Root finding for a convex 3D function

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Say I have a convex 2D function $f(x,y)$ ; x-absciss y-ordinate
Getting zeros of this function is ok for me with the Newton method.



Now say this time I still have a convex function but in 3D $f(x,y,z)$.
You have to imagine a function such as $f(x,y)=x^2-2$ And increasing in the z-dimension (depth).



I'd like to find the value of $z$ such as the whole function is positive.
I was thinking about using Newton and dichotomy but the function is not monotonic... Any clue ?



Edit:
I know the range of z where to search, say [5, 15]



I could do a Newton search on different points: 5,6,7,...,15 and then look at the optimal value but this would not be optimal.




roots




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    down vote

    favorite












    Say I have a convex 2D function $f(x,y)$ ; x-absciss y-ordinate
    Getting zeros of this function is ok for me with the Newton method.



    Now say this time I still have a convex function but in 3D $f(x,y,z)$.
    You have to imagine a function such as $f(x,y)=x^2-2$ And increasing in the z-dimension (depth).



    I'd like to find the value of $z$ such as the whole function is positive.
    I was thinking about using Newton and dichotomy but the function is not monotonic... Any clue ?



    Edit:
    I know the range of z where to search, say [5, 15]



    I could do a Newton search on different points: 5,6,7,...,15 and then look at the optimal value but this would not be optimal.




    roots




    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Say I have a convex 2D function $f(x,y)$ ; x-absciss y-ordinate
      Getting zeros of this function is ok for me with the Newton method.



      Now say this time I still have a convex function but in 3D $f(x,y,z)$.
      You have to imagine a function such as $f(x,y)=x^2-2$ And increasing in the z-dimension (depth).



      I'd like to find the value of $z$ such as the whole function is positive.
      I was thinking about using Newton and dichotomy but the function is not monotonic... Any clue ?



      Edit:
      I know the range of z where to search, say [5, 15]



      I could do a Newton search on different points: 5,6,7,...,15 and then look at the optimal value but this would not be optimal.




      roots




      share|cite|improve this question














      Say I have a convex 2D function $f(x,y)$ ; x-absciss y-ordinate
      Getting zeros of this function is ok for me with the Newton method.



      Now say this time I still have a convex function but in 3D $f(x,y,z)$.
      You have to imagine a function such as $f(x,y)=x^2-2$ And increasing in the z-dimension (depth).



      I'd like to find the value of $z$ such as the whole function is positive.
      I was thinking about using Newton and dichotomy but the function is not monotonic... Any clue ?



      Edit:
      I know the range of z where to search, say [5, 15]



      I could do a Newton search on different points: 5,6,7,...,15 and then look at the optimal value but this would not be optimal.





      roots





      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Aug 21 at 9:04

























      asked Aug 21 at 8:55









      Cedric_W

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