solving simultaneous equations trouble using solve()

aurelius_nero
85 ●2 ●2 ●10

Hi, I am having a lot of difficulty solving the following set of equations using solve. I am gonna insert my code below:

  sage:var('x1,x2,x3,w1,w2,w3')
  sage:f=w1+w2+w3==2;g=(w1*x1)+(w2*x2)+(w3*x3)==0;h=(w1*(x1)^2)+(w2*(x2)^2)+(w3*(x3)^2)==(2/3);j=(w1*(x1)^3)+(w2*(x2)^3)+(w3*(x3)^3)==0;k=(w1*(x1)^4)+(w2*(x2)^4)+(w3*(x3)^4)==(2/5);l=(w1*(x1)^5)+(w2*(x2)^5)+(w3*(x3)^5)==0
  sage: solve([f,g,h,j,k,l],w1,w2,w3,x1,x2,x3,solution_dict=True)

The output is:

 [{w1 + w2 + w3: 2},
 {w1*x1 + w2*x2 + w3*x3: 0},
 {w1*x1^2 + w2*x2^2 + w3*x3^2: 2/3},
 {w1*x1^3 + w2*x2^3 + w3*x3^3: 0},
 {w1*x1^4 + w2*x2^4 + w3*x3^4: 2/5},
 {w1*x1^5 + w2*x2^5 + w3*x3^5: 0}]

obviously this output doesn't help me as I need the solution for all six variables. Any help is appreciated please.

Comments

For systems of polynomial equations, do not use the symbolic ring, but ideals in polynomial rings. See https://doc.sagemath.org/html/en/refe...

FrédéricC ( 3 years ago )

@FrédéricC I am not sure how to do so. Could you elaborate a little more ?

aurelius_nero ( 3 years ago )

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answered 3 years ago

FrédéricC

5426 ●4 ●44 ●121

Here is a minimal illustrating example:

sage: R = PolynomialRing(QQbar,'x,y')
sage: x, y = R.gens()
sage: I = R.ideal([x*x+y*y-1, x-5*y])
sage: I.variety()
[{y: -0.1961161351381841?, x: -0.9805806756909202?},
 {y: 0.1961161351381841?, x: 0.9805806756909202?}]

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How do I even use any of this ? 1) what is the R.gens() 2) what is the R.ideal ? 3) what is I.variety ? 4) isn't there a simpler way to solve polynomials ?

aurelius_nero ( 3 years ago )

The way suggested by FredericCis the simplest way to tackle general polynomial systems in Sage (and in general...).

See chapters 7 and 9 of this book for an introduction to the subject. Chapter 9 points to a textbook, which turns out to be easily Googleable,

BTW, I'd recommend reading the whole Sage book (and a good Python tutorial) to gain a deeper understanding of Sage...

Emmanuel Charpentier ( 3 years ago )

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answered 3 years ago

aurelius_nero
85 ●2 ●2 ●10

For all those interested I have manged to do what I wanted using sympy.

 sage: var('x1,x2,x3,w1,w2,w3')
 sage: f=w1+w2+w3==2;g=(w1*x1)+(w2*x2)+(w3*x3)==0;h=(w1*(x1)^2)+(w2*(x2)^2)+(w3*(x3)^2)==(2/3);j=(w1*(x1)^3)+(w2*(x2)^3)+(w3*(x3)^3)==0;k=(w1*(x1)^4)+(w2*(x2)^4)+(w3*(x3)^4)==(2/5);l=(w1*(x1)^5)+(w2*(x2)^5)+(w3*(x3)^5)==0
 sage: import sympy
 sage sympy.solve([f,g,h,j,k,l],w1,w2,w3,x1,x2,x3)

output:

  [{w1: 5/9, w2: 5/9, w3: 8/9, x1: -sqrt(15)/5, x2: sqrt(15)/5, x3: 0},
  {w1: 5/9, w2: 5/9, w3: 8/9, x1: sqrt(15)/5, x2: -sqrt(15)/5, x3: 0},
  {w1: 5/9, w2: 8/9, w3: 5/9, x1: -sqrt(15)/5, x2: 0, x3: sqrt(15)/5},
  {w1: 5/9, w2: 8/9, w3: 5/9, x1: sqrt(15)/5, x2: 0, x3: -sqrt(15)/5},
  {w1: 8/9, w2: 5/9, w3: 5/9, x1: 0, x2: -sqrt(15)/5, x3: sqrt(15)/5},
  {w1: 8/9, w2: 5/9, w3: 5/9, x1: 0, x2: sqrt(15)/5, x3: -sqrt(15)/5}]

If anyone can enlighten me on how to use the polynomial rings to solve polynomial systems of equations I'd be very happy (I am also a noob at sagemath btw)

Thank you.

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