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.
reference-request complex-dynamics
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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.
reference-request complex-dynamics
add a comment |Â
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.
reference-request complex-dynamics
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
reference-request complex-dynamics
edited 47 mins ago
asked 1 hour ago
coudy
11.5k14591
11.5k14591
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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:
Those black regions are exactly where your assertion fails. Notice, also, the five regions converging to five simple roots.
Indeed, I forgot to add that the roots of $P$ must all be real in Lyubich's result. Corrected, thanks.
â coudy
49 mins ago
This answer does not address the question.
â Alexandre Eremenko
3 mins ago
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1
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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
2
active
oldest
votes
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:
Those black regions are exactly where your assertion fails. Notice, also, the five regions converging to five simple roots.
Indeed, I forgot to add that the roots of $P$ must all be real in Lyubich's result. Corrected, thanks.
â coudy
49 mins ago
This answer does not address the question.
â Alexandre Eremenko
3 mins ago
add a comment |Â
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:
Those black regions are exactly where your assertion fails. Notice, also, the five regions converging to five simple roots.
Indeed, I forgot to add that the roots of $P$ must all be real in Lyubich's result. Corrected, thanks.
â coudy
49 mins ago
This answer does not address the question.
â Alexandre Eremenko
3 mins ago
add a comment |Â
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:
Those black regions are exactly where your assertion fails. Notice, also, the five regions converging to five simple roots.
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:
Those black regions are exactly where your assertion fails. Notice, also, the five regions converging to five simple roots.
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.
â coudy
49 mins ago
This answer does not address the question.
â Alexandre Eremenko
3 mins ago
add a comment |Â
Indeed, I forgot to add that the roots of $P$ must all be real in Lyubich's result. Corrected, thanks.
â coudy
49 mins ago
This answer does not address the question.
â 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.
â coudy
49 mins ago
Indeed, I forgot to add that the roots of $P$ must all be real in Lyubich's result. Corrected, thanks.
â coudy
49 mins ago
This answer does not address the question.
â Alexandre Eremenko
3 mins ago
This answer does not address the question.
â Alexandre Eremenko
3 mins ago
add a comment |Â
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."
add a comment |Â
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."
add a comment |Â
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."
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."
answered 45 mins ago
Joe Silverman
29.8k178156
29.8k178156
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