, 10qo - 1] > , > > [10p1 - 1, 10p2 - 1, 10p3 - 1, 10p4 - 1, 10po - 1, 10q1 - 1, 10q2 - > 1, > 10q3 - 1, 10q4 - 1, 10qo - 1] > , > [10p1 + 1, 10p2 + 1, 10p3 + 1, 10p4 + 1, 5po - 2, 5q1 + 2, 10q2 - 1, > 10q3 - 1, 10q4 - 1, 10qo - 1] > , > > [10p1 - 1, 10p2 - 1, 10p3 - 1, 10p4 - 1, 5po - 3, 5q1 + 2, 10q2 - 1, > 10q3 - 1, 10q4 - 1, 10qo - 1] > , > [p1,p2,p3,p4,po,q1,q2,q3,q4,qo], [p1,p2,p3,p4,po - 1,q1,q2,q3,q4,qo] > > give 8 rational solutions. Other systems, like: > > 2 2 > [5p4 - 4q2 + 2q2,p1 + p4,2p2 - p4,2p3 + p4,2po - 1,2q1 + 2q2 - > 1,q3,q4,qo] > > > almost directly give 1 dimensional family of solutions (the first > polynomial > give a curve in (p4, q2) plane and other equations are linear and have > unique solution for any (p4, q2)). Other system require more work > to solve, but typically some variables are determined so should > be easier than your original system. Most seem to have infinite > number of solutions and FriCAS is of limited help with them. > OTOH, I do not know in which form you would like to see solutions: > in some cases solution set has nice parametrization, but this is > not always the case. BTW, there is system consisting of only 7 > equations, so you will get at least 3 dimensional solution set. > > -- > Waldek Hebisch > [email protected] <javascript:> > Thanks for your quick answer.
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