Convergence of Newton's method

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For a polynomial $P$ of degree $n$ with real coefficients and with $n$ distinct real roots, the Newton's method $z_n+1 = z_n + P(z_n) over P'(z_n)$ converges for almost all initial values $z_0$ in $bf C$ to a root of $P$. This is a result due to M. Lyubich (~ 1984).



I think I remember that for a polynomial with complex coefficients, almost all initial values $z_0$ has an orbit that converges to a periodic orbit in $bf C cup infty$, but there are examples where that orbit is not a root of $P$.



Unfortunately, I can't remember who is the author of that result and I would like to find a reference.










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    For a polynomial $P$ of degree $n$ with real coefficients and with $n$ distinct real roots, the Newton's method $z_n+1 = z_n + P(z_n) over P'(z_n)$ converges for almost all initial values $z_0$ in $bf C$ to a root of $P$. This is a result due to M. Lyubich (~ 1984).



    I think I remember that for a polynomial with complex coefficients, almost all initial values $z_0$ has an orbit that converges to a periodic orbit in $bf C cup infty$, but there are examples where that orbit is not a root of $P$.



    Unfortunately, I can't remember who is the author of that result and I would like to find a reference.










    share|cite|improve this question

























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      For a polynomial $P$ of degree $n$ with real coefficients and with $n$ distinct real roots, the Newton's method $z_n+1 = z_n + P(z_n) over P'(z_n)$ converges for almost all initial values $z_0$ in $bf C$ to a root of $P$. This is a result due to M. Lyubich (~ 1984).



      I think I remember that for a polynomial with complex coefficients, almost all initial values $z_0$ has an orbit that converges to a periodic orbit in $bf C cup infty$, but there are examples where that orbit is not a root of $P$.



      Unfortunately, I can't remember who is the author of that result and I would like to find a reference.










      share|cite|improve this question















      For a polynomial $P$ of degree $n$ with real coefficients and with $n$ distinct real roots, the Newton's method $z_n+1 = z_n + P(z_n) over P'(z_n)$ converges for almost all initial values $z_0$ in $bf C$ to a root of $P$. This is a result due to M. Lyubich (~ 1984).



      I think I remember that for a polynomial with complex coefficients, almost all initial values $z_0$ has an orbit that converges to a periodic orbit in $bf C cup infty$, but there are examples where that orbit is not a root of $P$.



      Unfortunately, I can't remember who is the author of that result and I would like to find a reference.







      reference-request complex-dynamics






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      edited 47 mins ago

























      asked 1 hour ago









      coudy

      11.5k14591




      11.5k14591




















          2 Answers
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          I don't think your initial assertion is accurate. Consider, for example, $f(z)=z^5-z-1$. If you iterate the Newton's method function $N(z) = z-f(z)/f'(z)$ from $z_0=0$, you'll quickly find an attractive orbit of period 3. The basin of attraction of that orbit is a positive measure set with no point converging to a root of $f$. The standard Newton method picture looks like so:



          enter image description here



          Those black regions are exactly where your assertion fails. Notice, also, the five regions converging to five simple roots.






          share|cite|improve this answer




















          • Indeed, I forgot to add that the roots of $P$ must all be real in Lyubich's result. Corrected, thanks.
            – coudy
            49 mins ago










          • This answer does not address the question.
            – Alexandre Eremenko
            3 mins ago

















          up vote
          1
          down vote













          For Newton's method (and more general iterative methods) for finding roots of complex polynomials, you may want to look at Curt McMullen's paper:



          Families of Rational Maps and Iterative Root-Finding Algorithms,
          Curt McMullen,
          Annals of Mathematics,
          Second Series, Vol. 125, No. 3 (May, 1987), pp. 467-493



          From the abstract: "In this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root-finding algorithms. We answer a question of Smale's by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more. We settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics; and we give a classification of all such algorithms."






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            2 Answers
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            active

            oldest

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            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            2
            down vote













            I don't think your initial assertion is accurate. Consider, for example, $f(z)=z^5-z-1$. If you iterate the Newton's method function $N(z) = z-f(z)/f'(z)$ from $z_0=0$, you'll quickly find an attractive orbit of period 3. The basin of attraction of that orbit is a positive measure set with no point converging to a root of $f$. The standard Newton method picture looks like so:



            enter image description here



            Those black regions are exactly where your assertion fails. Notice, also, the five regions converging to five simple roots.






            share|cite|improve this answer




















            • Indeed, I forgot to add that the roots of $P$ must all be real in Lyubich's result. Corrected, thanks.
              – coudy
              49 mins ago










            • This answer does not address the question.
              – Alexandre Eremenko
              3 mins ago














            up vote
            2
            down vote













            I don't think your initial assertion is accurate. Consider, for example, $f(z)=z^5-z-1$. If you iterate the Newton's method function $N(z) = z-f(z)/f'(z)$ from $z_0=0$, you'll quickly find an attractive orbit of period 3. The basin of attraction of that orbit is a positive measure set with no point converging to a root of $f$. The standard Newton method picture looks like so:



            enter image description here



            Those black regions are exactly where your assertion fails. Notice, also, the five regions converging to five simple roots.






            share|cite|improve this answer




















            • Indeed, I forgot to add that the roots of $P$ must all be real in Lyubich's result. Corrected, thanks.
              – coudy
              49 mins ago










            • This answer does not address the question.
              – Alexandre Eremenko
              3 mins ago












            up vote
            2
            down vote










            up vote
            2
            down vote









            I don't think your initial assertion is accurate. Consider, for example, $f(z)=z^5-z-1$. If you iterate the Newton's method function $N(z) = z-f(z)/f'(z)$ from $z_0=0$, you'll quickly find an attractive orbit of period 3. The basin of attraction of that orbit is a positive measure set with no point converging to a root of $f$. The standard Newton method picture looks like so:



            enter image description here



            Those black regions are exactly where your assertion fails. Notice, also, the five regions converging to five simple roots.






            share|cite|improve this answer












            I don't think your initial assertion is accurate. Consider, for example, $f(z)=z^5-z-1$. If you iterate the Newton's method function $N(z) = z-f(z)/f'(z)$ from $z_0=0$, you'll quickly find an attractive orbit of period 3. The basin of attraction of that orbit is a positive measure set with no point converging to a root of $f$. The standard Newton method picture looks like so:



            enter image description here



            Those black regions are exactly where your assertion fails. Notice, also, the five regions converging to five simple roots.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 1 hour ago









            Mark McClure

            1,438613




            1,438613











            • Indeed, I forgot to add that the roots of $P$ must all be real in Lyubich's result. Corrected, thanks.
              – coudy
              49 mins ago










            • This answer does not address the question.
              – Alexandre Eremenko
              3 mins ago
















            • Indeed, I forgot to add that the roots of $P$ must all be real in Lyubich's result. Corrected, thanks.
              – coudy
              49 mins ago










            • This answer does not address the question.
              – Alexandre Eremenko
              3 mins ago















            Indeed, I forgot to add that the roots of $P$ must all be real in Lyubich's result. Corrected, thanks.
            – coudy
            49 mins ago




            Indeed, I forgot to add that the roots of $P$ must all be real in Lyubich's result. Corrected, thanks.
            – coudy
            49 mins ago












            This answer does not address the question.
            – Alexandre Eremenko
            3 mins ago




            This answer does not address the question.
            – Alexandre Eremenko
            3 mins ago










            up vote
            1
            down vote













            For Newton's method (and more general iterative methods) for finding roots of complex polynomials, you may want to look at Curt McMullen's paper:



            Families of Rational Maps and Iterative Root-Finding Algorithms,
            Curt McMullen,
            Annals of Mathematics,
            Second Series, Vol. 125, No. 3 (May, 1987), pp. 467-493



            From the abstract: "In this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root-finding algorithms. We answer a question of Smale's by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more. We settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics; and we give a classification of all such algorithms."






            share|cite|improve this answer
























              up vote
              1
              down vote













              For Newton's method (and more general iterative methods) for finding roots of complex polynomials, you may want to look at Curt McMullen's paper:



              Families of Rational Maps and Iterative Root-Finding Algorithms,
              Curt McMullen,
              Annals of Mathematics,
              Second Series, Vol. 125, No. 3 (May, 1987), pp. 467-493



              From the abstract: "In this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root-finding algorithms. We answer a question of Smale's by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more. We settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics; and we give a classification of all such algorithms."






              share|cite|improve this answer






















                up vote
                1
                down vote










                up vote
                1
                down vote









                For Newton's method (and more general iterative methods) for finding roots of complex polynomials, you may want to look at Curt McMullen's paper:



                Families of Rational Maps and Iterative Root-Finding Algorithms,
                Curt McMullen,
                Annals of Mathematics,
                Second Series, Vol. 125, No. 3 (May, 1987), pp. 467-493



                From the abstract: "In this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root-finding algorithms. We answer a question of Smale's by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more. We settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics; and we give a classification of all such algorithms."






                share|cite|improve this answer












                For Newton's method (and more general iterative methods) for finding roots of complex polynomials, you may want to look at Curt McMullen's paper:



                Families of Rational Maps and Iterative Root-Finding Algorithms,
                Curt McMullen,
                Annals of Mathematics,
                Second Series, Vol. 125, No. 3 (May, 1987), pp. 467-493



                From the abstract: "In this paper we develop a rigidity theorem for algebraic families of rational maps and apply it to the study of iterative root-finding algorithms. We answer a question of Smale's by showing there is no generally convergent algorithm for finding the roots of a polynomial of degree 4 or more. We settle the case of degree 3 by exhibiting a generally convergent algorithm for cubics; and we give a classification of all such algorithms."







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 45 mins ago









                Joe Silverman

                29.8k178156




                29.8k178156



























                     

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