Analytic solution of quatric polynomial does not add up
Dear friends,
I currently ran into a math-issue when trying to get the analytical roots of a polynomial of order 4.
What I want to solve is:
The solution shall run on a microcontroller with dynamically changing parameters [a,d,e]. All parameters are real. I am searching for the real results, imaginary solutions are to be prevented. When applying Matlab symbolic, The parameters [a,d,e] are simplified and represent longer sub-terms. In accordance to the article Wiki: quartic function, I will receive quite a long solution, that is futile to reproduce here. I used the following short code to generate it:
syms x a d e
y= a*x^4+d*x+e
sol=solve(y==0,x,’MaxDegree’,4,’ReturnConditions’,true)
To check the solution, I tested it on an easier sub-problem: When d=0 is chosen follows:
When chosing a>0, e<0 this clearly yields some real results, besides the imaginary ones. Now the problem: the above given long solution cannot reproduce this result. Even if I force assumptions that should solve it, follows something that does not look equal:
syms x a d e
assume(a>=0)
assume(d>=0)
assume(e<=0)
y= a*x^4+d*x+e
sol=solve(y==0,x,’MaxDegree’,4,’ReturnConditions’,true)
testresult=simplify(subs(sol.x, d,0))
latex(testresult)
However, if I change the equation before solving the quartic function, the solution of it looks as on paper:
y= a*x^4+e
assume(a>=0)
assume(d>=0)
assume(e<=0)
sol=solve(y==0,x,’MaxDegree’,4,’ReturnConditions’,true)
testresult2=sol.x
latex(testresult2)
Numerical testing via inserting numbers in the two solutions according to the assumptions a>0, e<0 confirms the difference of the results. Chosing a=1 and e=-2 yields:
1) the general solution of the quartic polynomial (inserting d=0 after solving the quartic polynomial):
vpares1=vpa(subs(testresult,{a,e},{1,-2}))
vpares1 =
0
0
0
0
2) the solution of the simplified quartic polynomial (inserting d=0 before solving):
vpares2=vpa(subs(testresult2,{a,e},{1,-2}))
vpares2 =
1.1892071150027210667174999705605
-1.1892071150027210667174999705605
-1.1892071150027210667174999705605i
1.1892071150027210667174999705605i
Visibly, the trivial solution is not permissible given the solution of case 2). Interestingly, if the requirement is switched from {a>0,e<0} to {a<0, e>0} the solutions match again.
Similar difficulties have shown, when I hand-coded the general solution from the mentioned Wikipedia article. As I need the solution for , I now have trust-issues with the long solution. Therefore, I ask for your help or comment, why the "general solution" does not end in the same result as the simplified problem.
Thank you in advance for your help!Dear friends,
I currently ran into a math-issue when trying to get the analytical roots of a polynomial of order 4.
What I want to solve is:
The solution shall run on a microcontroller with dynamically changing parameters [a,d,e]. All parameters are real. I am searching for the real results, imaginary solutions are to be prevented. When applying Matlab symbolic, The parameters [a,d,e] are simplified and represent longer sub-terms. In accordance to the article Wiki: quartic function, I will receive quite a long solution, that is futile to reproduce here. I used the following short code to generate it:
syms x a d e
y= a*x^4+d*x+e
sol=solve(y==0,x,’MaxDegree’,4,’ReturnConditions’,true)
To check the solution, I tested it on an easier sub-problem: When d=0 is chosen follows:
When chosing a>0, e<0 this clearly yields some real results, besides the imaginary ones. Now the problem: the above given long solution cannot reproduce this result. Even if I force assumptions that should solve it, follows something that does not look equal:
syms x a d e
assume(a>=0)
assume(d>=0)
assume(e<=0)
y= a*x^4+d*x+e
sol=solve(y==0,x,’MaxDegree’,4,’ReturnConditions’,true)
testresult=simplify(subs(sol.x, d,0))
latex(testresult)
However, if I change the equation before solving the quartic function, the solution of it looks as on paper:
y= a*x^4+e
assume(a>=0)
assume(d>=0)
assume(e<=0)
sol=solve(y==0,x,’MaxDegree’,4,’ReturnConditions’,true)
testresult2=sol.x
latex(testresult2)
Numerical testing via inserting numbers in the two solutions according to the assumptions a>0, e<0 confirms the difference of the results. Chosing a=1 and e=-2 yields:
1) the general solution of the quartic polynomial (inserting d=0 after solving the quartic polynomial):
vpares1=vpa(subs(testresult,{a,e},{1,-2}))
vpares1 =
0
0
0
0
2) the solution of the simplified quartic polynomial (inserting d=0 before solving):
vpares2=vpa(subs(testresult2,{a,e},{1,-2}))
vpares2 =
1.1892071150027210667174999705605
-1.1892071150027210667174999705605
-1.1892071150027210667174999705605i
1.1892071150027210667174999705605i
Visibly, the trivial solution is not permissible given the solution of case 2). Interestingly, if the requirement is switched from {a>0,e<0} to {a<0, e>0} the solutions match again.
Similar difficulties have shown, when I hand-coded the general solution from the mentioned Wikipedia article. As I need the solution for , I now have trust-issues with the long solution. Therefore, I ask for your help or comment, why the "general solution" does not end in the same result as the simplified problem.
Thank you in advance for your help! Dear friends,
I currently ran into a math-issue when trying to get the analytical roots of a polynomial of order 4.
What I want to solve is:
The solution shall run on a microcontroller with dynamically changing parameters [a,d,e]. All parameters are real. I am searching for the real results, imaginary solutions are to be prevented. When applying Matlab symbolic, The parameters [a,d,e] are simplified and represent longer sub-terms. In accordance to the article Wiki: quartic function, I will receive quite a long solution, that is futile to reproduce here. I used the following short code to generate it:
syms x a d e
y= a*x^4+d*x+e
sol=solve(y==0,x,’MaxDegree’,4,’ReturnConditions’,true)
To check the solution, I tested it on an easier sub-problem: When d=0 is chosen follows:
When chosing a>0, e<0 this clearly yields some real results, besides the imaginary ones. Now the problem: the above given long solution cannot reproduce this result. Even if I force assumptions that should solve it, follows something that does not look equal:
syms x a d e
assume(a>=0)
assume(d>=0)
assume(e<=0)
y= a*x^4+d*x+e
sol=solve(y==0,x,’MaxDegree’,4,’ReturnConditions’,true)
testresult=simplify(subs(sol.x, d,0))
latex(testresult)
However, if I change the equation before solving the quartic function, the solution of it looks as on paper:
y= a*x^4+e
assume(a>=0)
assume(d>=0)
assume(e<=0)
sol=solve(y==0,x,’MaxDegree’,4,’ReturnConditions’,true)
testresult2=sol.x
latex(testresult2)
Numerical testing via inserting numbers in the two solutions according to the assumptions a>0, e<0 confirms the difference of the results. Chosing a=1 and e=-2 yields:
1) the general solution of the quartic polynomial (inserting d=0 after solving the quartic polynomial):
vpares1=vpa(subs(testresult,{a,e},{1,-2}))
vpares1 =
0
0
0
0
2) the solution of the simplified quartic polynomial (inserting d=0 before solving):
vpares2=vpa(subs(testresult2,{a,e},{1,-2}))
vpares2 =
1.1892071150027210667174999705605
-1.1892071150027210667174999705605
-1.1892071150027210667174999705605i
1.1892071150027210667174999705605i
Visibly, the trivial solution is not permissible given the solution of case 2). Interestingly, if the requirement is switched from {a>0,e<0} to {a<0, e>0} the solutions match again.
Similar difficulties have shown, when I hand-coded the general solution from the mentioned Wikipedia article. As I need the solution for , I now have trust-issues with the long solution. Therefore, I ask for your help or comment, why the "general solution" does not end in the same result as the simplified problem.
Thank you in advance for your help! mathematics, symbolic, 4th order polynomial, polynomial MATLAB Answers — New Questions
