Numerical method to approximate *all* the solutions of of nonlinear algebraic equation
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Suppose we have a nonlinear algebraic equation (by "algebraic" I mean it does not involve derivatives of the unknown) that does not have unique solution, e.g.
$$sin(x) = 0 qquad xin [-10, 10] $$
We could also think of more general problems
$$mathcalN(x) = 0 qquad xinOmega$$
where $mathcalN: Omega rightarrow mathbbR$ is a nonlinear algebraic operator defined on $Omega$, a compact subset of $mathbbR^d, d geq 1$. Also suppose $Omega$ is "well behaved", e.g. is the closure of an open set.
Are there numerical methods to approximate the whole set of solutions of such equations?
In other words, are there methods to approximate the kernel $K(mathcalN) = left xinOmega: mathcalN(x)=0right$ in the case this is not a singleton?
What if, in addiction, the operator $mathcalN$ is a "blackbox" (i.e. no explicit formula for $mathcalN$ or its gradient are available)?
The best I can think about at the moment is to turn the problem into a minimization problem with objective function:
$$vertmathcalN(x)vert qquad xinOmega$$
then run an appropriate optimization algorithm. However this approach is meant to determine one solution rather than a set of solution.
Another aspect of the problem I was considering is the fact that the kernel $K(mathcalN)$ may have connected components (that may not be singletons), however I do not know if/how to exploit this information to approach the problem.
numerical-methods roots nonlinear-system
asked Aug 18 at 8:24
funes
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1
down vote
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Suppose we have a nonlinear algebraic equation (by "algebraic" I mean it does not involve derivatives of the unknown) that does not have unique solution, e.g.
$$sin(x) = 0 qquad xin [-10, 10] $$
We could also think of more general problems
$$mathcalN(x) = 0 qquad xinOmega$$
where $mathcalN: Omega rightarrow mathbbR$ is a nonlinear algebraic operator defined on $Omega$, a compact subset of $mathbbR^d, d geq 1$. Also suppose $Omega$ is "well behaved", e.g. is the closure of an open set.
Are there numerical methods to approximate the whole set of solutions of such equations?
In other words, are there methods to approximate the kernel $K(mathcalN) = left xinOmega: mathcalN(x)=0right$ in the case this is not a singleton?
What if, in addiction, the operator $mathcalN$ is a "blackbox" (i.e. no explicit formula for $mathcalN$ or its gradient are available)?
The best I can think about at the moment is to turn the problem into a minimization problem with objective function:
$$vertmathcalN(x)vert qquad xinOmega$$
then run an appropriate optimization algorithm. However this approach is meant to determine one solution rather than a set of solution.
Another aspect of the problem I was considering is the fact that the kernel $K(mathcalN)$ may have connected components (that may not be singletons), however I do not know if/how to exploit this information to approach the problem.
numerical-methods roots nonlinear-system
asked Aug 18 at 8:24
funes
62
Your example problem has roots $npi$, $nin Bbb Z$. How do you propose to numerically treat/distinguish the roots for $n=10^17$ and $10^17+2$? And differentiate them from all the non-roots in-between? In floating point they get mapped to the same binary approximation.
– LutzL
Aug 19 at 15:23
Good point. The example is not good. I think the problem is interesting enough even if we consider only bounded domains (the size of which should not be problematic to handle with floating point arithmetic). I have edited the question.
– funes
Aug 19 at 18:42
If solutions are isolated, running global optimization algorithms for $minlimits_x in Omega | mathcalN(x) |$ is an okay idea. For example, DIRECT can solve this problem knowing only function values and not using derivatives.
– Evgeny
Aug 20 at 9:50
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose we have a nonlinear algebraic equation (by "algebraic" I mean it does not involve derivatives of the unknown) that does not have unique solution, e.g.
$$sin(x) = 0 qquad xin [-10, 10] $$
We could also think of more general problems
$$mathcalN(x) = 0 qquad xinOmega$$
where $mathcalN: Omega rightarrow mathbbR$ is a nonlinear algebraic operator defined on $Omega$, a compact subset of $mathbbR^d, d geq 1$. Also suppose $Omega$ is "well behaved", e.g. is the closure of an open set.
Are there numerical methods to approximate the whole set of solutions of such equations?
In other words, are there methods to approximate the kernel $K(mathcalN) = left xinOmega: mathcalN(x)=0right$ in the case this is not a singleton?
What if, in addiction, the operator $mathcalN$ is a "blackbox" (i.e. no explicit formula for $mathcalN$ or its gradient are available)?
The best I can think about at the moment is to turn the problem into a minimization problem with objective function:
$$vertmathcalN(x)vert qquad xinOmega$$
then run an appropriate optimization algorithm. However this approach is meant to determine one solution rather than a set of solution.
Another aspect of the problem I was considering is the fact that the kernel $K(mathcalN)$ may have connected components (that may not be singletons), however I do not know if/how to exploit this information to approach the problem.
numerical-methods roots nonlinear-system
asked Aug 18 at 8:24
funes
62
Suppose we have a nonlinear algebraic equation (by "algebraic" I mean it does not involve derivatives of the unknown) that does not have unique solution, e.g.
$$sin(x) = 0 qquad xin [-10, 10] $$
We could also think of more general problems
$$mathcalN(x) = 0 qquad xinOmega$$
where $mathcalN: Omega rightarrow mathbbR$ is a nonlinear algebraic operator defined on $Omega$, a compact subset of $mathbbR^d, d geq 1$. Also suppose $Omega$ is "well behaved", e.g. is the closure of an open set.
Are there numerical methods to approximate the whole set of solutions of such equations?
In other words, are there methods to approximate the kernel $K(mathcalN) = left xinOmega: mathcalN(x)=0right$ in the case this is not a singleton?
What if, in addiction, the operator $mathcalN$ is a "blackbox" (i.e. no explicit formula for $mathcalN$ or its gradient are available)?
The best I can think about at the moment is to turn the problem into a minimization problem with objective function:
$$vertmathcalN(x)vert qquad xinOmega$$
then run an appropriate optimization algorithm. However this approach is meant to determine one solution rather than a set of solution.
Another aspect of the problem I was considering is the fact that the kernel $K(mathcalN)$ may have connected components (that may not be singletons), however I do not know if/how to exploit this information to approach the problem.
numerical-methods roots nonlinear-system
asked Aug 18 at 8:24
funes
62
edited Aug 19 at 18:50
asked Aug 18 at 8:24
funes
62
asked Aug 18 at 8:24
funes
62
asked Aug 18 at 8:24
funes
62
62
Your example problem has roots $npi$, $nin Bbb Z$. How do you propose to numerically treat/distinguish the roots for $n=10^17$ and $10^17+2$? And differentiate them from all the non-roots in-between? In floating point they get mapped to the same binary approximation.
– LutzL
Aug 19 at 15:23
Good point. The example is not good. I think the problem is interesting enough even if we consider only bounded domains (the size of which should not be problematic to handle with floating point arithmetic). I have edited the question.
– funes
Aug 19 at 18:42
If solutions are isolated, running global optimization algorithms for $minlimits_x in Omega | mathcalN(x) |$ is an okay idea. For example, DIRECT can solve this problem knowing only function values and not using derivatives.
– Evgeny
Aug 20 at 9:50
add a comment |Â
Your example problem has roots $npi$, $nin Bbb Z$. How do you propose to numerically treat/distinguish the roots for $n=10^17$ and $10^17+2$? And differentiate them from all the non-roots in-between? In floating point they get mapped to the same binary approximation.
– LutzL
Aug 19 at 15:23
Good point. The example is not good. I think the problem is interesting enough even if we consider only bounded domains (the size of which should not be problematic to handle with floating point arithmetic). I have edited the question.
– funes
Aug 19 at 18:42
If solutions are isolated, running global optimization algorithms for $minlimits_x in Omega | mathcalN(x) |$ is an okay idea. For example, DIRECT can solve this problem knowing only function values and not using derivatives.
– Evgeny
Aug 20 at 9:50
Your example problem has roots $npi$, $nin Bbb Z$. How do you propose to numerically treat/distinguish the roots for $n=10^17$ and $10^17+2$? And differentiate them from all the non-roots in-between? In floating point they get mapped to the same binary approximation.
– LutzL
Aug 19 at 15:23
Your example problem has roots $npi$, $nin Bbb Z$. How do you propose to numerically treat/distinguish the roots for $n=10^17$ and $10^17+2$? And differentiate them from all the non-roots in-between? In floating point they get mapped to the same binary approximation.
– LutzL
Aug 19 at 15:23
Good point. The example is not good. I think the problem is interesting enough even if we consider only bounded domains (the size of which should not be problematic to handle with floating point arithmetic). I have edited the question.
– funes
Aug 19 at 18:42
Good point. The example is not good. I think the problem is interesting enough even if we consider only bounded domains (the size of which should not be problematic to handle with floating point arithmetic). I have edited the question.
– funes
Aug 19 at 18:42
If solutions are isolated, running global optimization algorithms for $minlimits_x in Omega | mathcalN(x) |$ is an okay idea. For example, DIRECT can solve this problem knowing only function values and not using derivatives.
– Evgeny
Aug 20 at 9:50
If solutions are isolated, running global optimization algorithms for $minlimits_x in Omega | mathcalN(x) |$ is an okay idea. For example, DIRECT can solve this problem knowing only function values and not using derivatives.
– Evgeny
Aug 20 at 9:50
add a comment |Â
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Your example problem has roots $npi$, $nin Bbb Z$. How do you propose to numerically treat/distinguish the roots for $n=10^17$ and $10^17+2$? And differentiate them from all the non-roots in-between? In floating point they get mapped to the same binary approximation.
– LutzL
Aug 19 at 15:23
Good point. The example is not good. I think the problem is interesting enough even if we consider only bounded domains (the size of which should not be problematic to handle with floating point arithmetic). I have edited the question.
– funes
Aug 19 at 18:42
If solutions are isolated, running global optimization algorithms for $minlimits_x in Omega | mathcalN(x) |$ is an okay idea. For example, DIRECT can solve this problem knowing only function values and not using derivatives.
– Evgeny
Aug 20 at 9:50